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Volume 16, Issue 12, December 2004
 LETTERS


Buoyant mixing of miscible fluids in tilted tubes
View Description Hide DescriptionBuoyant mixing of two fluids in tubes is studied experimentally as a function of the tilt angle from vertical, the density contrast and the common viscosity. At high contrasts and low , longitudinal mixing is macroscopically diffusive, with a diffusivity increasing strongly with and . At lower contrasts and higher , a counterflow of the two fluids with little transverse mixing sets in. The transition occurs at an angle increasing with density contrast and decreasing with . These results are discussed in terms of the dependence of transerse mixing on and an analogy with the Boycott effect.

Use of the particle image velocimetry technique to study the propagation of second sound shock in superfluid helium
View Description Hide DescriptionTransient counterflow velocity fields induced by second sound shock in helium II have been measured using the particle image velocimetry (PIV) technique. The arrival of the shock front, passage of the shock tail, and onset of heat diffusion are clearly visualized as velocity profiles of suspended micronsize particles. Measured particle velocities are compared to calculated normal fluid velocities. Other observations include a critical energy flux for the onset of quantum turbulence and random motion of particles behind the shock. These results indicate that PIV may be a useful tool in the study of second sound and superfluid turbulence.

 ARTICLES


Forced Couette flow simulations using direct simulation Monte Carlo method
View Description Hide DescriptionThreedimensional unsteady flows between two infinite walls are simulated by using the direct simulation Monte Carlo (DSMC) method. An artificial forcing that mimics the centrifugal force in the Taylor problem has been applied to the flow. The sampled behaviors of the resulting flow, including the long time average and the disturbance components, are studied. The computations have been preformed using parallel computer clusters. The results presented are for two different channel heights with various values for the forcing coefficient. The change in the channel height, which also results in changes in the flow Knudsen number and Reynolds number, affects the development of both the mean flows and the disturbances. Spatially coherent mean flow patterns, which are dominated by a hierarchy of harmonic modes, can be identified in the DSMC solutions. Temporally, the evolution of the Fourier amplitudes of the harmonic modes shows that these modes grow in a sequential manner. Disturbances with energy spectra that are significantly higher than the statistical noises are resolved. Their pathline patterns indicate that the disturbance flow fields are three dimensional and spatially coherent. These results suggest that the discrete DSMC approach is capable of capturing unsteady, threedimensional flow disturbances that evolve around a stationary mean flow.

Direct numerical simulation of equilibrium spatially localized structures in pipe flow
View Description Hide DescriptionThe possibility of intermittent turbulent shear flows’ description by means of nonstationary longwave threedimensional Navier–Stokes solutions is shown in principle. The classical problem of viscous incompressible fluid flows in a circular pipe at transitional Reynolds numbers is taken as a model one. By means of direct numerical simulation statistically stationary Navier–Stokes solutions which describe turbulent (at ), intermittent (at ), and laminar flow regimes are obtained. Numerical solutions at describe equilibrium selfsustained flow regimes in which turbulent structures surrounded by almost laminar flow propagate downstream while preserving their length. Thus, theoretical confirmation of the existence of particular transitional flow regimes—equilibrium puffs—is achieved. The spacetime structure of equilibrium puffs is examined with particular emphasis on flow visualization and calculation of propagation velocities. Basic turbulence statistics inside and outside the puff are computed and compared with the existing experimental data.

Grad’s equations and hydrodynamics for weakly inelastic granular flows
View Description Hide DescriptionWe introduce and discuss Grad’s moment equations for dilute granular systems of hard spheres with dissipative collisions and variable coefficient of restitution, under the assumption of weak inelasticity. An important byproduct is that in this way we obtain the hydrodynamic description of a system of nearly elastic particles by a direct procedure from the Boltzmann equation, without resorting to any homogeneous cooling state assumption. Several crucial results of the pertinent literature are recovered in the present physical context in which deviation from elastic scattering is of the same order as the Knudsen number. In particular, the correlation function plays a fundamental role in the decay of the temperature, and the latter is described asymptotically, in space homogeneous conditions, by a corrected Haff’s law.

