Volume 15, Issue 1, January 2003
 LETTERS


On the application of nonextensive statistics to Lagrangian turbulence
View Description Hide DescriptionA secondorder Lagrangianstochastic model formulated in terms of the mean dissipation rate and satisfying the wellmixed condition for a Tsallis distribution of Lagrangian accelerations is shown to be incompatible with Kolmogorov’s similarity theory. This difficulty does not arise when, following the approach advocated by Beck [Phys. Rev. Lett. 87, 180601 (2001)], the Tsallis distribution is recovered from a Gaussian model through the employment of a distribution of dissipation rates. The effects caused by ignoring fluctuations in dissipation along trajectories are evaluated in numerical simulations in which Lagrangian accelerations and dissipation histories evolve jointly as a Markovian process.

The effect of microbubbles on developed turbulence
View Description Hide DescriptionThe motion and the action of microbubbles in homogeneous and isotropic turbulence are investigated through (threedimensional) direct numerical simulations of the Navier–Stokes equations and applying the Lagrangian approach to track the bubble trajectories. The forces acting on the bubbles are added mass, drag, lift, and gravity. The bubbles are found to accumulate in vortices, preferably on the side with downward velocity. This effect, mainly caused by the lift force, leads to a reduced average bubble rise velocity. Once the reaction of the bubbles on the carrier flow is embodied using a pointforce approximation, an attenuation of the turbulence on large scales and an extra forcing on small scales is found.

Nonvertical ascension or fall of a free sphere in a Newtonian fluid
View Description Hide DescriptionIt is shown that the system represented by a free sphere ascending or falling in a Newtonian fluid under the action of gravity buoyancy undergoes a regular, symmetry breaking bifurcation making the trajectory deviate from the vertical direction. The instability threshold expressed in terms of the asymptotic Reynolds number lies below that of a fixed sphere wake. The instability is shown to saturate and reach a fixed point corresponding to a straight oblique ascension (fall).

Regularization modeling for largeeddy simulation
View Description Hide DescriptionA new modeling approach for largeeddy simulation(LES) is obtained by combining a “regularization principle” with an explicit filter and its inversion. This regularization approach allows a systematic derivation of the implied subgrid model, which resolves the closure problem. The central role of the filter in LES is fully restored, i.e., both the interpretation of LES predictions in terms of direct simulation results as well as the corresponding subgrid closure are specified by the filter. The regularization approach is illustrated with “Leraysmoothing” of the nonlinear convective terms. In turbulent mixing the new, implied subgrid model performs favorably compared to the dynamic eddyviscosity procedure. The model is robust at arbitrarily high Reynolds numbers and correctly predicts selfsimilar turbulent flow development.
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 ARTICLES


Granular flow down a rough inclined plane: Transition between thin and thick piles
View Description Hide DescriptionThe rheology of granular particles in an inclined plane geometry is studied using three dimensional molecular dynamics simulations. The flow–noflow boundary is determined for piles of varying heights over a range of inclination angles θ. Three angles determine the phase diagram: the angle of repose, is the angle at which a flowing system comes to rest; the maximum angle of stability, is the inclination required to induce flow in a static system; and is the maximum angle for which stable, steady state flow is observed. In the stable flow region three flow regimes can be distinguished that depend on how close θ is to (i) Bagnold rheology, characterized by a mean particle velocity in the direction of flow that scales as for a pile of height h, (ii) The slow flow regime, characterized by a linear velocity profile with depth, and (iii) Avalancheflow characterized by a slow underlying creepmotion combined with occasional free surface events and large energy fluctuations. We also probe the physics of the initiation and cessation of flow. The results are compared to several recent experimental studies on chute flows and suggest that differences between measuredvelocity profiles in these experiments may simply be a consequence of how far the system is from jamming.

Microchannel flow of a macromolecular suspension
View Description Hide DescriptionIn the delivery of DNA molecules by microfluidic devices, the channel width is very often in the same order as the size of the DNA molecules and the applicability of continuum mechanics at this level may be questioned. In this paper we use finitely extendable nonlinear elastic (FENE) chains to model the DNA molecules and employ the dissipative particle dynamics (DPD) method to simulate their behavior in the flow. Simple DPD fluids are found to behave just like a Newtonian fluid in Poiseuille flow. However, the velocity profiles of FENE chain suspensions can be fitted with powerlaw curves, especially for dilute suspensions. Some results on the conformation and migration of FENE chains are also reported.

Analytical solutions for distributed multipolar vortex equilibria on a sphere
View Description Hide DescriptionAnalytical solutions of the steady Euler equations corresponding to stationary multipolar vortices on a sphere are derived. The solutions represent localized regions of distributed vorticity consisting of uniform vortex patches with a finite set of superposed point vortices. The mathematical method combines stereographic projection with conformal mapping theory to generalize a class of exact solutions for planar multipolar vortices developed by Crowdy [Phys. Fluids 11, 2556 (1999)] to the physically more important scenario of multipolar vortices on a spherical surface. The solutions are believed to be the first examples of analytical solutions of the Euler equations on a sphere involving patches of distributed vorticity with nontrivial shape.

