Volume 13, Issue 10, October 2001
 LETTERS


Threedimensional instability during vortex merging
View Description Hide DescriptionThe interaction of two parallel vortices of equal circulation is observed experimentally. For low Reynolds numbers (Re), the vortices remain two dimensional and merge into a single one, when their timedependent core size exceeds approximately 30% of the vortex separation distance. At higher Re, a threedimensional (3D) instability is discovered, showing the characteristics of an elliptic instability of the vortex cores. The instability rapidly generates smallscale turbulent motion, which initiates merging for smaller core sizes and produces a bigger final vortex than for laminar 2D flow.
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 ARTICLES


Free surface Stokes flow over topography
View Description Hide DescriptionWe study twodimensional viscousflow over topographical features under the action of an external body force. The Stokes equations are written as a set of harmonic and biharmonic equations for vorticity and stream function. A direct biharmonic boundary integralequation method is used to transform these equations to a pair of integral equations on the boundary of the domain. These equations are solved with a preassumed free surface profile to obtain values of the flow variables on the boundary. The location of the free boundary is then updated by considering the normal stress condition and using an iteration technique. We have studied the flow over steps and trenches of different depths and for a wide range of capillary numbers, Ca. Our computation shows that for small Ca, the free surface develops a ridge before the entrance to a step down and a depression region right before a step up. The magnitude and location of these features depend on the capillary number and the step depth. For large capillary numbers the free surface nearly follows the topography and the ridge and depression are found to be exponentially small in the capillary number. On the other hand, our results agree well with lubrication theory for small capillary numbers on the order of or less, even for steep features.

Creeping flow through a pipe of varying radius
View Description Hide DescriptionCreeping flow of a Newtonian fluid through tubes of varying radius is studied. Using an asymptotic series solution for low Reynolds number flow, velocity profiles and streamlines are obtained for constricted tubes, for various values of constriction wavelength and amplitude. A closedform expression is derived to estimate the pressure drop through this type of tube. The results obtained with this new expression are compared to data from previous experimental and numerical studies for sinusoidally constricted tubes. Good agreement is found in the creeping flow regime for the pressure drop versus flow rate relationship. Our method offers an improvement over the integrated form of the Hagen–Poiseuille equation (i.e., lubrication approximation), which does not account for the wavelength of the constrictions.

Buoyantthermocapillary instabilities in extended liquid layers subjected to a horizontal temperature gradient
View Description Hide DescriptionWe report experiments on buoyantthermocapillary instabilities in differentially heated liquid layers. The results are obtained for a fluid of Prandtl number 10 in a rectangular geometry with different aspect ratios. Depending on the height of liquid and on the aspect ratios, the twodimensional basic flow destabilizes into oblique traveling waves or longitudinal stationary rolls, respectively, for small and large fluid heights. Temperature measurements and space–time recordings reveal the waves to correspond to the hydrothermal waves predicted by the linear stability analysis of Smith and Davis [J. Fluid Mech. 132, 119 (1983)]. Moreover, the transition between traveling and stationary modes agrees with the work by Mercier and Normand [Phys. Fluids8, 1433 (1996)] even if the exact characteristics of longitudinal rolls differ from theoretical predictions. A discussion about the relevant nondimensional parameters is included. In the stability domain of the waves, two types of sources have been evidenced. For larger heights, the source is a line and generally evolves towards one end of the container leaving a single wave whereas for smaller heights, the source looks like a point and emits a circular wave which becomes almost planar farther from the source in both directions.

Thermal effects on liquid film flow during spin coating
View Description Hide DescriptionA theoretical analysis of the thermal effects on the freesurface filmflow on a flat rotating disk is presented. Assuming a small aspect ratio of the initial film thickness to the disk radius and neglecting peripheral effects of the liquid film, the evolution equation describing the shape of thin liquid film interface is obtained as a function of space and time and is solved using perturbation analysis. The results reveal the effects of inertial, gravitational, surfacetension and thermocapillary forces and of variable viscosity on the film planarization and thinning. Among these, it was found that the variable viscosity has the most profound effect on the transient film thickness.

