Index of content:
Volume 6, Issue 5, May 1994

Nonlinear stability of counter‐rotating vortices
View Description Hide DescriptionRecently, Mallier and Maslowe [Phys. Fluids A 5, 1074 (1993)] found an exact nonlinear solution of the inviscid, incompressible, two‐dimensional Navier–Stokes equations, representing an infinite row of counter‐rotating vortices, which extended the previous Kelvin–Stuart vortices. The aim of this work is to establish explicit sufficient conditions for the nonlinear stability of this solution. The result is derived with the energy‐Casimir stability method as a function of the parameters of the solution and the domain size. The size of the domain over which the street of vortices is unstable is exhibited.

The rheology of bimodal hard‐sphere dispersions
View Description Hide DescriptionThe viscosity of a monolayer of a suspension of bimodally distributed hard spheres is determined for area fractions, φ_{ a }, from 0.15 to 0.74 with different particle size ratios, λ (diameter of large sphere/diameter of small spheres=1, 2, and 4), and different fractions of small spheres of total solids, ξ (0.07, 0.27, 0.49, 0.64, and 0.83). Particle distributions are generated by a Monte Carlo technique and the hydrodynamic interactions are calculated by Stokesian dynamics. These results, which correspond to the high‐frequency dynamicviscosities, are compared with those from the dynamic simulation of hydrodynamically interacting spherical particles [Chang and Powell, J. Fluid Mech. 253, 1 (1993)]. Dynamic simulation is found to yield higher relative viscosities, η_{ r }, as compared with the results of Monte Carlo simulation at high concentrations. This results from the absence of long clusters that completely cross a periodic cell in the Monte Carlo simulations that are present in the dynamic simulations. When φ_{ a } is normalized by the maximum packing fraction, φ_{ m } ^{2‐D}, all the viscosity data fall onto a master curve. This is the same trend as that found in dynamic simulations, except that the Monte Carlo simulation gives lower relative viscosities for φ_{ a }/φ_{ m } ^{2‐D} ≳ 0.3. When λ and φ_{ a } are fixed, η_{ r } decreases as ξ increases from zero, reaches a minimum, and then increases as ξ→1, similar to the trend found in the dynamic simulations. Good agreement is found among the results of two‐dimensional simulations, experiments, and three‐dimensional simulations for monodispersed suspensions.

A method for computing Stokes flow interactions among spherical objects and its application to suspensions of drops and porous particles
View Description Hide DescriptionA method for computing Stokes flow interactions in suspensions of spherical objects is described in detail and applied to the suspensions of porous particles, drops, and bubbles to determine their hydrodynamic transport coefficients.

Inclusion of lubrication forces in dynamic simulations
View Description Hide DescriptionA new method is described for incorporating close‐field, lubrication forces between pairs of particles into the multiparticle Stokes flow calculations. The method is applied to the suspensions of both spherical as well as cylindrical particles, and results computed by the method are shown to be in excellent agreement with the exact known results available in the literature.

Bubble growth and stability in an effective porous medium
View Description Hide DescriptionThis paper examines the in situ growth and stability of a single bubble in an effective porous medium, where the pore microstructure is neglected. The main objective of the study is the effect of mass transfer and flow parameters on bubble stability. Because it is necessary for flow in both phases to be considered, this is actually a generalized phase‐change problem involving two fluids in a porous medium. Similarity solutions are developed in both three‐dimensional (3‐D) and two‐dimensional (2‐D) geometries, the stability of which is subsequently analyzed. In the absence of capillarity, an instability analogous to Mullins–Sekerka is found, which is independent of the process parameters, including the viscosity ratio, M. Capillary effects are next considered for the 2‐D geometry of a Hele–Shaw cell (the analysis is also extended to 3‐D geometries). At the large M limit, the stability condition is identical to Patterson’s [J. Fluid Mech. 113, 513 (1981)] for the displacement of a liquid by a gas. This theory is found consistent with bubble growth experiments in Hele–Shaw cells. At finite M, the theory predicts that the interface becomes more stable as M increases, which is directly opposite to the effect of M in viscous fingering. This is a novel mechanism of convective stabilization and it is intrinsic to in situ phase growth. Higher solubility and diffusion coefficients also have a stabilizing effect. For conditions typical to bubble growth, however, the sensitivity to these parameters is very weak.

Flow beneath a rotating annulus
View Description Hide DescriptionAir flow between a spinning annular disk and a flat baseplate has been studied both experimentally and computationally. The Ekman number, Ek, has been determined to be the single important dimensionless parameter required to characterize this problem. A numerical solution which predicts spacing‐dependent axisymmetric flow has been developed. It has been found to agree with experimental data over a wide range of parameter space. The calculated flow predicts unusual complexity; counter‐rotating recirculation cells are likely to be present.

