Volume 15, Issue 7, July 2003
 LETTERS


Vortex ring pinchoff in the presence of simultaneously initiated uniform background coflow
View Description Hide DescriptionVortex rings were formed with a pistoncylinder mechanism in the presence of uniform background coflow supplied through a shroud surrounding the cylinder. The jet and coflow were started simultaneously. Ratios of the coflow to jet velocity in the range 0–1 were considered. The formation number as a function of was determined using the procedure of Gharib et al. [J. Fluid Mech. 360, 121 (1998)] and a generalized definition of formation time. The results show a sharp decrease in as increases from 0.5–0.75, suggesting possible performance limitations for pulsedjet propulsion.
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 ARTICLES


Flows past a tiny circular cylinder at high temperature ratios and slight compressible effects on the vortex shedding
View Description Hide DescriptionThe present study contributes to the numerical simulations of weak compressible effects for laminar flows around heated circular cylinders. Our aim in this work is to understand the physics of the flow around a heated cylinder and the reasons for the decrease in the frequency of vortex shedding in the wake behind the cylinder as the ratio of surface temperature to freestream temperature increases for moderate Re numbers. The previous achievements on this subject rely mainly on experimental investigations by using a “representative” or “film temperature” and therefore should be investigated more accurately by the numerical considerations. In that context, numerical experiments are carried out. A technique which was previously tested and validated on a variety of flow problems is utilized in the numerical experiments. The results show that the temperature is not a passive contaminant especially for flows with temperature ratios greater than 1.1. The temperature boundary layer surrounding the wall introduces weak compressible effects on the velocity and on the pressure field especially for Re numbers below 100. As a consequence of this result, the local Ma number effects are shown on a Re–Nu curve for the laminar unsteady vortex shedding regime.

Chaotic advection of finitesize bodies in a cavity flow
View Description Hide DescriptionWe considered advection of neutrally buoyant discs in twodimensional chaotic Stokes flow. The goal of the study is to explore a possibility to enhance laminar mixing in batchflow mixers. Addition of freely moving bodies to periodically driven chaotic flow renders the flowfield nonperiodic [D. F. Zhang and D. A. Zumbrunnen, AIChE J. 42, 3301 (1996)], i.e., the Lagrangianchaos of the bodies motion induces Eulerian chaos of the flow that makes mixing more intensive. The presence of three bodies creates new topological features that do not exist in “pure” fluid. The trajectories of the discs in the augmented phase space tangle and form a braid that leads to socalled topological chaos [P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)]. Simulations were performed using a new variant of the immersed boundaries method that allows the direct numerical simulation of fluid–solid flows on a regular rectangular grid without explicit calculation of the forces that the particles exert on the fluid.

Spreading of droplets under the influence of intermolecular forces
View Description Hide DescriptionThe motion of fluid droplets under the influence of short and long range intermolecular forces is examined using a lubrication model. Surface energies as well as the microscopic contact line structure are identified in the model. A physically constructed precursor film prevents the usual stress singularity associated with a moving contact line. In the quasistatic limit, an analysis of the energy and its dissipation yield an ordinary differential equation for the rate of spreading. Two dimensional and axisymmetric solutions are found and compared to numerical simulations. The motion of the contact line is found to be both a function of the local contact angle and the overall droplet geometry.

Dynamics and mixing in jet/vortex interactions
View Description Hide DescriptionThis study describes large eddy simulations of the interaction between an exhaust jet and a trailing vortex, in the nearfield of an aircraft wake. Two cases are analyzed: in the first one, typical of cruise flight, the jet and the vortex axes are sufficiently well separated to study first the jet dynamics before considering its interaction with the vortex.Dynamics and mixing are controlled both by the jet diffusion and its entrainment around the vortex core. In the second case the jet partially blows in the vortex core, making the flow similar to a Batchelor vortex. The strong perturbations injected into the core cause an instability of the system which is continuously fed by the jet elements wrapping around the core. This leads to a strong decay of angular momentum and diffusion of the core. Global mixing properties, such as plume area and global mixedness evolutions, are analyzed and two applications to environmental problems are finally discussed.