A computational evaluation of the Ergun and Forchheimer equations for fibrous porous media
View Description Hide DescriptionThe results of a comprehensive computational evaluation of the Ergun and Forchheimer equations for the permeability of fibrous porous media are reported in this study. Square and hexagonal arrays of uniform fibers have been considered, as well as arrays in which the fiber size is allowed to change in a regular manner, expressed by a size variation parameter (δ). The range of porosity (φ) examined is from 0.30 to 0.60, the Reynolds number ranges between 0 and 160, and the size variation parameter (δ) between 0 (corresponding to the uniform array) and 0.90 (in which case the diameter of the large fibers in the array is 19 times that of the small ones). We obtain computational results for pressure drop and flow rate in a total of 440 cases mapping the (φ,δ,Re) space; these are presented in terms of a friction factor and are compared to the predictions of the Ergun and Forchheimer equations, both widely used models for the permeability of porous media. In the limit of creeping flow (Re<1), the Forchheimer equation is in excellent agreement with the computational results, while the Ergun equation is unable to capture the behavior of fiber arrays in which the flow has a strong contracting/expanding element. The Forchheimer equation, in its original form, is in closer agreement with the computational results. When the Forchheimer term (F) is expressed as a function of porosity, we obtain a modified form of the Forchheimer equation that is in excellent agreement with computational results for the entire range of (φ,δ,Re) examined.

Collapse and rebound of a laserinduced cavitation bubble
View Description Hide DescriptionA strong laser pulse that is focused into a liquid produces a vapor cavity, which first expands and then collapses with subsequent rebounds. In this paper a mathematical model of the spherically symmetric motion of a laserinduced bubble is proposed. It describes gas and liquid dynamics including compressibility, heat, and mass transfer effects and nonequilibrium processes of evaporation and condensation on the bubble wall. It accounts also for the occurrence of supercritical conditions at collapse. Numerical investigations of the collapse and first rebound have been carried out for different bubble sizes. The results show a fairly good agreement with experimental measurements of the bubble radius evolution and the intensity of the outgoing shock wave emitted at collapse. Calculations with a small amount of noncondensable gas inside the bubble show its strong influence on the dynamics.

Chaotic oscillation of a bubble in a weakly viscous dielectric fluid under electric fields
View Description Hide DescriptionThe dynamics of a bubble in a weakly viscousdielectric fluid under electric fields is studied. The dynamical equations for the volume and shape mode oscillations are derived using the domain perturbation method with firstorder accuracy in deformation. For the volume mode oscillation, we obtain the modified Rayleigh–Plesset equation which includes a forcing term due to the effect of electric field. For the shape mode oscillations, the Prosperetti–Seminara equation [Phys. Fluids 21, 1465 (1978)] is also extended. The dynamical equations are analyzed with two types of electric fields: the uniform field and the axisymmetric straining field. Equilibrium analysis is performed to find the equilibrium points in the phase planes and their stabilities in static electric fields. Then, the effects of timeperiodic electric fields on the bubble dynamics are considered at two levels of viscosity effect [the inviscid limit and the case of ]. The nonlinear dynamics theory is used for analysis of the complicated volume and shape mode oscillations.

Centrifugal instabilities in a curved rectangular duct of small aspect ratio
View Description Hide DescriptionWe report experimental results on the stability of the flow occurring in a curved rectangular duct of small aspect ratio, where the centrifugational forces act along the largest dimension. The basic flow is threedimensional as in a square or circular curved duct, and above a critical flow rate, streamwise vortices are observed. The threshold of the instability is found to be controlled by a Dean number.

