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Volume 6, Issue 2, February 1994

Flow through arrays of cylinders: Lattice gas cellular automata simulations
View Description Hide DescriptionA presentation is given of the results of two‐dimensional lattice gas cellular automata (LGA) simulations for calculating the permeability for flow through periodic arrays of cylinders, where the cylinders have either a rhomboid or cruciform cross section. For the case of rhombi, the results are related via a semiempirical shape factor to analytical solutions for flow through arrays of circular cylinders. Finite size effects are studied by comparing simulation results with theory. This calibration of the LGA model then validates its use for the study of flow around cruciforms, for which there is no analytical solution. This has an important application in the mathematical modeling of flow through arrays of dendrites in the mushy region during the solidification of alloy steel.

Linear oscillations of weakly dissipative axisymmetric liquid bridges
View Description Hide DescriptionLinear oscillations of axisymmetric capillary bridges are analyzed for large values of the modified Reynolds numberC ^{−1}. There are two kinds of oscillating modes. For nearly inviscid modes (the flow being potential, except in boundary layers), it is seen that the damping rate −Ω_{ R } and the frequency Ω_{ I } are of the form Ω_{ R }=ω_{1} C ^{1/2}+ω_{2} C+O(C ^{3/2}) and Ω_{ I }=ω_{0}+ω_{1} C ^{1/2}+O(C ^{3/2}), where the coefficients ω_{0}≳0, ω_{1}<0, and ω_{2}<0 depend on the aspect ratio of the bridge and the mode being excited. This result compares well with numerical results if C≲0.01, while the leading term in the expansion of the damping rate (that was already known) gives a bad approximation, except for unrealistically large values of the modified Reynolds number (C≲10^{−6}). Viscous modes (involving a nonvanishing vorticity distribution everywhere in the liquid bridge), providing damping rates of the order of C, are also considered.

Drop motion with surfactant transfer in a homogeneous surrounding
View Description Hide DescriptionThe influence of a solublesurfactant on the stationary motion of a drop in an infinite motionless homogeneous surfactantsolution is considered when the surfactant undergoes a first‐order isothermal chemical reaction on the surface of the drop. The study is carried out for the asymptotic case of small Reynolds and Péclet numbers. First, the linear approximation is considered. It appears, in particular, that the hydrodynamical force acting on the drop provides either thrust or drag according to the parameter values. It also appears that the motion can be unstable and critical Marangoni numbers corresponding to instability thresholds of the motionless state of the fluids in the absence of buoyancy are provided. Emphasis is given to instability of the drop to its translations. A weakly nonlinear analysis past translational instability shows the possibility of multiple stationary states. On the one hand the hydrodynamical force dependence on the drop velocity can be nonmonotonous. Then, in particular, one may have either the rest state or self‐sustained (autonomous), Marangoni‐driven motion in the absence of buoyancy or any other external forcing factors. On the other hand, the velocity dependence on the hydrodynamical force can also be nonmonotonous. Then, in particular, in the absence of any external compensating forces one may have the drop levitating even under nonzero buoyancy. Finally, the relative stability of these nonlinear regimes and possible experimental observations are discussed.

Exit flows from highly porous media
View Description Hide DescriptionThis paper presents laser velocimetrymeasurements of the streamwise component of mean velocities and turbulence intensities measured downstream from the exit plane of porous ceramicfoams through which air is flowed. The recent development and commercial availability of porous ceramicfoams has lead to their application in many fields. Their uses have extended to combustion, high‐temperature fluid filtering, biotechnology, and as support matrix for catalysts. These applications have created an interest in their pore scale fluid mechanics, both within the porous matrix and along surfaces open to flow. One emerging application is porous ceramic burners which combust liquid or gaseous fuels within the pore matrix or along the surface of the ceramic. The ceramicfoams have pore sizes ranging from 4 to 12 pores per cm (ppcm) and porosities of 85%. Mean velocities between 0.3 and 1.5 m/s were examined. Radial distributions of mean velocities show a jet‐like structure through the pores, with local mean velocities reaching maximum values over two times the area mean velocity. Negative mean velocities were often observed between pores, suggesting that recirculation zones are present above the web‐like struts surrounding the pores. Levels of turbulence intensities normalized by the area mean velocity ranged from 0.05 to 0.6 for the various flow rates and pore sizes. Turbulence intensities were found to increase with increasing pore size for a given flow rate.

Fluid cusps at the molecular scale
View Description Hide DescriptionRecent experiments and calculations have exhibited apparent steady cusps in certain fluid free surfaces driven by a converging subsurface flow.Molecular dynamics simulations are used to elucidate some of the issues raised by this phenomenon, with emphasis on the behavior of the fluid on very small scales. The similar but different situation of two immiscible fluids of equal viscosity driven by solid rollers is considered. Molecular scale simulations of the evolution of the separating interface exhibit a gradual increase of curvature with rotation rate. However, high curvature interfaces do not reach a steady state, but instead drops of the fluid above the free surface are detached. In no case does a true cusp form, and the stress tensor is never unusually large in the near‐cusp region.