Widnall instabilities in vortex pairs
View Description Hide DescriptionIn this article we analyze the cooperative threedimensional shortwave instabilities developing on concentrated vortex dipoles that have been obtained by means of twodimensional direct numerical simulations. These dipoles are characterized by their aspect ratio where is the radius of the vortices based on the polar moments of vorticity and is the separation between the vortex centroids. In the inviscid case, we show that the selection of the antisymmetric eigenmode smoothly increases with for the amplification rate of the antisymmetric eigenmode is only 1.4% larger than the amplification rate of the symmetric eigenmode. When this difference increases up to 7%. The results of the normal mode analysis may be compared to those of an asymptotic stability analysis of a Lamb–Oseen vortex subjected to a weak straining field, following Moore and Saffman [Proc. R. Soc. London, Ser. A 346, 413 (1975)]. This theory shows that the instability may occur whenever two Kelvin waves exist with the same frequency ω, the same axial wavenumber and with azimuthal wavenumbers and Contrary to the case of a Rankine vortex [Tsai and Widnall, J. Fluid Mech. 73, 721 (1976)], the presence of critical layers in a Lamb–Oseen vortex prevents a large number of possible resonances. For example, resonances between and modes lead to damped modes. The only resonances that occur are related to the stationary bending waves obtained for specific values of the axial wavenumber. All these predictions are found to be in good agreement with the results obtained by the stability analysis of the considered vortex pairs. At last, we present a nonautonomous amplitude equation which takes into account all effects of viscosity, i.e., the viscous damping of the amplification rate of the perturbation but also the increase of the dipole aspect ratio due to the viscousdiffusion of the basic flowfield. The lowReynolds number experiment of Leweke and Williamson [J. Fluid Mech. 360, 85 (1998)] is revisited under the light of these theoretical results. We show that these theoretical results yield predictions for the amplification rate and for the wavenumber that agree with the experimental observations.

Spreading fronts and fluctuations in sedimentation
View Description Hide DescriptionA diffuse interface or “front” at the top of the suspension is investigated experimentally and numerically. The width of the front is found to grow linearly in time, mainly due to a polydispersity of particle size in the very dilute experiments, and due only to fluctuations in particle density in the simulations. Away from the front, the fluctuations in the particle velocities are found not to decay.

The constitutive relation of suspensions of noncolloidal particles in viscous fluids
View Description Hide DescriptionThe motion of noncolloidal particles convected by a nonhomogeneous and nonstationary viscousfluid flow is investigated, assuming that inertial effects can be neglected. It appears that the particle volumetric flux is the sum of a convective part, and a diffusive term, where is the Eulerian mean velocity of a test particle, φ is the particle volume fraction, and is the coefficient of selfdiffusion. The latter measures the local temporal growth of the mean square displacement of a tracer particle from its average position and can be written as where the crossdiffusion tensor is the time integral of the velocity crosscorrelation function. On the other hand, the Eulerian mean velocity is the sum of the coarsegrained average particle velocity, and a drift velocity, This last term, which is identically zero when the suspended particles are passive tracers, indicates that the suspended particles tend to move toward regions with smaller diffusivities. This result demonstrates that the motion of each suspended particle is a random process satisfying a generalized nonlinear Langevin equation, where the fluctuating term is described through the cross diffusivity D.

A new matched asymptotic expansion for the intermediate and far flow behind a finite body
View Description Hide DescriptionAn approximated Navier–Stokes steady solution is here presented for the two dimensional bluff body wake region that is intermediate between the field on the body scale which includes the two symmetric counterrotating eddies, and the ultimate far wake. The nonparallelism of the streamlines in the intermediate wake cannot yet be considered negligible. The R is of the order of the critical value for the onset of the first instability and the limiting behavior for large R is not considered. The solution is obtained by matching an inner solution—a Navier–Stokes expansion in powers of the inverse of the longitudinal coordinate—and an outer solution, which is a Navier–Stokes asymptotic expansion in powers of the inverse of the distance from the body. The matching is built on the criteria that, where the two solutions meet, the longitunal pressure gradients and the vorticities must be equal and the flow toward the inner layer must be equal to the outflow from the external stream. At high orders in the inner expansion solution, the lateral decay turns out to be algebraic. This approximate solution is here examined in relation to the class of asymptotic solutions that, in the past, were obtained by adopting the rapid decay principle, which implies an irrotational outer flow. The theme running through this paper is the necessity of the addition of this criterion to the equations of motion to build a solution that describes the intermediate wake. The present solution has been obtained by relaxing the imposition of the rapid decay principle. It can be concluded that, at Reynolds numbers as low as the first critical value and where the nonparallelism of the streamlines is not yet negligible, the division of the field into two basic parts—an inner vortical boundary layer flow and an outer potential flow—is spontaneously shown up to the second order of accuracy: at higher orders in the expansion solution the vorticity is first convected and then diffused in the outer field. If exploited to represent the basic flow of bluff body wakes, the analytical simplicity of this asymptotic expansion could be useful for the nonparallel analysis of the instability of twodimensional wakes.

