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Volume 6, Issue 8, August 1994

Effect of local property smearing on global variables: Implication for numerical simulations of multiphase flows
View Description Hide DescriptionThis work addresses the sensitivity of a calculated global quantity to the smoothing of discontinuous variation of properties. The analysis is based on the very simple model problem of a fully developed, thermocapillary channel flow. The predictions of this simple model are shown to be in excellent agreement with the results for the Navier–Stokes simulations of the thermocapillary migration of a spherical drop. The main conclusion is that the migration velocity depends very strongly on the smearing of the driving interfacial force whereas it is rather insensitive to the smearing of the discontinuous viscosity distribution. This hints at the indispensability of grid adaption for modeling interfacial forces.

The formation of spilling breaking water waves
View Description Hide DescriptionPhotographs from high‐speed movies of the profiles of a mechanically generated, gentle spilling breaking water wave are presented. It is found that as the wave steepens a bulge forms on the forward face of the wave near the crest and capillary waves form on the water surface ahead of the ‘toe’ of the bulge (see Fig. 1). The toe of the bulge then moves rapidly down the forward face of the wave and a train of large‐amplitude waves with short wavelength grows rapidly on the surface of the bulge. These waves quickly break down into a random pattern indicating that the flow has become turbulent.

Shear‐induced particle migration in Couette and parallel‐plate viscometers: NMR imaging and stress measurements
View Description Hide DescriptionCouette and parallel plate viscometers are two commonly used flow geometries to characterize shear viscosity of concentrated suspensions. In Couette flow, it is well documented that prolonged shearing causes a decrease in the apparent viscosity of concentrated suspensions due to shear‐induced particle migration from the annulus region to the stagnant region under the bob. In this study, the technique of nuclear magnetic resonance imaging(NMRI) was used to measure the evolution of suspension concentration profiles in Couette and parallel‐plate flow devices upon shearing. Neutrally buoyant suspensions of nearly monodisperse, non‐Brownian spherical particles at a volume fraction of 0.5 in a Newtonian fluid were used. The same flowcells and suspensions were also used in a rheometer to measure the changes in shear stress under identical experimental conditions such that a direct comparison can be made between the stress and concentration data. For Couette flow, the NMRI data correlated very well with the stress measurements and directly confirmed the Leighton–Acrivos [J. Fluid. Mech. 181, 415 (1987)] shear‐induced migration theory. In torsional flow between parallel plates, no detectable change was found in particle concentration in the radial direction, but some decrease in the apparent viscosity was observed. These results provide some important clues for developing and evaluating more general descriptions of particle migration for nonrectilinear shear flows.

Roll waves on a layer of a muddy fluid flowing down a gentle slope—A Bingham model
View Description Hide DescriptionTo study rapid and laminar flow of a mud layer down a gentle slope, a shallow layer of Bingham fluid is considered. Attention is focused on long waves of finite amplitude. Convective inertia is kept in the approximate equations which are simplified by Kármán’s momentum integral approach with an assumed velocity profile. A linearized instability analysis is first carried out for periodic disturbances. The nonlinear development of unstable disturbances is computed numerically to study the formation of periodic shocks, or roll waves. Nonlinear motion caused by external influx to an otherwise steadily flowing mud layer is also investigated.

Collisions of slightly deformable, high Reynolds number bubbles with short‐range repulsive forces
View Description Hide DescriptionA collision between two bubbles of radii a with an initial relative velocity 2U _{∞} is studied in the limit, Re≡ρU _{∞} a/μ≫1 and We≡ρU ^{2} _{∞} a/γ≪1. Here ρ and μ are the density and viscosity of the liquid and γ is the surface tension of the air–liquid interface. In the absence of colloidal forces, Chesters and Hoffman [Appl. Sci. Res. 38, 353 (1982)] showed that this situation leads to coalescence. The case where there is a nonhydrodynamic repulsive force between the gas–liquid interfaces of the two bubbles that is sufficient to prevent coalescence is considered. Such a force may arise due to ionic salts in aqueous solution or due to surfactants. The analysis shows the bubbles undergo a nearly elastic bounce on an O(a We^{1/2}/U _{∞}) time scale and the repulsive force per unit area must be at least 2γ/a for rebound. Also, experimental observations of bubble collisions in aqueous solutions using high‐speed video photography are presented.