Distributed acoustic receptivity in laminar flow control configurations
View Description Hide DescriptionA model problem related to distributed receptivity to free‐stream acoustic waves in laminar flowcontrol (LFC) configurations is studied within the Orr–Sommerfeld framework, by developing a suitable extension of the Goldstein–Ruban theory for receptivity due to localized disturbances on the airfoil surface. One advantage of the proposed method is that it easily permits the prediction of receptivity due to a wide class of noncompact distributions for surface nonuniformities, such as small‐amplitude variations in suction velocity and surface admittance. The results, thus, complement the earlier work on the receptivity produced by local variations in these two quantities. It is shown that the cumulative effect of distributed receptivity can be substantially larger than that of a single, isolated suction strip or slot. Furthermore, even if the receptivity is spread out over very large distances, the most effective contributions come from a relatively short region in the vicinity of the lower branch of the neutral stability curve. Finally, it is found that the receptivity is effectively dominated by a narrow band of Fourier components from the wall‐suction and admittance distributions, roughly corresponding to a detuning of less than 10% with respect to the neutral instability wave number at the frequency under consideration. The results also indicate that the drop‐off in receptivity magnitudes away from the resonant wave number is nearly independent of the frequency parameter. Differences in receptivity due to the concentration of wall suction in different parts of the airfoil chord are also investigated.

An improved analytic solution for analysis of particle trajectories in fibrous, two‐dimensional filters
View Description Hide DescriptionThe Kuwabara solution for creeping fluid flow through periodic arrangement of cylinders is widely used in analytic and numerical studies of fibrous filters. Numerical solutions have shown that the Kuwabara solution has systematic errors, and when used for the particle trajectories in filters it results in some error in the predicted filter efficiency. The numerical solutions, although accurate, preclude further analytic treatments, and are not as compact and convenient to use as the Kuwabara solution. By reexamining the outer boundary conditions of the Kuwabara solution, a correction term to the Kuwabara solution has been derived to obtain an extended solution that is more accurate and improves prediction of the filter efficiency. By comparison with the numerical solutions, it is shown that the Kuwabara solution is the high porosity asymptote, and that the extended solution has an improved porosity dependence. A rectification is explained that can make particle collection less efficient for periodic, in‐line arrangements of fibers with particle diffusion or body force. This rectification also results in the alignment of particles with inertia (i.e., high Stokes number particles).

The dynamics of fibrous aerosols in a quiescent atmosphere
View Description Hide DescriptionMeasurements of the orientational properties of small fibrous graphiteaerosols in free fall through air have been performed. The fibers have lengths on the order of millimeters and diameters on the order of microns. Estimates of the Reynolds number based on the fiber’s half‐length and terminal velocity are on the order of 1. Measurements of the relative extinction in selective polarizations of microwave radiation upon transmission through a cloud of falling fibers in dilute concentration as well as direct photographic measurements indicate the fibers exhibit a tendency to assume a stable orientation with their long axes horizontal under quiescent conditions. Terminal velocities were found to range from 2.5 to 3 cm/s. Photographic measurements on single fibers were also made of the rate at which the fibers settle into their stable orientation given an initial arbitrary orientation. Orientational stabilization times, otherwise referred to as orientational relaxation time, are typically on the order of 3 to 5 s; this is mostly dependent on initial orientation. The measured transient angular motion of the fibers is compared with a proposed model of the motion. The model shows qualitative agreement with observations.

Flow structure from an oscillating nonuniform cylinder: Generation of patterned vorticity concentrations
View Description Hide DescriptionForced oscillation of a mildly nonuniform cylinder generates patterns of vorticity concentrations oriented orthogonally to its axis. These complex, but ordered, patterns are repeatable at subharmonics of the cylinder oscillation frequency; they therefore provide a means of identifying the first and second period‐doubled states of the wake response.

Observations of secondary flows generated by a torsionally oscillating sphere in linearly stratified fluids
View Description Hide DescriptionObservations have been made of a cellular flow pattern in the secondary flow field generated by a sphere, 2.52 cm in radius, torsionally oscillating in linearly stratified fluids (Brunt–Väisälä frequencies, N, between 0.9 and 2.1 rad/s). Shadowgraphs were made of the secondary flow for frequencies of oscillation in the range 2.0 rad/s≤ω≤13.5 rad/s, and amplitudes of oscillation θ≤1 rad. The size and number of flow cells were determined for fixed values of the parameter (ω/N)θ^{2}, and bifurcation‐like transitions occurred during continuous variations of this parameter. Flow cells were mapped using a laser‐Doppler velocimeter and compared to the secondary flow for homogeneous fluids. A phase diagram is presented to show the range of critical values of parameters where flow transitions occur.