The stability of noncolumnar swirling flows in diverging streamtubes
View Description Hide DescriptionA linear stability analysis of a family of steady, noncolumnar and axisymmetric, swirling flows that may develop in a finitelength slightly diverging pipe is presented. These flow states are described by the asymptotic analysis of Rusak et al. (1998). There exists a limit level of the incoming flow swirl ratio which is the corrected critical swirl as a result of the pipe divergence. When the swirl ratio is in a certain range below two steady states can exist for the same inlet, outlet, and wall conditions: One which describes a nearcolumnar vortex state and another which describes a swirling flow with a largeamplitude disturbance. When the swirl level is above no nearcolumnar, steady, and axisymmetric state exists. The stability of this family of flows is examined by studying the linearized dynamics of an unsteady and axially symmetric perturbation which also satisfies the boundary conditions. The stability analysis shows that is a point of exchange of stability for the family of the noncolumnar vortex flows. The nearcolumnar states have a linearly stable mode of disturbance whereas the states with large disturbances are unstable. Also, the nearcolumnar states lose their stability characteristics as the swirl level approaches Therefore, the analysis implies that when the swirl level of the incoming flow is above the flow in the pipe must develop a transition process that involves largeamplitude perturbations and may lead to vortex breakdown states. The effect of the increase of pipe divergence on the flowdynamics and transition to breakdown states is also discussed.

Temporal instability of swirling gas jets injected in liquids
View Description Hide DescriptionA temporal linear stability analysis of an inviscid incompressible swirling gas jet injected into a coflowing liquid was conducted. The ratio of the tangential velocity to the axial velocity (swirl number) played a significant role in the destabilization of the gas jet. Even at small gas Weber numbers, the presence of swirl caused the higher order azimuthal modes to become unstable; the growth rates, and the dominant and limiting wave numbers of the higher order modes were greater than those of the varicose and sinuous modes. The differences in growth rates, limiting and dominant wave numbers of the various azimuthal modes became significant at large gas Weber numbers. An increase in liquid viscosity resulted in a reduction in the growth rates and the dominant wave numbers. The liquid coflow velocity controlled the phase velocity of the unstable modes.

Threedimensional simulations of hydrodynamic instability in liquid bridges: Influence of temperaturedependent viscosity
View Description Hide DescriptionThe development of thermocapillary convection inside a cylindrical liquid bridge is investigated by using a direct numerical simulation of the threedimensional (3D), timedependent problem for a wide range of Prandtl numbers, Pr=1,3,4,5 and Pr=35. Above the critical value of temperature difference between the supporting disks, two counterpropagating hydrothermal waves bifurcate from the twodimensional (2D) steady state. The existence of standing and traveling waves is discussed. The dependence of viscosity upon temperature is taken into account. The critical Reynolds number and critical frequency at which the system undergoes a transition from a 2D steady state to a 3D oscillatory flow decreases if the viscosity diminishes with temperature. The stability boundary is determined for Pr=3–5 with a viscosity contrast up to a factor 10. Near the threshold of instability the flow organization is similar for the constant and variable viscosity cases despite the large difference in critical Reynolds numbers. The influence of variable viscosity on the flow pattern is increased when going into the supercritical region. The study of spatialtemporal behavior of oscillatory convection for the high Prandtl number, Pr=35, demonstrates a good agreement with previously published experimental results. For this high Prandtl number liquids instability begins as a standing wave with an azimuthal wave number which then switches to an oblique traveling wave ≈4%–5% above the onset of instability.

Asymptotic behavior of threedimensional bubbles in the Richtmyer–Meshkov instability
View Description Hide DescriptionWe report an analysis to the problem of nonlinear motion of bubbles and spikes generated by the Richtmyer–Meshkov instability. The flow is threedimensional (3D), periodic and anisotropic in the plane normal to the direction of shock. We show that in the traditional Layzertype approach, regular asymptotic solutions to the problem are absent in the general case. We propose yet another approach and find a family of regular asymptotic solutions parameterized by the principal curvatures at the bubble top. In the expanded functional space the interplay of harmonics is well captured. For solutions of this family, a bubble with a flattened surface is faster than a bubble with finite curvatures in both 3D and twodimensional (2D) cases, while highly symmetric 3D bubbles are faster than anisotropic 3D and 2D bubbles. For nearly symmetric 3D flows, the Layzertype solution is the point of bifurcation.