Surface strain modulation of insoluble surface film properties
View Description Hide DescriptionThe complex dispersion relation of capillary waves on water has been measured in a wavelength regime (λ∼1–2 cm) of interest to radar imaging. Two noninvasive sensors: a capacitive wave height antenna and a one‐dimensional scanning laser slope gauge, are used to study the alteration of surface tension and surfaceelastic modulus in the presence of monomolecular surface films. On a static water surface, these sensors are compared to accepted standard techniques. The surface tension results are compared to a Wilhelmy plate while the elastic modulus results are compared to equations of state obtained by Langmuir trough techniques. The sensors are then employed to measure the response of surface film properties to surface areal strains for film pressures and strain rates, which are comparable to natural films in the presence of ship‐generated surface currents. In the experiments reported in this paper, only insoluble surfactants are used. The monolayer surface films considered in this study are found to exhibit higher elastic modulus (∼8 mN/m) at small surface pressure (π≤1 mN/m) than expected from previous work reported in the literature. The response of these films to surface strain is consistent with our static surface measurements. The modulation of film parameters is found to be in phase with internal wave‐generated oscillatory surface strain with periods on the order of 9 s. This implies that the film conforms to the strained water surface with negligible slippage in this regime. These measurements demonstrate the capability of this approach to continuously monitor surface films in situ with detector system response time of the order of 1 s. The technique is suited to large or deep containers and is applicable to sheltered ocean environments.

The stability of nonaxisymmetric circular Couette flow with a radial temperature gradient
View Description Hide DescriptionA numerical solution of linear differential equations governing the stability of the nonaxisymmetric wide‐gap circular Couette flow was present in this study. The numerical solution took into account the presence of a radial temperature gradient between the two rotating cylinders. Boussinesq approximation was applied in the analysis so as to examine the interaction of radial temperature difference with the centrifugal potential but neglect the gravity effect. The critical Taylor number and corresponding critical axial and azimuthal wave numbers were shown graphically for various values of radius ratio, ratio of angular velocities of the inner and outer cylinders, and temperature gradient parameter N for Prandtl number 0.7. Not only the critical Taylor number but the oscillatory onset mode of nonaxisymmetric disturbances could be altered by the effects of a temperature gradient, as indicated by the calculated results.

The viscous stability analysis of Long’s vortex
View Description Hide DescriptionThe viscousstability analysis of Long’s vortex is performed. Some quantitative disagreements with the inviscid results of previous investigators are reported. The truncation of the infinite radial domain into a small computational domain utilized in past investigations is found to be the source of these discrepancies. More important, for small flow forces, a type‐II Long’s vortex is shown to be destabilized by perturbations having positive azimuthal wave number. Based on previous numerical computations, it had been conjectured that these unstable eigenmodes may not exist. The stability characteristics of the unstable modes are fully mapped and are presented in detail.

On the Hopf bifurcation occurring in the two‐layer Rayleigh–Bénard convective instability
View Description Hide DescriptionThe oscillating convective structures appearing at the threshold of the two‐layer Rayleigh–Bénard instability are analyzed in the nonlinear regime. By deriving the amplitude equations for left‐ and right‐traveling waves from the infinite Prandtl number Boussinesq equations, it is shown that one of these waves should generally appear, rather than standing waves, in sufficiently large cells. Numerical results show that these waves have a limited range of existence, because a hysteretic transition to stationary convection occurs when the Rayleigh number is increased (via approach of a heteroclinic orbit for standing waves, and steady‐state bifurcation for traveling waves). From numerical evidence and by comparison with similar behaviors encountered in the one‐layer two‐component problem, it is inferred that the overall behavior is typical of a codimension‐2 Takens–Bogdanov bifurcation.