The modeling of turbulent reactive flows based on multiple mapping conditioning
View Description Hide DescriptionA new modeling approach—multiple mapping conditioning (MMC)—is introduced to treat mixing and reaction in turbulent flows. The model combines the advantages of the probability density function and the conditional moment closure methods and is based on a certain generalization of the mapping closure concept. An equivalent stochastic formulation of the MMC model is given. The validity of the closuring hypothesis of the model is demonstrated by a comparison with direct numerical simulation results for the threestream mixing problem.

Variable scale filtered Navier–Stokes equations: A new procedure to deal with the associated commutation error
View Description Hide DescriptionA simple procedure to approximate the noncommutation terms that arise whenever it is necessary to use a variable scale filtering of the motion equations and to compensate directly the flowsolutions from the commutation error is here presented. Such a situation usually concerns large eddy simulation of nonhomogeneous turbulent flows. The noncommutation of the average and differentiation operations leads to nonhomogeneous terms in the motion equations, that act as source terms of intensity which depend on the gradient of the filter scale δ and which, if neglected, induce a systematic error throughout the solution. Here the different noncommutation terms of the motion equation are determined as functions of the δ gradient and of the δ derivatives of the filtered variables. It is shown here that approximated noncommutation terms of the fourth order of accuracy, with respect to the filtering scale, can be obtained using series expansions in the filter width of approximations based on finite differences and introducing successive levels of filtering, which makes it suitable to use in conjunction with dynamic or mixed subgrid models. The procedure operates in a way which is independent of the type of filter in use and without increasing the differential order of the equations, which, on the contrary, would require additional boundary conditions. It is not necessary to introduce a mapping function of the nonuniform grid in the physical domain into a uniform grid in an infinite domain. A priori tests on the turbulent channel flow 180 and 590) highlight the approximation capability of the present procedure. A numerical example is given, which draws attention to the nonlocal effects on the solution due to the lack of noncommutation terms in the motion equation and to the efficiency of the present procedure in reducing the commutation error on the solution.

Alternative statisticalmechanical descriptions of decaying twodimensional turbulence in terms of “patches” and “points”
View Description Hide DescriptionNumerical and analytical studies of decaying, twodimensional Navier–Stokes (NS) turbulence at high Reynolds numbers are reported. The effort is to determine computable distinctions between two different formulations of maximum entropy predictions for the decayed, latetime state. Though these predictions might be thought to apply only to the ideal Euler equations, there have been surprising and imperfectly understood correspondences between the longtime computations of decaying states of NS flows and the results of the maximum entropy analyses. Both formulations define an entropy using a somewhat ad hoc discretization of vorticity into “particles.” Pointparticle statistical methods are used to define an entropy, before passing to a meanfield approximation. In one case, the particles are deltafunction parallel “line” vortices (“points,” in two dimensions), and in the other, they are finitearea, mutually exclusive convected “patches” of vorticity which only in the limit of zero area become “points.” The former are assumed to obey Boltzmann statistics, and the latter, LyndenBell statistics. Clearly, there is no unique way to reach a continuous, differentiable vorticity distribution as a meanfield limit by either method. The simplest method of taking equalstrength points and equalstrength, equalarea patches is chosen here, no reason being apparent for attempting anything more complicated. In both cases, a nonlinear partial differential equation results for the stream function of the “most probable,” or maximum entropy, state, compatible with conserved total energy and positive and negative velocity fluxes. These amount to generalizations of the “sinhPoisson” equation which has become familiar from the “point” formulation. They have many solutions and only one of them maximizes the entropy from which it was derived, globally. These predictions can differ for the point and patch discretizations. The intent here is to use timedependent, spectralmethod direct numerical simulation of the Navier–Stokes equation to see if initial conditions which should relax toward the different latetime states under the two formulations actually do so.