Linear instability of curved free shear layers
View Description Hide DescriptionThe linear inviscid hydrodynamicstability of slightly curved free mixing layers is studied in this paper. The disturbance equation is solved numerically using a shooting technique. Two mean velocity profiles that represent stably and unstably curved free mixing layers are considered. Results are shown for cases of five curvature Richardson numbers. The stability characteristics of the shear layer are found to vary significantly with the introduction of the curvature effects. The results also indicate that, in a manner similar to the Görtler vortices observed in a boundary layer along a concave wall, instability modes of spatially developing streamwise vortex pairs may appear in centrifugally unstable curved mixing layers.

Toward a thermodynamic theory of hydrodynamics: The Lorenz equations
View Description Hide DescriptionEarlier work on the thermodynamics of nonlinear systems is extended to the Lorenz model in a first attempt to apply the theory to hydrodynamics. An excess work, Φ, related to the work necessary for displacement of the system from stationary states is defined. The excess work Φ is shown to have the following properties: (1) The differential of Φ is expressed in terms of thermodynamic functions: the energy for viscousflow and the entropy for thermal conduction when taken separately; (2) Φ is an extremum at all stationary states, a minimum (maximum) at stable (unstable) stationary states, and thus yields necessary and sufficient criteria for stability; (3) Φ describes the bifurcation from homogeneous to inhomogeneous stationary states; (4) Φ is a Lyapunov function with physical significance parallel to that of the Gibbs free energy change (maximum work) on relaxation to an equilibrium state; (5) Φ is the thermodynamic ‘‘driving force’’ (potential) toward stable stationary states; (6) Φ̇ is a component of the total dissipation during the relaxation toward a stable stationary state; (7) for constraints leading to equilibrium Φ reduces to the known thermodynamic function, which is the work of displacing the system from the equilibrium for those given constraints; and (8) Φ qualitatively explains the positive energy release in both the destruction and formation of a convective structure in a Rayleigh–Bénard experiment.

The three‐dimensional vortex structure of an impacting water drop
View Description Hide DescriptionObservations are reported for the three‐dimensional vortex structure created by a dyed water dropimpacting a pool of water. The structure and evolution of the vorticity is studied for a Weber number of 22–25 and a Froude number of 25–28. The drop and pool do not make first contact at the bottom of the drop but at latitudes away from the bottom pole of the drop. This traps a thin, curved, pancake‐shaped air bubble beneath the drop which rapidly contracts into a sphere. As the dropimpacts the pool its impulse produces vorticity which rolls up into a primary vortex ring. As the vortex ring travels down through the pool, vortex filaments extend from the central axis of the vortex ring to form a ‘‘stalk.’’ This reaches from the primary ring to another ring of vorticity which has formed in the now reversing free surfaceimpact crater. As the primary ring convects downward some vortex filaments undergo an azimuthal instability which grows until the filaments escape the trapped orbits of the primary vortex ring and are ‘‘shed.’’ This results in three to five loops or ‘‘petals’’ left behind the primary ring. A three‐dimensional vortex skeleton of this structure is presented. The results confirm the hypothesis that the structure is topologically similar to that of the separation bubble on a blunt flat plate. The structure’s appearance is also strikingly similar to the appearance of an aboveground atomic blast.

Flow visualization of the three‐dimensional, time‐evolving structure of a turbulent boundary layer
View Description Hide DescriptionExperiments were performed using volumetric flow visualization to study the evolution of three‐dimensional boundary layer turbulence, and the interactions between inner and outer layer coherent structures. In the experiment, the inner and outer layers of a low Reynolds number (R _{θ}=705) turbulent boundary layer in air were seeded with smoke. A section of the boundary layer was rapidly scanned with a high intensity pulsed laser sheet, and the individual images were recorded using a high‐speed video camera. The video images were digitized and assembled into three‐dimensional volumes. A variety of image processing techniques and conditional sampling and statistical methods were then used to extract quantitative information from the three‐dimensional, time‐evolving visualization data.