A numerical study of cryogenic fluid injection and mixing under supercritical conditions
View Description Hide DescriptionThe evolution of a cryogenicfluid jet initially at a subcritical temperature and injected into a supercritical environment, in which both the pressure and temperature exceed the thermodynamic critical state, has been investigated numerically. The model accommodates full conservation laws and realfluid thermodynamics and transport phenomena. All of the thermophysical properties are determined directly from fundamental thermodynamics theories, along with the use of the corresponding state principles. Turbulence closure is achieved using a largeeddysimulation technique. As a specific example, the dynamics of a nitrogen fluid jet is studied systematically over a broad range of ambient pressure. Owing to the differences of fluid states and flow conditions between the jet and surroundings, a string of strong densitygradient regimes is generated around the jet surface and exerts a stabilizing effect on the flow development. The surface layer acts like a solid wall that transfers the turbulent kinetic energy from its axial to radial component. The spatial growth rate of the surface instability wave increases with increasing pressure. The frequency of the most unstable mode exhibits a weak pressure dependence at high pressures. It, however, decreases significantly in the nearcritical regime due to the enhanced effect of density stratification and increased mixinglayer momentum thickness. The result agrees well with the linear stability analysis. The jet dynamics is largely dictated by the local thermodynamic state through its influence on the fluid thermophysical properties. When the fluid temperature transits across the inflection point on an isobaric densitytemperature curve, the resultant rapid property variations may qualitatively modify the jet behavior compared with its counterpart at low pressures. An increase in the ambient pressure results in an earlier transition of the jet into the selfsimilar regime.

Higher modes of the mixed buoyantMarangoni unstable convection originated from a droplet dissolving in a liquid/liquid system with miscibility gap
View Description Hide DescriptionSome phenomena, never observed before, concerning a system composed by two organicliquid bicomponent phases with a miscibility gap, used as transparent surrogates for immiscible metal alloys, are discussed and elucidated in the framework of experimental analyses and numerical simulations. It is shown that a single dissolving droplet at the bottom of a test cell behaves as an intriguing patternforming dynamical system leading to a wealth of different spatiotemporal modes of convection when the imposed temperature gradient is increased. The last part of the analysis is devoted to comparison with other similar phenomena (the flow instability pertaining to the Marangoni convection around bubbles surrounded by a liquid heated from above, and the case of rising buoyant jets), showing analogies and differences. Such a comparison is also used as a means to focus on the intrinsic nature of the present instability.

A Bhatnagar–Gross–Krooktype approach for chemically reacting gas mixtures
View Description Hide DescriptionA recently proposed consistent approach for elastically scattering gas mixtures, of the type introduced by Bhatnagar, Gross, and Krook (BGK), has been extended to deal with a four species gas undergoing reversible bimolecular chemical reactions. The single BGK collision operator introduced for each species must take into account also transfer of mass and of energy of chemical bond. Suitable auxiliary fields have then to be introduced not only for temperatures and velocities, but also for densities, in order to fulfill correctly balance equations for mass, momentum, and total energy. The exact collision equilibrium, satisfying the mass action law of chemistry, is also recovered, and the proper choice of collision frequencies is discussed. Preliminary numerical results for the relaxation problem in spacehomogeneous conditions are reported and briefly commented on.

Study of the longtime dynamics of a viscous vortex sheet with a fully adaptive nonstiff method
View Description Hide DescriptionA numerical investigation of the longtime dynamics of two immiscible twodimensional fluids shearing past one another is presented. The fluids are incompressible and the interface between the bulk phases is subjected to surface tension. The simple case of density and viscosity matched fluids is considered. The twodimensional Navier–Stokes equations are solved numerically with a fully adaptive nonstiff strategy based on the immersed boundary method. Dynamically adaptive mesh refinements are used to cover at all times the separately tracked fluid interface at the finest grid level. In addition, by combining adaptive front tracking, in the form of continuous interface marker equidistribution, with a predictor–corrector discretization an efficient method is introduced to successfully treat the wellknown numerical difficulties associated with surface tension. The resulting numerical method can be used to compute stably and with high resolution the flow for wideranging Weber numbers but this study focuses on the computationally challenging cases for which elongated fingering and interface rollup are observed. To assess the importance of the viscous and vortical effects in the interfacial dynamics the full viscous flow simulations are compared with inviscid counterparts computed with a stateoftheart boundary integral method. In the examined cases of rollup, it is found that in contrast to the inviscid flow in which the interface undergoes a topological reconfiguration, the viscousinterface remarkably escapes selfintersection and rich longtime dynamics due to separation, transport, and diffusion of vorticity is observed. An even more striking motion occurs at an intermediate Weber number for which elongated interpenetrating fingers of fluid develop. In this case, it is found that the Kelvin–Helmholtz instability weakens due to shedding of vorticity and unlike the inviscid counterpart in which there is indefinite finger growth the viscousinterface is pulled back by surface tension. As the interface recedes, thin necks connecting pockets of fluid with the rest of the fingers form. Narrow jets are observed at the necking regions but the vorticity there ultimately appears to be insufficient to drain all the fluid and cause reconnection. However, at another point, two disparate portions of the interface come in close proximity as the interface continues to contract. Large curvature points and an intense concentration of vorticity are observed in this region and then the motion is abruptly terminated by the collapse of the interface.