Threedimensional acoustic scattering by vortical flows. I. General theory
View Description Hide DescriptionWhen an acoustic wave is incident on a threedimensional vortical structure, with length scale small compared with the acoustic wavelength, what is the scatteredsound field that results? A frequently used approach is to solve a forced wave equation for the acoustic pressure, with nonlinear terms on the righthand side approximated by the bilinear product of the incident wave and the undisturbed vortex: we refer to this as the “acoustic analogy” approximation. In this paper, we show using matched asymptotic expansions that the acoustic analogy approximation always predicts the leadingorder scatteredsound field correctly, provided the Mach number of the vortex is small, and the acoustic wavelength is a factor of order larger than the scale of the vortex. The leadingorder scattered field depends only on the vortexdipole moment. Our analysis is valid for acoustic frequencies of the same order or smaller than the vorticity of the vortex. Over long times, the vortex may become significantly disturbed by the incident acoustic wave. Additional conditions are derived to maintain validity of the acoustic analogy approximation over times of order long enough for motion of the vortex to be significant on the length scale of the acoustic waves.

Threedimensional acoustic scattering by vortical flows. II. Axisymmetric scattering by Hill’s spherical vortex
View Description Hide DescriptionThe general theory of Part I is applied to the the specific case of scattering of a wave incident along the axis of Hill’s spherical vortex. The full asymptotic solution to the initialvalue problem is calculated. Results agree with the general approach, showing that the conditions required for the latter to hold apply in the case of Hill’s spherical vortex.

Nonlinear analysis of oscillatory flow in the annulus of an elastic tube: Application to catheterized artery
View Description Hide DescriptionThe changed flow pattern of pulsatile blood flow in an annulus with elastic outer wall has been studied through a mathematical model. The main objective is to apply the model to study the combined effect of introduction of the catheter and elastic properties of the arterial wall on the pulsatile nature of the blood flow. The diameter variation of the wall is considered small for the perturbation analysis to be valid. The steady streaming effect brings into focus the existence of a nonzero mean pressure gradient in addition to the one predicted by the linear theory—a fact overlooked by previous authors. Thus, our results are intended to provide a correction to the “mean pressure gradient–flow rate relationship” usually calculated by neglecting the nonlinear inertia terms. This correction depends on the amplitude of the diameter variation, flow rate wave forms, and the phase difference between them. The calculations based on the geometry and the flow conditions representing a real physiological situation as closely as possible suggest that mean pressure gradient changes with catheter size for any frequency parameter. The results obtained for arbritrary frequency parameter and for small steady streaming Reynolds number, show that the geometry of the wall plays an important role in the dynamics of the flow even for small catheter radius. The interaction of the amplitude of catheter oscillation and the amplitude of the wall movement is first manifested through the induced mean pressure gradient and induced mean velocity. Further, the results are sensitive to the elastic nature of the wall reflected by the phase difference between the diameter variation and the flow rate. Interesting streamline patterns depict distinct boundary layer characteristics both at artery wall and catheter wall. Depending upon the material properties, a thin catheter experiencing small oscillations due to the flow conditions is likely to have a similar influence to a thicker catheter which remains fairly stationary inside the artery. Finally, the effect of catheterization on various physiologically important flow rate characteristics—mean velocity, mean pressure gradient, wall shear stress—is studied for a range of different catheter sizes and frequency parameters.

Dynamics of anisotropy on decaying homogeneous turbulence subjected to system rotation
View Description Hide DescriptionThe dynamics of the anisotropy of the Reynolds stresstensor and its behavior in decaying homogeneous turbulence subjected to system rotation are investigated in this study. Theoretical analysis shows that the anisotropy can be split into two parts: polarization and directional anisotropies. The former can be further separated into a linear part and a nonlinear part. The corresponding linear solution of the polarizationanisotropy is derived in this paper. This solution is found to be equivalent to the linear solution of the anisotropy. While proposing a method to introduce the polarizationanisotropy into an isotropic turbulence, direct numerical simulation (DNS) of the rotating turbulence with or without the initial anisotropy is carried out. The linear solution of the anisotropy agrees very well with the DNS result, showing that the evolution of the polarizationanisotropy is mainly dominated by the linear effect of the system rotation. With an immediate rotation rate, the coupling effect between the system rotation and nonlinear interactions causes an energy transfer from the region near the pole to the region near the equator in wave space. This type of transfer causes an anisotropic distribution of the kinetic energy between the pole and equator, which relates closely to the directional anisotropy and the twodimensionalization. In addition, we find that the presence of the initial polarizationanisotropy does not affect the evolution of the directional anisotropy, while the presence of the initial directional anisotropy greatly influences the evolution of the polarizationanisotropy.

