Volume 7, Issue 5, May 1995
Index of content:

Locally isotropic pressure Hessian in a high‐symmetry flow
View Description Hide DescriptionRegions in a high‐symmetry flow [J. Phys. Soc. Jpn. 54, 2132 (1985); Phys. Fluids 6, 2757 (1994)] where the pressure Hessian term remains diagonal for all times are identified. It is found that vorticity also vanishes in such regions. The components of the pressure Hessian tensor are equal only at the origin, toward which the vortices approach. For the case of isotropic pressure Hessian with no vorticity, and in the absence of viscosity, the resulting equations for the evolution of the eigenvalues and the Q invariant of the strain rate tensor are integrated numerically and analytically respectively. It is found that all the eigenvalues and the Q invariant diverge to infinity in finite time. In a recent simulation of a high‐symmetry flow [Phys. Fluids 6, 2757 (1994)], it is found that the two (positive) eigenvalues of the strain rate tensor become equal as all the eigenvalues and the Q invariant show trends of divergence and as the vortices approach to the origin.

Dispersion in zero‐mean periodic flows
View Description Hide DescriptionThe dispersion of tracer particles in zero‐mean shear flows with no‐slip boundaries is found to be diffusive when the particle trajectories are chaotic and ergodically explore the flow domain. This diffusive spreading occurs even when the molecular diffusivity vanishes. The long‐time behavior for zero‐mean shear flows thus differs quantitatively from non‐zero mean shear flows. A model problem is formulated and the moment equations solved using techniques from the theory of random walks. Dispersion of ‘‘perfect’’ tracers in oscillatory Rayleigh–Bénard convection is also found to behave diffusively.

Streamwise vortices in plane Couette flow
View Description Hide DescriptionExperimental observations of various flows have led to the conclusion of the existence of streamwise vortices involved in the destabilization process of these flows. In the plane Couette flow, a linear shear flow, such structures have never been observed, because of the linear stability of the flow. The flow was slightly modified by introducing a wire in its central plane, parallel to the spanwise direction. A destabilization then occurs. It generates streamwise structures periodically spaced in the spanwise direction. These structures have been identified as pairs of counter‐rotating streamwise vortices. This Letter characterizes the dependence on the Reynolds number of the behavior of these vortices and how their destabilization leads to turbulence.

Magnetic resonance imaging study of sedimenting suspensions of noncolloidal spheres
View Description Hide DescriptionBatch sedimentation experiments were conducted with suspensions of noncolloidal spherical particles. Using nuclear magnetic resonance imaging(NMRI), the time evolution of the volume fraction versus height profile was measured for initial suspension volume fractions, φ_{ i }, ranging from 0.08 to 0.44. NMRI clearly delineates the clear fluid layer at the top of the suspension, below which there is a transition to a region having the initial mean particle concentration. The hindered settling function determined from these data corresponds well with previous results. The spreading of the interface in excess of that expected from the combined effects of polydispersity and self‐sharpening was analyzed as a diffusion process. The measured values of the self‐induced hydrodynamic diffusivity agreed with those reported previously, and they decreased sharply for φ_{ i }≳0.15. The concentration profile was also measured through the fan region into the sediment, where the volume fraction is near maximum packing, and determined the fan thickness as a function of time. The fan thickness is found to increase as the initial suspension volume fraction is increased.

Convective instability mechanisms in thermocapillary liquid bridges
View Description Hide DescriptionThe primary instability of axisymmetric steady thermocapillary flow in a cylindrical liquid bridge with non‐deformable free surface is calculated by a mixed Chebychev‐finite difference method. For unit aspect ratio the most dangerous mode has an azimuthal wavenumber m=2. The physical instability mechanisms are studied by analyzing the linear energy balance of the neutral mode. If the Prandtl number is small (Pr≪1), the bifurcation is stationary. The associated neutral mode is amplified in the shear layer close to the free surface. For large Prandtl number (Pr=4), the basic state becomes linearly unstable to a pair of hydrothermal waves propagating nearly azimuthally. Both mechanisms are compared with those previously proposed in the literature.