Dynamic breakup of liquid–liquid jets
View Description Hide DescriptionThe axisymmetric, dynamic breakup of a Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from jet start‐up to breakup into drops was simulated numerically by solving the time‐dependent axisymmetric equations of motion and continuity using an algorithm based on the Volume of Fluid (VOF) method that was previously proven successful in simulations of steady‐state liquid jets (i.e., of the jet region close to the nozzle before breakup). The algorithm has been further refined here based on its performance on transient problems such as the solution of the free liquid–liquid capillary jet breakup problem. The comparison of the simulation results with previous experimental measurements of jet length under conditions where all forces, i.e., viscous, inertial, buoyancy, and surface tension, are important, can be judged satisfactory given the sensitive dependence of the results on details of the experimental setup that are not available. The comparison involves the jet length till breakup as well as the jet and drop shapes, often far from regular. In comparison with experiment, the results of the present numerical method show a greater sensitivity of the jet length to the Reynolds number than the best predictions previously available based on the linear stability analysis of the free liquid–liquid capillary jet breakup problem.

Two‐way coupling in shear layers with dilute bubble concentrations
View Description Hide DescriptionDirect numerical simulations are used to analyze the evolution of a temporally growing two‐dimensional shear layer seeded with dilute concentrations of bubbles under gravity. The bubble concentrations are dilute enough so that bubble–bubble interactions can be neglected, but are large enough for cumulative effects of bubbles to influence the flow. The evolution of the bubble field is determined by tracking many individual bubbles, and the flow field is advanced by using the Navier–Stokes equations with a coupling term representing the effect of the bubbles on the flow. The results are interpreted in terms of the vorticity, density, and pressure fields relative to the one‐way coupled or passive case. For the coupled case, a reduction in the magnitude of the vorticity and pressure gradients near the vortex center is observed. In addition to modification of the flow, it is observed that the accumulation of bubbles is smaller and the location of the equilibrium points are shifted farther from the vortex center as a result of the coupling. It is explored how these changes are modified by different Froude numbers and bubble sizes. The differences between passive and coupled cases usually increase due to larger accumulations as larger bubbles are considered. However, for certain Froude numbers an optimum coupling is observed at intermediate bubble sizes due to the absence of equilibrium points for large bubbles.

A branching liquid jet
View Description Hide DescriptionA new phenomenon of successive branching of a liquid jet emanating from a nozzle into the atmosphere has been reported [Phys. Fluids A 3, 241 (1991)]. It was observed that an intact jet can be made to bifurcate successively into a two‐, three‐, and multiple‐pronged jet by oscillating the nozzle along its axis at a successively higher frequency. It is shown here that the observed multiple‐pronged jet was actually multiple streams of droplets after the jet breakup generated by the lateral component of the nozzle oscillation, which was induced at the free end when the piezoelectric nozzle was axially vibrated. A plausible theoretical explanation of the formation of the multiple streams is given.

One‐dimensional models for slender axisymmetric viscous liquid jets
View Description Hide DescriptionA one‐dimensional analysis of slender axisymmetric viscousliquid jets is considered. A set of one‐dimensional models is derived by substituting a truncated Taylor series in the radial coordinate into the Navier–Stokes equations and boundary conditions at the interface. The relative error, defined as the order of magnitude of the neglected terms divided by the order of the retained ones, is small if the dimensionless wave number k is small enough. The Lee slice model is generalized to take into account viscosity, the relative error being k ^{2}. A new model having a parabolic radial dependence for the axial velocity is developed, with a relative error k ^{4}. The Cosserat model comes from the introduction of the mean axial velocity into the previous one, but an inconsistency arises from neglecting some viscous terms of the same order as those retained. A new model for the mean axial velocity is derived. It conserves the same inertial contribution but avoids the above‐mentioned problem by estimating the involved terms instead of neglecting them. Therefore the relative error is k ^{4} for any value of viscosity. Linear stability analysis is performed for the infinite jet. Results are compared with the exact linear solution given by Lord Rayleigh. The main features predicted in the derivation of the one‐dimensional models manifest themselves in the linear case.