Physical mechanisms of instability in a liquid layer subjected to an electric field and a thermal gradient
View Description Hide DescriptionThe linear stability of a plane layer of dielectric liquid subjected to an electric field and a thermal gradient is studied. The liquid is supposed to have a non‐negligible residual conductivity and to exhibit an injection of charge from one of the electrodes. Both effects are due to the dissolution of a given salt in the liquid. The ionic mobility and the dielectric constant are temperature dependent. The stability of this configuration is studied with the help of an heuristic model and a precise discussion of the physical mechanisms of instability is made. Complete sets of the stability maps corresponding to different cases are presented.

Solitary wave dynamics of film flows
View Description Hide DescriptionThe development and interaction of solitary wave pulses is critical to understanding wavy film flows on an inclined (or vertical) surface. Sufficiently far downstream, the wave structure consists of a generally irregular sequence of solitary waves independent of the conditions at the inlet. The velocity of periodic solitary waves is found to depend on their frequency and amplitude. Larger pulses travel faster; this property, plus a strong inelasticity, causes larger pulses to absorb others during interactions, leaving a nearly flat interface behind. These wave interactions lead to the production of solitary wave trains from periodic small amplitude waves. The spacings between solitary waves can be irregular for several different reasons, including the amplification of ambient noise, and the interaction process itself. On the other hand, this irregularity is suppressed by the addition of periodic forcing.

Nonlinear stability analysis of molten flow in laser cutting
View Description Hide DescriptionThe method of multiple scales is used to investigate the nonlinear stability behavior of a thin molten laser cutting. In this analysis, the balance of momentum and energy with the effects of phase change is considered. Under the assumption of long wavelength, a corresponding nonlinear generalized kinematic equation for the film thickness is thereby derived. The result shows that there exist supercritical stability in the linearly unstable region and subcritical instability in the linearly stable region. For higher cutting speeds, the surface roughness could be improved by decreasing the cutting speed or increasing the gas velocity due to the decreasing of the maximum equilibrium finite amplitude. On the other hand, for lower cutting speeds, the surface roughness could be improved by increasing the cutting speed or decreasing the gas velocity due to the increasing of the minimum threshold amplitude.

Stochastic transition intermittency in pipe flows: Experiment and model
View Description Hide DescriptionNew experimental results at the onset of turbulence in a gravity‐driven pipe flow are presented, and a simple phenomenological model is introduced to describe the intermittent behavior observed. In this model slugs are stochastically produced at the pipe inlet, and the decrease in velocity due to turbulentfriction is taken into account. The present approach shows that stochastic arguments account well for several experimental observations at low intermittency factors. In particular, it is shown that special intermittency routes to chaos are not needed to explain the exponentially decaying inverse cumulative distribution of laminar times.

Laboratory‐generated, shallow‐water surface waves: Analysis using the periodic, inverse scattering transform
View Description Hide DescriptionThe space‐time evolution of laboratory‐generated, shallow‐water wave trains is considered in one‐space and one‐time dimensions. An electronic control‐and‐feedback system is used to generate periodic/quasiperiodic wave trains in a laboratory flume (0.76×0.8×46 m^{3}) in which five spatially distributed, resistance wave gauges simultaneously record time series of the surface wave field. A concrete ramp with a slope of 0.02 minimizes reflections, and thus ensures near‐unidirectional wave motion. The data are analyzed using both (1) linear Fourier analysis and (2) a relatively new kind of nonlinearFourier analysis based upon the inverse scatteringtransform (IST) for the periodic Korteweg–de Vries (KdV) equation. The periodic IST formalism consists of a linear superposition of the ‘‘hyperelliptic‐function oscillation modes,’’ which are intrinsically nonlinear, while simultaneously undergoing nonlinear interactions with each other. The KdV oscillation modes may be viewed as the ‘‘sine waves’’ of the periodic scattering transform, although they are generally nonsinusoidal in shape. The amplitudes of the oscillation modes are constants of the motion for wave motion governed purely by KdV. For a series of experiments in the Ursell number range 0.30<U<0.99, it is found that the linear Fourier modes have amplitudes that vary substantially in space and time, while the inverse scattering modes are found to be nearly constant. It is therefore suggested that the inverse scattering formulation may be more appropriate than linear Fourier analysis for describing the nonlinear dynamical motions studied experimentally herein. Introducing the concept of ‘‘phase locking’’ among the IST modes, features in the data that are referred to as ‘‘coherent structures’’ are identified. Such structures are found not to be strictly solitons (they constitute multiple cnoidal wave interactions), although they have many of the properties normally identified with solitons, including preservation of their amplitudes after collisions with other waves.