Asymptotic study and weakly nonlinear analysis at the onset of Rayleigh–Bénard convection in Hele–Shaw cell
View Description Hide DescriptionThe aim of this paper is the derivation of the Ginzburg–Landau equation [as introduced by A. C. Newell and J. A. Whitehead, J. Fluid Mech. 38, 279 (1969)] from the hydrodynamicequations for an infinite Hele–Shaw cell. The dimensional analysis and the asymptotic study allow one to distinguish two nonlinear formulations, each one depends on the order of magnitude of the Prandtl number. The first formulation corresponds to the case Pr=O(1) or Pr≫1, whereas the second corresponds to the case Pr=O(ε*^{2}), where ε*≪1 denotes the aspect ratio of the cell. Here a weakly nonlinear analysis is performed for the two formulations.

On the transition to columnar convection
View Description Hide DescriptionConvection in a rotating annulus with no‐slip sidewalls, stress‐free ends, radial gravity, and sideways heating is considered. The transition from fully three‐dimensional convection cells to Taylor columns with increasing rotation rate is studied and its dependence on the annulus parameters is established.

Theory of drop formation
View Description Hide DescriptionThe motion of an axisymmetric column of Navier–Stokes fluid with a free surface is considered. Due to surface tension, the thickness of the fluid neck goes to zero in finite time. After the singularity, the fluid consists of two halves, which constitute a unique continuation of the Navier–Stokes equation through the singular point. The asymptotic solutions of the Navier–Stokes equation are calculated, both before and after the singularity. The solutions have scaling form, characterized by universal exponents as well as universal scaling functions, which are computed without adjustable parameters.

Arbitrarily oriented capillary‐viscous planar jets in the presence of gravity
View Description Hide DescriptionNumerical solutions for the steady flow of an arbitrarily oriented incompressible capillary‐viscous jet in the presence of gravity are presented. A combination of the finite element method and an elliptic mesh generation technique are used to compute solutions over a wide range of the four relevant physical parameters: Reynolds number, Stokes number, capillary number, and angle of the jet nozzle with respect to gravity. It is shown that both gravity and surface‐tension forces can be instrumental in the mechanism of jet turning. In addition, it is found that multiple solutions are possible when the jet nozzle is directed close to vertically upward. The linear stability of the steady solutions to two‐dimensional disturbances is briefly considered.

Nonlinear modeling of jet atomization in the wind‐induced regime
View Description Hide DescriptionA boundary element method(BEM) has been developed to solve for the nonlinear evolution of a liquid jet acting under the influence of both surface tension and the aerodynamicinteractions with the surrounding atmosphere. For longer waves, aerodynamic effects are shown to cause a ‘‘swelling’’ of the liquid surface in the trough region. The model predicts the presence of satellite drops in the first wind‐induced regime, and predicts the evolution of a ‘‘spiked’’ surface at the periphery of the jet for conditions consistent with the second wind‐induced regime. The effects of the disturbance wave number, the liquid Weber number, and the density ratio between the liquid jet and the surrounding gas on the breakup of the jet have been examined. Transition points between various flow regimes have also been identified.

Vortex breakdown incipience: Theoretical considerations
View Description Hide DescriptionThe sensitivity of the onset and the location of vortex breakdowns in concentrated vortex cores, and the pronounced tendency of the breakdowns to move upstream have been characteristic observations of experimental investigations; they have also been features of numerical simulations and led to questions about the validity of these simulations. This movement upstream may be a migration in time for fixed values of the relevant parameters, or the movement of the breakdown location closer to the entrance to the flow or computational domain with small changes in these parameters. This behavior seems to be inconsistent with the strong time‐like axial evolution of the flow, as expressed explicitly, for example, by the quasicylindrical approximate equations for this flow. An order‐of‐magnitude analysis of the equations of motion near breakdown leads to a modified set of governing equations, analysis of which demonstrates that the interplay between radial inertial, pressure, and viscous forces gives an elliptic character to these concentrated swirling flows. Analytical, asymptotic, and numerical solutions of a simplified nonlinear equation are presented; these qualitatively exhibit the features of vortex onset and location noted above.