Conditions for static bubbles in viscoplastic fluids
View Description Hide DescriptionWe consider the slow motion of a gas bubble in a cylindrical column filled with a viscoplastic fluid, modeled here as a Herschel–Bulkley fluid. Because of the yield stress of the fluid, it is possible that a bubble will remain trapped in the fluid indefinitely. We adapt Prager’s two variational principles to our problem. From these variational principles we develop two general stopping conditions, i.e., for a given bubble we can calculate a critical Bingham number above which the bubble will not move. The first condition is derived by bounding the velocity field and the second condition by bounding the stress field. We illustrate these conditions by considering specific bubble shapes, e.g., axisymmetric bubbles. We also develop a condition for bubble motion.

Heating effect on steady and unsteady horizontal laminar flow of air past a circular cylinder
View Description Hide DescriptionExtensive numerical experiments were carried out to study the effect of cylinder heating on the characteristics of the flow and heat transfer in a twodimensional horizontal laminar flow of air past a heated circular cylinder for the range of Reynolds numbers. The fluid was treated as incompressible (density is independent of the pressure) while the variation of the fluid properties with temperature was taken into account. By including the transient density term of the continuity equation, which was neglected in a previous study by Lange, Durst, and Breuer [Int. J. Heat Mass Transfer41, 3409 (1998)], we were able to predict correctly the vortex shedding frequency at various overheat ratios using an incompressible flow solver. The effect of dynamic viscosity and density variations on the flow dynamics occurring with the cylinder heating was analyzed separately. Another emphasis of the work was to investigate the physical mechanism behind the “effective Reynolds number” concept widely applied in engineering correlations. Similarity was discovered for the distribution of the local dimensionless viscous force, the vorticity and the Nusselt number at the cylinder surface and the pressure force in the rear part of the cylinder. Two characteristic temperatures, for the flow dynamics and for the heat transfer, were identified.

Electrokinetic aspects of turbulent drag reduction in surfactant solutions
View Description Hide DescriptionElectrokinetic mechanism of drag reduction of turbulent flow of surfactantsolutions is proposed. The surfactantmicelles with surface charge attract ions of the solvent or counterions, forming an electric double layer. The ions in the layer are “frozen” in a strongly bonded structure, which reduces the local flow around micelle. The local flow is the strain of small turbulent scale fluctuation. Numerical example indicates that electrokinetic mechanism supplies a reasonable explanation of the drag reduction in surfactantsolutions. The proposed model is only the preliminary one, which points on the electrokinetics as an additional mechanism which contributes to the hydrodynamics of surfactantsolutions.

On the simulation of particle trajectories in turbulent flows
View Description Hide DescriptionA different class of stochastic model, comprising of the Langevin equation with a random time scale, for the simulation of fluid velocities along particle trajectories in high Reynoldsnumber turbulent flows is formulated. These velocities are neither purely Lagrangian nor purely Eulerian in character. The distribution of time scales is chosen to ensure that the modeled form of the fluidvelocity structure function and spectral functions are compatible with Kolmogorov similarity scaling and with the scaling analysis of Fung, Hunt, and Perkins [Proc. R. Soc. London, Ser. A 459, 445 (2003)]. It is shown that the model accounts naturally for the crossing trajectory effect and integral time scales are compatible with the much used parameterizations advocated by Csanady [J. Atmos. Sci. 20, 201 (1963)] and by Frenkiel [Adv. Appl. Mech. 3, 61 (1953)]. Model predictions for particle dispersion in grid generated turbulence are shown to be in close accord with the experimental data of Snyder and Lumley [J. Fluid Mech. 48, 41 (1971)].