Linear stability of lid‐driven cavity flow
View Description Hide DescriptionPrevious experimental studies indicate that the steady two‐dimensional flow in a lid‐driven cavity becomes unstable and goes through a sequence of transitions before becoming turbulent. In this study, an analysis of this instability is undertaken. The two‐dimensional base flow is computed numerically over a range of Reynolds numbers and is perturbed with three‐dimensional disturbances. The partial differential equations governing the evolution of these perturbations are then obtained using linear stability analysis and normal modeanalysis. Using a finite difference discretization, a generalized eigenvalue problem is formulated from these equations whose solution gives the dispersion relation between complex growth rate and wave number. An eigenvalue solver using simultaneous iteration is employed to identify the dominant eigenvalue which is indicative of the growth rate of these perturbations and the associated eigenfunction which characterizes the secondary state. This paper presents stability curves to identify the critical Reynolds number and the critical wavelength of the neutral mode and discusses the mechanism of instability through energy calculations. This paper finds that the loss of stability of the base flow is due to a long wavelength mode at a critical Reynolds number (Re) of 594. The mechanism is analyzed through a novel application of the Reynolds–Orr equations and shown to be due to a Goertler type instability. The stability curves are relatively flat indicating that this state will be challenged by many shorter wavelength modes at a slightly higher Reynolds number. In fact, a second competing mode with a wavelength close to the cavity width was found to be unstable at Re=730. The present results of the reconstructed flow based on these eigenfunctions at the neutral state, show striking similarities to the experimental observations.

Flows induced in a cylinder with both end walls rotating
View Description Hide DescriptionThe flow field inside a cylindrical container induced by the rotation of the top and bottom end walls with a fixed sidewall is described. For this problem, this paper shows that stagnation points occur along the axis of rotation between the midplane of symmetry and the rotating end walls for appropriate values of the characteristic parameters, viz., the Reynolds number and the aspect ratio of the container. Aspect ratios of 0.5, 0.8, 1.0, and 1.5 were examined over a range of Reynolds numbers from 100 to 2000. As the Reynolds number increased beyond a critical value a recirculation zone surrounding a columnar vortex core in the meridional‐plane flow pattern is predicted to occur near the midplane. This toroidal vortex is different from the type B vortex breakdown phenomenon that occurs in cylindrical containers with only one end wall rotating.

Discrete shedding modes of the cylinder wake in a jet with a homogeneous core
View Description Hide DescriptionThe von Kármán vortex street behind a circular cylinder in a laminar homogeneous jet core is experimentally investigated. The Strouhal–Reynolds number relationships are measured for various shear‐layer thicknesses and aspect ratios. The experimental Strouhal number values are found to collapse with the discrete vortex shedding modes, which were observed for boundary‐layer end conditions. The results indicate that the shedding modes are independent from the end conditions, but are an intrinsic feature of the shedding process of an infinitely long cylinder. The experimentally assumed shedding mode, however, is strongly dependent of the geometric parameters, like the shear‐layer thickness and the aspect ratio. The observed tendencies can be made physically plausible.