Chaotic oscillations and breakup of quasigeostrophic vortices in the N‐layer approximation
View Description Hide DescriptionThis paper examines the onset of chaotic oscillations in the N‐layer model of stratified quasigeostrophic flow due to interaction of Nvortex sections in different layers, and then considers the possible role of chaotic self‐induced motion in the breakup of quasigeostrophic vortices. In order to investigate the nature of the long‐time evolution of the vortex using a low‐dimensional dynamical system, attention is restricted to vortices with uniform potential vorticity within the core and sufficiently large values of the stratification parameter that the vortex cross section remains nearly circular with time. Computations using contour dynamics in many‐layered systems are performed to check that the deformation of the vortex cross section is small. It is shown that for finite‐amplitude perturbations of the vortex centerline, the barotropic interaction between sections of the vortex in different layers can cause the vortex to oscillate chaotically. The approach to chaotic motion is initially investigated using a system with only four layers, in which the vortex is approximated by equal‐strength segments of either point vortices or circular patches. The onset of chaos is shown to be sensitive to the stratification parameter, the vortex core radius, and the initial configuration. The consequences of chaos on the vortex evolution and breakup are then demonstrated by numerical computations with a large number of layers for vortices with finite‐core area.

Flow structure from an oscillating cylinder with a localized nonuniformity: Patterns of coherent vorticity concentrations
View Description Hide DescriptionForced oscillation of a uniform cylinder with a localized, fluid‐induced nonuniformity at its midspan, in the form of either blowing or suction, generates well‐defined patterns of vorticity concentrations. The spanwise extent of these patterns abruptly grows to an order of magnitude larger than the initially induced nonuniformity. Such patterns exhibit ordered changes in the number, arrangement, and circulation of the vorticity concentrations as the magnitude of the nonuniformity is altered.

Dipole formation in the transient planar wall jet
View Description Hide DescriptionAn initially quiescent quarter‐plane of fluid is set into motion by the action of a wall jet, i.e., a jet parallel and adjacent to a wall. The circumstances for which the jet separates from the wall and forms a dipole at the upstream head of the jet are studied numerically. The streamfunction‐vorticity formulation is used to track the time evolution of vorticity over a range of Reynolds numbers, 50≤Re≤1000, based on jet velocity and width. By implementing both a no‐slip and a slip boundary condition and comparing the results for the time evolution of vorticity, it is found that the boundary condition has a negligible effect on dipole development. By contrast, the relative magnitude of positive ω_{+} and negative ω_{−}vorticity present in the inlet jet flow controls whether dipole formation occurs. In particular, dipole formation requires a sufficiently large magnitude of negative vorticity ω_{−}/ω_{+}≳0.65. The results refine a conjecture by Yushina (‘‘Evolution of the near‐wall jet,’’ in General Circulations of the Oceans, Technical Report No. WHOI of the 1989 Summer Study Program in GeophysicalFluid Dynamics, Woods Hole, Oceanographic Institute, Woods Hole, MA, 1989, pp. 470–512) that negative vorticity is necessary for dipole creation. The results further indicate that the criterion for the existence of a moving separation point concurrently serves as a criterion for dipole formation. This suggests that, in some situations in an unsteady boundary layer, dipole generation will be associated with unsteady separation.

Distortion and evolution of a localized vortex in an irrotational flow
View Description Hide DescriptionThis paper examines the interaction of an axisymmetric vortex monopole, such as a Lamb vortex, with a background irrotational flow. At leading order, the monopole is advected with the background flow velocity at the center of vorticity. However, inhomogeneities of the flow will cause the monopole to distort. It is shown that a shear‐diffusion mechanism, familiar from the study of mixing of passive scalars, plays an important role in the evolution of the vorticity distribution. Through this mechanism, nonaxisymmetric vorticity perturbations which do not shift the center of vorticity are homogenized along streamlines on a Re^{1/3} time scale, much faster than the Re decay time scale of an axisymmetric monopole. This separation of time scales leads to the quasisteady evolution of a monopole in a slowly varying flow. The asymptotic theory is verified by numerically computing the linear response of a Lamb monopole to a time‐periodic straining flow and it is shown that a large amplitude, O(Re^{1/3}), distortion results when the monopole is forced at its resonant frequency.