Scalar decay in twodimensional chaotic advection and Batchelorregime turbulence
View Description Hide DescriptionThis paper considers the decay in time of an advected passive scalar in a largescale flow. The relation between the decay predicted by “Lagrangian stretching theories,” which consider evolution of the scalar field within a small fluid element and then average over many such elements, and that observed at large times in numerical simulations, associated with emergence of a “strange eigenmode” is discussed. Qualitative arguments are supported by results from numerical simulations of scalar evolution in twodimensional spatially periodic, time aperiodic flows, which highlight the differences between the actual behavior and that predicted by the Lagrangian stretching theories. In some cases the decay rate of the scalar variance is different from the theoretical prediction and determined globally and in other cases it apparently matches the theoretical prediction. An updated theory for the wavenumber spectrum of the scalar field and a theory for the probability distribution of the scalar concentration are presented. The wavenumber spectrum and the probability density function both depend on the decay rate of the variance, but can otherwise be calculated from the statistics of the Lagrangian stretching history. In cases where the variance decay rate is not determined by the Lagrangian stretching theory, the wavenumber spectrum for scales that are much smaller than the length scale of the flow but much larger than the diffusive scale is argued to vary as , where is wavenumber, and is a positive number which depends on the decay rate of the variance and on the Lagrangian stretching statistics. The probability density function for the scalar concentration is argued to have algebraic tails, with exponent roughly −3 and with a cutoff that is determined by diffusivity and scales roughly as and these predictions are shown to be in good agreement with numerical simulations.

Semianalytical solutions of laminar fully developed pulsating flows through ducts of arbitrary cross sections
View Description Hide DescriptionA semianalyticalanalysis of fully developed pulsating flows in pipes of noncircular cross section is presented. The flow is assumed to be pressure gradient driven. Details of the analytical treatment of the flow are presented and it is shown that the analysis can be employed for any arbitrary crosssectional shape. Special considerations are given to laminar pipe flows with circular cross sections and results of the present analysis are compared with those of conventional analytical treatments of the flow. Comparison of the velocity distribution, mass flowrate pulsation, ratio of amplitudes of the mass flow rate and the pressure gradient, and the corresponding phase lag obtained by the present treatment show excellent agreement with data in the literature. Similar comparisons are also performed for oscillating flow through ducts of square and rectangular cross sections. Numerical data are introduced to confirm the resultant analysis. Finally, a few test cases are presented for sinusoidally pulsating flows through ducts of rectangular and elliptical cross sections with varying aspect ratios.

Direct numerical simulations of isotropic compressible turbulence: Influence of compressibility on dynamics and structures
View Description Hide DescriptionIn the present paper the statistical properties of compressible isotropic turbulence are analyzed by means of direct numerical simulations. The scope of the work is to evaluate the influence of compressibility on the time evolution of mean turbulenceproperties and to quantify the statistical properties of turbulent structures, their dynamics and similarities with the incompressible case. Simulations have been carried out at various turbulentMach numbers and compressibility ratios by using a conservative hybrid scheme that relies on an optimized weighted essentially nonoscillatory approach for the convective terms and compact differencing for the viscous contributions. In order to identify similarities with incompressible turbulence we have also carried out an analysis in the plane of the second and third invariants of the anisotropic part of the deformation rate tensor. The simulations show that the joint probability density function has a universal structure, as found in incompressible turbulence. The study confirms that the enstrophy obeys a twostage evolution, due to the competing mechanisms of vortex stretching and viscous dissipation; however, at high turbulentMach numbers, compressibility effects, associated to the occurrence of shocklets, become important. Furthermore, the analysis of the controlling mechanisms of vorticity generation has shown that even for compressible turbulence the growth of enstrophy is associated to a preferential alignment of the vorticity with the intermediate eigenvector of the anisotropic part of the strainrate tensor.

Stabilizing viscosity contrast effect on miscible displacement in heterogeneous porous media, using lattice Bhatnagar–Gross–Krook simulations
View Description Hide DescriptionWe analyze the displacement of a viscous fluid by a miscible more viscous one in heterogeneous porous media. We performed lattice Bhatnagar–Gross–Krook simulations, which were previously successfully applied to the study of the dispersion of a passive tracer in a stochastic heterogeneous porous medium. In the present situation, the flow is stable (no viscous fingering) and leads to an overall Gaussian dispersion, the coefficient of which decreases as the viscosity ratio increases. The results are in reasonable agreement with the stochastic approach of Welty and Gelhar.