Resonant triads of capillary–gravity waves in the presence of a current
View Description Hide DescriptionResonant triads of capillary–gravity waves propagating on the surface of inviscid fluid of infinite depth, in which the basic state is a plane, parallel current of the wind‐drift type, are examined in the case that one of the waves propagates parallel to the current. The velocity profiles chosen for the current have nonconstant vorticity, necessitating numerical computation of the dispersion relation and the coefficients of the amplitude equations. A wave propagating in the direction of the current is found to participate in two families of triads, corresponding to those described by Simmons [Proc. R. Soc. London Ser. A 309, 551 (1969)] for a stagnant basic state. Only triads from one of the families will be excited, and this family only exists when the original wave has a sufficiently short wavelength. The presence of a current increases the maximum wavelength at which this family of resonances occurs and increases the maximum possible angle between the original wave and an oblique, excited, resonant wave. When a wave propagates against the current the maximum wavelength and the maximum angle are decreased by the presence of the current. The amplitude equations governing temporal modulation are derived, and in the cases examined, the coefficients are found to be of differing sign, which excludes the possibility of explosive instability.

Turbulence renormalization group calculations using statistical mechanics methods
View Description Hide DescriptionThe renormalizationgroup theory of fluid turbulence is developed from a statistical mechanical viewpoint using an exact expression for the functional probability distribution of the velocity field. The latter is similar in form to an equilibrium Gibbs distribution, and is derived by combining Lagrangianstatistical mechanics with an Eulerian fluid description. It is shown that this distribution enables an RG transformation to be defined, evaluated, and analyzed using statistical mechanical techniques. The method of determining amplitudes used here also differs from previous work, in that the indeterminacy of the relevant stirring force parameters is resolved through the essential requirement that all coarse‐grained distributions must yield the actual dissipation rate E. Consequently, although the fixed point solutions of the dynamic RG approach are recovered, slightly different numerical results are obtained for amplitudes. The present approach yields, for example, for Kolmogorov’s constant, C _{K}=1.44.

A simple theory of conditional mean velocity in turbulent scalar‐mixing layer
View Description Hide DescriptionThis paper presents the experimental results on conditional mean transverse velocity across a turbulent scalar mixing layer. The velocity is conditioned on mixture fraction. It is found that the distribution of the conditional mean transverse velocity is closely related to the local mean‐mixture fraction in physical space. Near the local mean‐mixture fraction a linear relationship exists between the conditional mean transverse velocity and the mixture‐fraction fluctuations. Departure from the linear relationship at large mixture‐fraction fluctuations is mainly due to the nonlinear distribution of the mean mixture‐fraction profile and it is not, as has been suggested by others, due to the limiting velocity of the large eddies in the flow.

Scale resolved intermittency in turbulence
View Description Hide DescriptionThe deviations δζ_{ m } (‘‘intermittency corrections’’) from classical (‘‘K41’’) scaling ζ_{ m }=m/3 of the mth moments 〈‖u(p)‖^{ m }〉 in high Reynolds number turbulence are calculated, extending a method to approximately solve the Navier–Stokes equation described earlier. It is suggested to introduce the notion of scale resolved intermittency corrections δζ_{ m }(p), because these δζ_{ m }(p) are found to be large in the viscous subrange, moderate in the nonuniversal stirring subrange but, surprisingly, extremely small if not zero in the inertial subrange. If ISR intermittency corrections persisted in experiment up to the large Reynolds number limit, it would show by calculation that this could be due to the opening of phase space for larger wave vectors. In the higher order velocity moments 〈‖u(p)‖^{ m }〉 the crossover between inertial and viscous subrange is (10ηm/2)^{−1}, thus the inertial subrange is smaller for higher order moments.

Stochastic equations with multifractal random increments for modeling turbulent dispersion
View Description Hide DescriptionPreviously studied stochastic models of one‐particle dispersion in stationary, isotropic, and homogeneous turbulence are reconsidered and intermittency corrections sought. Known Lagrangianintermittency effects, in the form of multifractal scaling, independently derived from Eulerian measurements [M. S. Borgas, Philos. Trans. R. Soc. London Ser. A 342, 379 (1993)], are used to develop a new model. The previous models and approaches are shown to be inadequate. The new model incorporating Lagrangianintermittency satisfies Thomson’s well‐mixed criterion [J. Fluid Mech. 180, 529 (1987)] and gives almost‐Gaussian mean‐concentration distributions for Gaussian sources. The trajectories generated by the model are not fractal, in agreement with the results of Borgas. The practical impact of intermittency upon dispersion is found to be small.

On the generation of turbulent wall friction
View Description Hide DescriptionThe formation of streaky velocity structures in the near wall region of turbulent boundary layers is studied through a simplified two‐dimensional computational model in the plane normal to the average velocity. It is shown that the redistribution of the longitudinal velocity by streamwise vortices produces features very similar to those observed in the experiments, and that compact streamwise vortices form naturally from more general vorticity distributions. It is also shown, both numerically and analytically, that one effect of the formation of the streaks is to increase the average wall friction, and it is suggested that this effect is responsible for the higher friction in turbulent boundary layers, as opposed to laminar ones. An approximate quantitative analysis of the process supports this assumption.