Interaction of two flapping filaments in a flowing soap film
View Description Hide DescriptionUsing the immersed boundary method, we have simulated the motion of two flexible filaments in a flowing soap film, which was experimentally studied by Zhang et al. [Nature (London) 408, 835 (2000)]. We found numerically two distinct modes of sustained oscillation of the two filaments: parallel flapping and mirrorimage clapping, depending on the separation distance between the two filaments. Our simulation results agree with that of the experiment at least qualitatively, even though the Reynolds number of the flowing soap film in our simulation is about two orders of magnitude lower than that of the experiment.

On the mechanism of optimal disturbances: The role of a pair of nearly parallel modes
View Description Hide DescriptionIn the present work the role of a pair of nearly parallel least stable modes (having opposite phases and almost identical amplitude distributions), as a key element of optimal transient growth in shear flows, is explored. The general character of this mechanism is demonstrated by four examples. The first two examples are the temporal and spatial growth of optimal disturbances in circular pipe flow. The time and distance, at which the maximum energy amplification of an initial disturbance is achieved, are well predicted analytically by considering only the pair of least stable modes. Furthermore, the dependence of the maximum energy amplification on the Reynolds number matches previous numerical results based on the analysis of many modes. In the temporal case the predicted amplification factor agrees well with these numerical results. The other two examples are concerned with a twodimensional potential shear layer over a compliant surface, and with a twodimensional walljet. In these examples, a similar growth mechanism takes place near a point where two kinds of twodimensional modes bifurcate. In the potential shear layer, the maximum optimal growth achieved by two nearly parallel modes is solved analytically for temporal disturbances, whereas in the walljet case the distance, at which the maximum amplification is achieved, is well predicted for spatial disturbances. Finally, it appears that the transient growth mechanism based on the interference between two nearly parallel modes is a general case which includes the direct resonance mechanism as a limit for short times.

Transient displacement of a Newtonian fluid by air in straight or suddenly constricted tubes
View Description Hide DescriptionWe study the displacement of a viscous fluid by highly pressurized air in a straight or a suddenly constricted cylindrical tube of finite length. In contrast to previous efforts, the transient situation is examined. A long, narrower than the tube and roundended bubble is created during the process. This is sometimes called “fingering instability” and is often encountered in several applications, but we will focus on process parameters that are relevant to the gasassisted injection molding. For our numerical simulations we have combined the mixed finite element method with an appropriate system of elliptic partial differential equations and boundary conditions, capable of generating a boundaryfitted finite element mesh. The bubble front and the thickness of the deposited film on the tube wall are affected by the properties of the fluid being displaced and the flow conditions. Specifically, in straight tubes, the bubble keeps accelerating due to the decreasing fluid mass ahead of it. Increasing the Reynolds number decreases the film thickness and makes the bubble front steeper. When inertia becomes significant a tipsplitting instability arises. For sufficiently low Reynolds numbers, but still large applied pressures, a steady bubble shape is attained even in a straight tube and the fraction of the liquid deposited on the wall of the tube reaches the asymptotic value of 0.60, as observed by Taylor [J. Fluid Mech. 10, 161 (1961)] and Cox [J. Fluid Mech. 14, 81 (1962)]. In a constricted tube, the bubble temporarily attains a nearly constant velocity. When its front approaches the tube constriction it becomes pointed, due to the extensional flow that prevails there, but it reassumes its fingerlike profile, after it goes through the constriction.

Prediction of electrokinetic and pressure flow in a microchannel Tjunction
View Description Hide DescriptionIt is generally accepted that in simple microchannel flows the electrical double layer at walls is thin enough for “slip velocity” boundary conditions to be used with good approximation. Recent theoretical work by one of the authors has considered the limits of this approach in cases characterized by nonuniform liquid properties and complex channel geometries. In that work, the chemically reacting flow in an arbitrary channel geometry produced by electric potential and pressure differences with heat transfer and electrophoresis is considered. The present work undertakes a broad test of the model approach in a complex channel network geometry, in the case of nonreacting uniformproperty liquid. Velocity is measured by particle tracking with correction for electrophoreticmotion.Measured and predicted velocities in a threedimensional experimental Tjunction within a network of five channel segments are compared for three cases of steady flow including electrically driven flow, pressuredriven flow, and mixed pressure and electrical flow. All conditions of the experiment required to determine the flow uniquely have been measured. The computational methodology used combines local threedimensional representation with overall circuit analysis of the channel network. Comparisons are found to be within the 5% experimental scatter of the velocity measurement method used. The work emphasizes the care required in fabrication, measurement,flow control, and numerical solution if prediction of actual fabricated devices is to be achieved.