Interaction of two circular cylinders in inviscid fluid
View Description Hide DescriptionThe unsteady problem of two parallel circular cylinders, moving in an inviscid fluid, is analyzed analytically by exploring a conformal mapping. Exact solutions are obtained for the flow induced and the hydrodynamic forces acting on them, as the two circular cylinders, with any radii and at any locations, expand (contract) and translate arbitrarily at time dependent speeds. As the two bodies are far apart, the solutions of the flow and forces reduce to those for a single circular cylinder moving in an unbounded inviscid flow. The force components along the line of centers are inversely proportional to the distance between them, whereas the force components perpendicular to the line of centers are inversely proportional to the square of the distance between them. Numerical analyses are performed for the two circular cylinders deforming and translating at constant speeds and in various ways, as well as a circular cylinder falling to a wall. It has been noticed that they are attracted to each other, as one of them expands and the other contracts, or as they translate perpendicular to the line of centers; whereas they are repelled from each other, as both of them expand, contract, or as they translate along the line of centers. The force on one of them increases with the size and speed of the other, as well as their proximity.

Stability of twolayer Newtonian plane Couette flow past a deformable solid layer
View Description Hide DescriptionThe linear stability of twolayer plane Couette flow of Newtonian fluids (designated by labels and ) of thicknesses and , and viscosities and past a soft, deformablelinear viscoelasticsolid of thickness , shear modulus, and viscosity is determined using a combination of low wavenumber asymptotic analysis and a numerical method. There are two qualitatively different interfacial modes in this system, viz., the twofluid interfacial mode due to viscosity stratification [“mode 1;” C. S. Yih “Instability due to viscosity stratification,” J. Fluid Mech. 27, 337 (1967)], and the fluid–solid interfacial mode [“mode 2;” Kumaran, Fredrickson, and Pincus, “Flow induced instability of the interface between a fluid and a gel at low Reynolds number,” J. Phys II 4, 893 (1994)]. The respective effects of solid layer deformability and fluid viscosity stratification on mode 1 and mode 2 are analyzed in detail using both asymptotic and numerical methods. Results of our low wavenumber asymptotic analysis show that the deformability of the solid layer has a dramatic effect on the interfacial instability (mode 1) between the two Newtonian fluids: When the more viscous fluid is of smaller thickness (an unstable configuration for the twofluid mode 1 instability), the solid layer could completely stabilize the twofluid interfacial instability, when the nondimensional elasticity parameter increases beyond a critical value. Here is the dimensional velocity of the top moving plate. When the more viscous fluid is of larger thickness compared to the less viscous fluid (a stable configuration in rigid channels), it is shown that the solid layer could destabilize or stabilize the twofluid interfacial mode, depending on the solid layer thickness . Numerical results at finite values of wavenumber reveal that the stabilization of the twofluid interfacial mode predicted by the low wavenumber analysis extends to moderate values of . For high values of , the perturbations are localized near the twofluid interface. Increase in therefore does not have any effect on the high unstable modes, which are stabilized by the presence of nonzero interfacial tension in the twofluid interface. When is further increased, the interfacial mode between fluid and the solid layer becomes unstable. It is demonstrated here that the parameters (representing the shear modulus of the solid),solid layer thickness , and the solid layer viscosity can be chosen such that both the interfacial modes are stabilized at all wavenumbers, for a fixed top plate velocity.

Direct numerical simulation of transitional and turbulent buoyant planar jet flames
View Description Hide DescriptionThe effects of gravity on the physical and compositional structures of transitional and turbulent diffusionflames are studied via analysis of the data generated by direct numerical simulation of a planar jet flame at various gravity conditions. A fully compressible, highorder compact, finitedifference computational scheme is used together with a global kinetics model for chemical reaction. The results of our nonreacting turbulent jet simulations are in good agreement with the available experimental data. The results of our reacting jet simulations are also consistent with previous findings and indicate that in the absence of gravity, combustion damps the flow instability, and hence reduces “turbulence production” and jet growth. However, in the “finitegravity” conditions, combustion generated density variations may promote turbulence and enhance both the mixing and the combustion through buoyancy effects. Our results also indicate that the gravity effects on a transitional/turbulent jet flame is not limited to largescale flame flickering, and there is a significant impact on smallscale turbulence and mixing as well. Furthermore, the analysis of compositional flame structures suggests that the finiterate chemistry effects are more significant in finitegravity conditions than in zerogravity.