Mixing properties of three‐dimensional (3‐D) stationary convection
View Description Hide DescriptionPassive tracers in steady‐state three‐dimensional (3‐D) convective flows with infinite Prandtl number, which is relevant for the Earth’s mantle, show a remarkable flow structure. Individual flowlines as shown by Poincare sections of the tracer paths lie on a two‐dimensional (2‐D) surface with distorted toroidal topology. Furthermore, the space occupied by the convecting fluid is filled by a set of these toroidal surfaces nested one within another. The small radius of the innermost toroidal surface approaches zero, defining a closed streamline whose location we have determined in specific cases using numerical solutions. The outermost of the toroidal surfaces coincides with the upper and lower surfaces of the layer and with vertical symmetry planes which separate the flow between neighboring cells. Both square and hexagonal convection planforms show a triangular cellular structure with triangles defined by (π/2,π/4,π/4) and (π/2,π/6,π/3), respectively. The outer toroidal surface is closed by a horizontal flow line through the middle of the cell. The numerical experiments suggest that streamlines are not generally closed in any small number of orbits. Instead the toroidal surface appears to be progressively filled in by the trace of a single streamline which, in successive orbits, is displaced across the surface without returning to the same path. This flow structure ensures that, while extreme shear strains can occur, particularly in the vicinity of the cell separatrices, mixing of the material only occurs in 2D. Tracers initially on one toroidal surface remain on that surface indefinitely. Like for 2‐D convective flow, time dependence of the solution appears to be a necessary prerequisite for thorough spatial mixing to occur.

Critical dissipation rates in density stratified turbulence
View Description Hide DescriptionThe estimation of a critical dissipation rate, ε_{ c }, below which vertical transport ρw is completely suppressed, is examined in the context of laboratory experiments on shear‐free, decaying, and stably stratified grid‐generated turbulence. It is shown how the use of a criterion based on ρw=0 in these transient flows is not appropriate for obtaining a universal value for the critical dissipation rate expressed nondimensionally as ε_{ c }/νN ^{2}. It remains unclear how any universal value of ε_{ c } may be inferred from these laboratory experiments.

On the onset of three‐dimensionality in mixing layers
View Description Hide DescriptionUsing a nonlinear critical layer analysis, a set of nonlinear integrodifferential evolution equations is derived, governing the temporal growth of a disturbance consisting of both a two‐dimensional wave and a pair of oblique waves with the same streamwise wave number as the two‐dimensional wave superimposed on a tanh y mixing layer [a so‐called Benney–Lin (BL) triad]. This is a model of so‐called K‐type transition. Numerical solutions to these equations develop a singularity at a finite time, which is confirmed by asymptotic analysis. The relevance of this analysis to the mixing layer experiments of Lasheras and Choi [J. Fluid Mech. 189, 53 (1988)] and numerical simulations of Ashurt and Meiburg [J. Fluid Mech. 189, 87 (1988)] and Rogers and Moser [J. Fluid Mech. 243, 183 (1992); NASA TM No. 103856, 1991] are discussed. As with similar equations derived for related problems, these equations are shown to have solutions which become singular after a finite time.

Solitary waves on water of finite depth with a surface or bottom shear layer
View Description Hide DescriptionTwo‐dimensional, gravity solitary waves on water of finite depth with a surface layer of uniform vorticity are considered. Accurate numerical solutions are computed by a boundary integralequation method. It is found that the waves have a limiting configuration with a 120° angle at the surface crest and that the shapes of the limiting profiles near the surface crest are different for positive and negative vorticity. The effects of the shear layer strength and thickness on the wave profiles are discussed. In addition, a related configuration when the shear layer is near the bottom is also considered. It is shown that some of the branches of solutions have a limiting configuration with a 120° angle at the surface crest and that others ultimately approach a solitary wave without gravity. This is to be contrasted with the case of a surface shear layer for which all the corresponding branches approach a limiting configuration with a 120° angle at the surface crest.

Two‐level solitary waves as generalized solutions of the KdV equation
View Description Hide DescriptionAn analytical class of standing wave solutions to the Korteweg–de Vries (KdV) equation is obtained in the framework of continuous finite generalized functions. This paper shows how this class can be used to describe a great variety of wave shapes, especially bores and jumps. These new solutions are built by appropriately combining parts of two ordinary KdV waves. These represent a system with a steep transition between different energy levels of two potential wells. A number of specific cases of the generalized solutions are identical to those obtained recently within the theory of the KdV equation forced by a Dirac delta function. Numerical simulations of both stationary and transient KdV equations are carried out in a few cases. The weak formulation used in the numerical scheme is equivalent to the analytical generalized functions approach. Simulations of initially perturbed wave fronts prove the high degree of stability of many of these solutions.