Impact of drops of polymer solutions on small targets
View Description Hide DescriptionThe collision of drops of polymer solutions with small targets was studied experimentally. The tested liquids were aqueous solutions of polyethylene oxide (MW=4 000 000) at concentrations of 10, 100, 1000 wt ppm. The drop impact velocity was about 3.5 m/s, and the drop diameters were in the range of 2.6–3.8 mm. The target was a stainless steel disk of 3.9 mm diameter. The collision was monitored by means of highspeed photography technique. As in the case of pure water, a circular liquid lamella was formed, and then it retracted with formation of outwardsdirected secondary jets. There was no significant difference between the values of the maximum diameter and the retraction velocity of the lamella in the cases of water and polymeric liquids. On the contrary, the polymeric additives drastically changed the character of the lamellaretraction. The secondary jets were transformed into thinning filaments submitted to elastic stresses with an attached droplet. Then, depending on the polymer concentration, the filaments ruptured and the attached droplets escaped, or the liquid filaments pulled the attached droplets back and the whole liquid returned to the target. A splashing threshold has been derived for polymeric liquids based on the liquid relaxation time and the impact conditions.

Magneticfield effects on secondmode instability of a weakly ionized Mach 4.5 boundary layer
View Description Hide DescriptionThis paper investigates, by numerical simulation, the effects of an imposed magnetic field on a weakly ionized Mach 4.5 boundary layer. The main emphasis of the study is on magnetohydrodynamic (MHD) effects on the second mode instability in supersonic boundary layer. The imposed magnetic fields are generated by placing twodimensional magnetic dipoles below the flat plate surface. The gas is assumed to have a constant electrical conductivity of 100 mho/m. The magnetic Reynolds number of the flow is small so that the induced magnetic field in the flow is neglected. The governing equations of the MHD flow, which are the Navier–Stokes equations with the applied magnetic force terms, are computed by a fifthorder shockfitting numerical scheme. A series of cases with different imposed magnetic fields have been investigated on the influences of imposed magnetic field on both the mean flow and on the second mode stability. It is found that the imposed magnetic fields significantly retard the streamwise velocity and reduce the local skin friction in the mean flow. For the case of a strong imposed magnetic field, a local separation region is generated in the mean flow with a strong adverse pressure gradient. Meanwhile, the second mode wave disturbances are found to be stabilized by the imposed magnetic fields, even for the case with strong adverse pressure gradient and a local separated flow region. This strong overall stabilization of the second mode wave is believed to be caused by the alteration of the steady base flow by the magnetic field. The results presented in this paper are the first concrete results on the interaction of second instability mode with magnetic field in a supersonic boundary layer.

Breakdown of a magnetohydrodynamic flow structure
View Description Hide DescriptionThe breakdown of a swirling blob of fluid subject to a magnetic field parallel to its axis is investigated experimentally in the present work. A flow structure in an electrically conducting fluid always tends to elongate in the direction of a strong, external magnetic field. However, if the magnetic field is sufficiently weak, the poloidal recirculation that is created at the expense of the swirl flow is not damped fast enough. While a vortex may diffuse weakly along the magneticfield lines, it begins to propagate radially outward in the form of a ring. The transition from a purely diffusive regime to a centrifugally unstable regime has been observed in the present study.

Dynamics of turbulence strongly influenced by buoyancy
View Description Hide DescriptionThe dynamics of quasihorizontal motions in a stably stratified fluid have been simulated for Froude numbers of order 1, so that the flows are strongly affected by the stable density stratification, and for a range of Reynolds numbers. It is found that the horizontal scales of the motion grow continuously in time. The vertical scales decrease and the vertical shearing increases with time, maintaining the Richardson number of order 1, as suggested by Lilly [J. Atmos. Sci. 40, 749 (1983)] and Babin et al. [Theor. Comput. Fluid Dyn. 9, 223 (1997)]. Smallscale instabilities and turbulentlike motions are observed to occur in the high shearing regions, while the largerscale motions appear to evolve somewhat independently of the Reynolds number. The results suggest that the largerscale, quasihorizontal motions would be a continuous source of smallerscale turbulence until the local Reynolds number drops below a critical value, which is estimated. Finally, a Froude number based upon a vertical differential scale and used in previous scaling arguments and theories is estimated in terms of other parameters.