Measurements of conserved scalar filtered density function in a turbulent jet
View Description Hide DescriptionConserved scalar (temperature) filtered density function (FDF) is studied experimentally in the fully developed region of a turbulent jet with Taylormicroscale Reynolds numbers of 293 and 190. We obtain the FDFs using onedimensional box filters of widths Δ ranging from 30 to 248 scalar dissipation scales as well as a twodimensional box filter which consists of three discrete sensors. Taylor’s hypothesis is used to perform streamwise filtering operations. The mean conserved scalar FDF conditioned on the resolvablescale scalar fluctuations and the subgrid scale (SGS) variance (lognormally distributed) is found to be bimodal when is large, indicating that the conditional SGS mixing is nearly binary. For small (<1) the conditional FDF is approximately Gaussian. The kurtosis of the conditional FDF decreases with increasing SGS variance and is independent of the filter widths for large SGS variance. The bimodal distribution can be symmetric or asymmetric depending on the curvature of the resolvablescale scalar. As the SGS variance increases, the conditional scalar differences for separations comparable to the filter widths also change from Gaussian to bimodal distributions. At the same time the conditional SGS scalar changes from approximately isotropic to strongly anisotropic. The results show that the contributions to the bimodal distributions come primarily from the SGS scales comparable to the filter width. It is remarkable that similarities exist between the bimodal conditional FDFs obtained here in a fully developed jet and bimodal probability density functions observed in the early stages of binary scalar mixing. The present study provides a physical basis for the assumed FDF method for conserved scalars used in largeeddy simulation of turbulent combustion.

Stokes and Reynolds number dependence of preferential particle concentration in simulated threedimensional turbulence
View Description Hide DescriptionAn analysis of particle concentrations formed in direct numerical simulations of forced threedimensional (3D) turbulence is described. Up to 48 million particles responding passively to the flow with response times ranging from 0.2 to 6 times the dissipation time of the fluid were evolved together until the concentration field reached a statistical stationary state. The Stokes number (St), defined at the dissipation time scale, was the sole parameter used to characterize the particle–fluid coupling in the regime where particles preferentially concentrate. Concentrations resulting from three simulations equilibrating at the Taylor microscale Reynolds numbers 40, 80, and 140 were studied. We present several new results for concentration measures utilized in previous studies as well as measures introduced in this paper. The measures are compared and contrasted on a finer St grid than presented in previous work and are analyzed as functions of and spatial binning scale. The measures are based on (a) deviations of the concentration PDF (probability density function) from the PDF of a uniform particle field, (b) the correlation dimension for both 3D and twodimensional concentrations, and (c) the relative Stdependent concentrations contained in a localized region of space. Measure (c) is motivated by the observation that the total and Stdependent concentrations are linearly correlated. The concentration measures reveal St dependencies that are insensitive to the Reynolds number of the flow with each measure having its own characteristic shape. The widths and maxima of St functions for measures explicitly constructed on a single spatial binning scale showed a very weak dependence on bin sizes ranging from 2 to 6 times the Kolmogorov length scale. We conclude that the measures studied in this paper reveal a universality that may persist to much higher Reynolds numbers.

Lagrangian approaches for particle collisions: The colliding particle velocity correlation in the multiple particles tracking method and in the stochastic approach
View Description Hide DescriptionTwo different Lagrangian approaches for particle/particle collisions are described. The first model is based on the simultaneous tracking of several particles and suitable treatment is developed on particle pairs to detect collisions on each time step of the particle trajectory realization. The second method is based on a stochastic approach where one single particle is tracked, and successive random processes are applied to generate a fictitious partner of collision. In order to validate both approaches, simulations have been carried out in homogeneous isotropic turbulence and they have been compared with LES data. The particle/particle correlated motion through the surrounding fluid is proved to be a key parameter in a particle/particle collision process.