The plane wake of a cylinder: An estimate of the pressure–strain rate tensor
View Description Hide DescriptionThe modeling of the pressure strain rate terms is an important issue in the improvement of the generality of closure models for the Reynolds stress transport (RST) equations. A part of these efforts is to provide accurate experimental information on the pressure–strain rate tensor, which in turn requires equally accurate information on the dissipation and the diffusiontensors. Here the far wake of a cylinder was studied in order to enable the required balances of the RST equations. The experimental results indicate a nonisotropic dissipation tensor, and show the energy redistribution between the different components as described by the pressure–strain rate correlations. Comparisons are made between the experimentally determined pressure–strain rate distributions and the corresponding distributions predicted by closure models.

Thermomechanical and magnetohydrodynamic stability of elongating plastic jets
View Description Hide DescriptionIn this study the stability characteristics of uniformly elongating plastic jets exposed to axial electric currents are investigated. The objective of this study is to expand the results of previous analyses by Littlefield [‘‘The effect of electromagnetic fields on the stability of a uniformly elongating plastic jet,’’ Phys. Fluids A 2, 2240 (1990); ‘‘Finite conductivity effects on the MHD instabilities in uniformly elongating plastic jets,’’ ibid. 3, 166 (1991); ‘‘Enhancement of stability in uniformly elongating plastic jets with electromagnetic fields,’’ ibid. 3, 2927 (1991)] to include high levels of electric current, where thermal energy effects must be included. Coupling of the magnetohydrodynamic and thermal characteristics of the flow is accomplished through the variation of mechanical, thermal, and electrical properties with temperature. Phase changeeffects are also considered. The jet is assumed incompressible and perfectly plastic, with the Levy–von Mises criterion imposed to limit the effective stress. Solutions to the appropriate base flow are subjected to small axisymmetric disturbances, and linear perturbation theory is employed to determine the time evolution of these disturbances. Perturbations that grow the fastest in magnitude as time progresses are identified as the most unstable. Results of the analysis indicate that thermal effects can dramatically alter both the base and perturbed flow fields, as well as the growth rate of perturbations.

Topological fluid dynamics of interfacial flows
View Description Hide DescriptionThe topological description of flows in the vicinity of a solid boundary, that is familiar from the aerodynamics literature, has recently been extended to the case of flow at a liquid–gas interface or a free surface by Lugt [Phys. Fluids 30, 3647 (1987)]. Lugt’s work is revisited in a more general setting, including nonconstant curvature of the interface and gradients of surface tension, using tools of modern nonlinear dynamics.Bifurcations of the flow pattern occur at degenerate configurations. Using the theory of unfolding, this paper gives a complete description of the bifurcations that depend on terms up to the second order. The general theory of this paper is applied to the topology of streamlines during the breaking of a wave and to the flow below a stagnant surface film.

Modification of the Euler equations for ‘‘vorticity confinement’’: Application to the computation of interacting vortex rings
View Description Hide DescriptionA new ‘‘vorticity confinement’’ method is described which involves adding a term to the momentum conservationequations of fluid dynamics. This term depends only on local variables and is zero outside vortical regions. The partial differential equations with this extra term admit solutions that consist of Lagrangian‐like confined vortical regions, or covons, in the shape of two‐dimensional (2‐D) vortex ‘‘blobs’’ and three‐dimensional (3‐D) vortex filaments, which convect in a constant external velocity field with a fixed internal structure, without spreading, even if the equations contain diffusive terms. Solutions of the discretized equations on a fixed Eulerian grid show the same behavior, in spite of numerical diffusion. Effectively, the new term, together with diffusive terms, constitute a new type of regularization of the inviscid equations which appears to be very useful in the numerical solution of flow problems involving thin vortical regions. The discretized Euler equations with the extra term can be solved on fairly coarse, Eulerian computational grids with simple low‐order (first‐ or second‐) accurate numerical methods, but will still yield concentrated vortices which convect without spreading due to numerical diffusion. Since only a fixed grid is used with local variables, the vorticity confinement method is quite general and can automatically accommodate changes in vortex topology, such as merging. Applications are presented for incompressible flow in 3D, where pairs of thin vortex ringsinteract and, in some cases, merge.