On the dynamics of ultrathin vortex filaments
View Description Hide DescriptionThe equations of motion for ultrathin cored vortex filaments with exponentially small core size are examined, with emphasis placed on the connection between the asymptotic equations of Klein and Majda [Physica D 49, 323 (1991); 53, 267 (1991)] and the desingularized cutoff equation for thin cored filaments. It is shown that, for the particular case of a helical filament of large pitch, the ultrathin asymptotic equations and the cutoff equation give the same answer when the latter is carefully calculated to include the usually neglected terms of order γ^{2} ln(a/ρ), where 1/γ is the pitch, a is the core size, and ρ is the radius of curvature. Disagreement with the Kelvin [Philos. Mag. 10, 152 (1880)] analysis of waves on vortex columns is attributed to the ultrathin requirement not allowing an overlap regime in which both analyses are valid. Ultrathin filaments of more general shape are considered, and it is shown that the cutoff equation, valid when a/ρ≪1, leads generally to the Klein and Majda equation in the ultrathin regime.

Chaotic phenomena in the interaction of vortex rings
View Description Hide DescriptionThe problem of interaction of several coaxial vortex rings in an inviscid fluid is investigated numerically. It is assumed that the core shape of the vortex rings remain circular. At the initial time the rings are located at the same distance ρ_{0} from the center of the system. This distance is a control parameter of the problem. The cases of interaction of three, four, and five vortex rings are studied. It is shown, that in spite of the nonintegrability of the problem, there are certain domains of values ρ_{0} where the motion of the vortex rings is quasiperiodical. The transition from one such domain to another occurs always through a domain of chaotic interaction. The results of the interaction of the vortex rings are compared with the interaction of their plane analogies–vortex pairs.

Direct numerical simulations of round jets: Vortex induction and side jets
View Description Hide DescriptionIn this paper, a numerical investigation of three‐dimensional round jets subjected to streamwise and azimuthal perturbations is reported. The main objective of the study is to give a consistent scenario for the breaking of rotational symmetry in such flows which may ultimately lead to the production of intense side jets. In particular it is shown that the development of the Widnall instability on the primary vortex rings and the evolution of the Bernal and Roshko [J. Fluid Mech. 170, 499 (1986)] streamwise vortices generated by the instability of the braid could be deeply intertwined. A comprehensive discussion of the vortex induction mechanisms leading to the reorientation of the initial vorticity both in the ring and braid regions and to the deformation of the rings is presented. The recent analysis by Monkewitz and Pfizenmaier [Phys. Fluids A 3, 1356 (1991)] is confirmed in the sense that strong radial ejection of fluid is not directly linked to the deformation of the vortex rings but rather to the occurrence of coherent streamwise vortex pairs. However, the final relative position of the streamwise vortex pairs with respect to the deformations of the vortex rings differs slightly from Monkewitz and Pfizenmaier’s proposition.

An adaptive turbulence filter for decomposition of organized turbulent flows
View Description Hide DescriptionA new decomposition has been developed in which turbulent processes in shear flows may be represented as a combination of organized and more random turbulent motions. Each component is modeled as a summation of its characteristic eddies, of strength that varies in time and space as a function of the entire process. The contribution of all turbulenteddies of the more random component are estimated with an adaptive turbulence filter, which recognizes this component as the orthogonal partner to organized motion, with a power density spectrum of appropriate shape. The decomposition recovers organized motion from time and space series of data in a physically meaningful way, and can be used to characterize interaction between coherent and more random motions. It also provides an estimate for the turbulence in shear flows that are too complex for a meaningful average motion to be identified.

Reynolds stresses and one‐dimensional spectra for a vortex model of homogeneous anisotropic turbulence
View Description Hide DescriptionHomogeneous anisotropicturbulence consisting of a collection of straight vortex structures is considered, each with a cylindrically unidirectional, but otherwise arbitrary, internal vorticity field. The orientations of the structures are given by a distribution P of appropriate Euler angles describing the transformation from laboratory to structure‐fixed axes. One‐dimensional spectra of the velocity components are calculated in terms of P, and the shell‐summed energy spectrum. An exact kinematic relation is found in which volume‐averaged Reynolds stresses are proportional to the turbulent kinetic energy of the vortex collection times a tensor moment of P. A class of large‐eddy simulation models for nonhomogeneous turbulence is proposed based on application of the present results to the calculation of subgrid Reynolds stresses. These are illustrated by the development of a simplified model using a rapid‐distortion‐like approximation.

On the behavior of two‐equation models at the edge of a turbulent region
View Description Hide DescriptionThe flow structure predicted in the vicinity of free‐stream edges by two‐equation eddy‐viscosity turbulencemodels is examined. Analytical expansions, previously used by several authors, are shown to be weak solutions to the pure nonlinear diffusion problem, connecting with trivial solutions in the nonturbulent region. They remain locally valid solutions to the full one‐dimensional system of modelequations in the vicinity of the edge, provided that some constraints on the turbulent ‘‘Prandtl numbers’’ are satisfied. Calculations performed with the (k,ε) turbulencemodel for a time‐evolving mixing layer and a flat‐plate boundary layer in zero pressure gradient are fully consistent with the analysis. In contradiction of a prior study by Lele [Phys. Fluids 28, 64 (1985)], the modeled turbulent‐kinetic‐energy, dissipation‐rate, and shear‐stress fronts are found to propagate into the nonturbulent region at the same velocity, with no need for any special relationship between the model constants. Implications related to the calibration of models are discussed.

Reynolds stress description of opposed and impinging turbulent jets. II. Axisymmetric jets impinging on nearby walls
View Description Hide DescriptionThe flow arising from an axisymmetric turbulent jet with an exit plane in close proximity to the wall against which it impinges is analyzed. The reciprocal of a Reynolds number based on the kinematicviscosity adds here to the two parameters identified in Part I [Phys. Fluids A 5, 203 (1993)] as determining the flow characteristics of two opposed jets, namely the ratio of the turbulence scale to the separation of the jet exit from the wall and the ratio of the turbulence intensity to the mean exit velocity. With all three of these parameters suitably small there is indicated an asymptotic analysis which describes separately the flow in three distinct regions. The temperature of the impinging fluid is assumed to be slightly different from that of the wall so that the mean velocity components, the mean temperature, and the various Reynolds stresses and fluxes vary within a thin wall layer which consists of a viscous sublayer and a turbulent shear layer. When the Reynolds number is suitably high, the sublayer accounts for all of the change in the mean radial velocity and in the mean temperature. The analysis of the turbulence external to the wall layer is compared with experimental data on a turbulent jet impinging on a wall. Agreement with respect to the mean axial velocity and the intensities of the axial and radial velocity components is quite satisfactory. Comparison is made with experimental data on heat transfer to the wall; the predicted dependence of Nusselt number on Reynolds number is confirmed.

Turbulent mixing of a passive scalar
View Description Hide DescriptionThe statistically stationary state of a turbulently advected passive scalar is studied, with an imposed linear mean gradient in two dimensions, via a number of numerical experiments. For a synthetic Gaussian velocity field, which is generated by a linear stochastic process, and whose spectra and Eulerian correlation time follow Kolmogorov scaling on all scales, the exponents of the scalar spectra are consistent with 5/3 or 17/3 depending on the diffusivity. For large Péclet numbers (Pe), the probability density function (PDF) of the scalar gradients perpendicular to the mean is well fit, from about 0.1–10 times the root‐mean‐square value, by a stretched exponential with exponent ∼0.6. The PDF for gradients parallel to the mean has similar tails and a O(1) skewness for all Pe studied. The scalar has a ramp‐and‐cliff structure similar to that first seen in shear‐flow experiments with scalars. A physical picture of the mechanism by which the ramp‐and‐cliff features form is given. A second model with the velocity evolving under the Euler equations restricted to a band of wave numbers produces the k ^{−1} Batchelor spectrum when the scalar is dissipated with a hyperdiffusivity (∝k ^{4}). For physical dissipation (∝k ^{2}), the PDF of the scalar has exponential tails, and for gradients less than the cutoff set by the maximum strain, the PDF of the gradients is similar to that obtained with the stochastic velocity model. The PDF of the dissipation is approximately stretched exponential like the gradient PDFs and not lognormal. The skewness of the gradients parallel to the mean decreases with decreasing autocorrelation time of the velocity, and the gradient PDFs assume a limiting form in the white‐noise limit.

An improved measure of strain state probability in turbulent flows
View Description Hide DescriptionProbability density functions(PDFs) of the strain‐rate tensoreigenvalues are examined. It is found that the accepted normalization used to bound the intermediate eigenvalue between ±1 leads to a PDF that must vanish at the end points for a non‐singular distribution of strain states. This purely kinematic constraint has led previous investigators to conclude incorrectly that locally axisymmetric deformations do not exist in turbulent flows. An alternative normalization is presented that does not bias the probability distribution near the axisymmetric limits. This alternative normalization is shown to lead to the expected flat PDF in a Gaussian velocity field and to a PDF that indicates the presence of axisymmetric strain states in a turbulent field. Extension of the new measure to compressible flow is discussed. Several earlier results concerning the likelihood of various strain states and the correlation of these with elevated kinetic energy dissipation rate are reinterpreted in terms of the new normalization. Most importantly, it is found that regions of axisymmetric expansion play a much more dominant role in the turbulent dissipation process than was previously believed.