Analysis of the smallscale structure of turbulence on smooth and rough walls
View Description Hide DescriptionEnergy spectra and structure functions of the streamwise and wallnormal turbulent velocity components in an open channel flow are experimentally investigated as a function of the distance from the wall. Both smooth and roughwall conditions are considered, with special attention to the zone close to the wall. In the core region, the smallscale turbulent flow field is always characterized by a high level of isotropy, while a strong anisotropy is found at small scales in the nearwall region. In the smoothwall case, the extent of the scaling range increases as the wall is approached, but with exponents which are different from the classical ones. In the roughwall case, the roughness strongly interacts with turbulence, destroying the scaling regions at small scales through the imposition of its characteristic scales. A lower level of intermittency and anisotropy is also observed at the small scales for roughwall conditions. Energy spectra and structure functions suggest a connection of these behaviors with the turbulent energy directly injected into the flow by the roughness elements. The integral structure functions show that roughness effects exceed the sole modification of local shear and reveal the direct impact of wakes and vorticesgenerated by the roughness in the nearwall region.

Asymptotic analysis of a surfaceinterfacial wave interaction
View Description Hide DescriptionThe threedimensional interaction of a surface wave with two oblique interfacial waves in a horizontally infinite twolayer fluid is analyzed asymptotically. The nondimensional density difference is taken as a perturbation parameter and simple expressions for the growth rates and kinematic properties of the waves are obtained. The results show that the interfacial wavelengths are an order smaller than the surface wavelength. Also, to the leadingorder approximation, the interfacial waves have a frequency half that of the surface wave, and their directions differ by 180° in the horizontal plane. The interaction coefficients are found to be equal at the leading order. The asymptotic solution is compared with the exact solution, and an excellent agreement is obtained for the range of applicability of the asymptotic theory. The analysis is extended to interactions in a medium with sidewalls. A previous laboratory flume study is addressed, and the asymptotic theory is used to explain the experimental observations.

Effect of horizontal divergence on the geostrophic turbulence on a betaplane: Suppression of the Rhines effect
View Description Hide DescriptionAn investigation is made on the effect of strong stratification, or horizontal divergence, on almost freely decaying geostrophic turbulence on a βplane. In a model ocean, the surface layer is assumed to be active above the quiet deep layer, so that barotropification is prohibited a priori to purify the effect of horizontal divergence. Spectral evolution is accelerated numerically by an adjustment to keep kinetic energy constant against the retarding effect of horizontal divergence. First, numerical experiment is carried out for small or moderate horizontal divergence as control runs for comparison; where U is the characteristic velocity of turbulence and F the squared inverse of the radius of deformation. As has been reported repeatedly, the βeffect induces a highly anisotropic field characterized by a band of zonal currents. It is confirmed also that kinetic energy has a onedimensional spectrum approximately proportional to at high wave numbers k. Then, horizontal divergence is enlarged enough so that for which geostrophic turbulence turns out to behave just as on an f plane: (1) the field becomes isotropic with no significant zonal currents; (2) the inverse cascade of energy is not hindered by the β effect though it takes a longer time for turbulence to transfer energy to longer scales; and (3) the spectrum of kinetic energy (not total energy) is proportional to at high wave numbers. An argument based on the physics of long baroclinic Rossby waves is presented to explain why strong horizontal divergence suppresses the β effect. Furthermore a transform of variables leads to a modified governing equation, which clearly shows that the β effect should disappear for large horizontal divergence.

Apparent diffusion due to topographic microstructure in shallow waters
View Description Hide DescriptionWave propagation in disordered (random) media is the underlying theme. We study the effective behavior of long coastal waves that travel over rough topographies. The topographies analyzed contain a smooth slowly varying profile together with disordered smallscale features. The mathematical model is a conservation law with random coefficients. The main asymptotic (stochastic theory) result is that the medium fluctuations cause the propagating pulse to broaden as it travels. The so called apparent diffusion (or pulse shaping) depends only on the traveling distance and the statistics of the random medium fluctuations. Thus, the broadening can be described in a deterministic way independently of the particular medium realization. This is confirmed numerically. Numerical experiments also show that the theory describing pulse shaping is very robust. Nonlinear shallow water simulations show that small amplitude pulse shaping is not affected by higher order terms. The robustness of the theory is observed numerically for a wide parameter regime. We vary both the microscale fluctuation level as well as the horizontal length scales of the topography. The numerical experiments produce very good results regarding the prediction for the wavefront attenuation.

On smoothed turbulent shear flows: Bounds, numerics and stressreducing additives
View Description Hide DescriptionWe derive a rigorous upper bound on the energy dissipation rate in a smoothed version of plane Poiseuille flow in which a minimum length scale parallel to the walls is imposed. Due to the smaller number of degrees of freedom, much higher Reynolds numbers can be reached in numerical simulations of this system. These indicate that the turbulent energy dissipation rate achieves the upper bound scaling except for an extra logarithmic factor of Assuming that this discrepancy in scaling between the best known upper bound and true dissipation carries over to the full plane Poiseuille problem, this result presents further evidence that the viscous dissipation rate there is (in inertial units) and hence vanishes as the Reynolds number The mean flow profiles which emerge from this smoothed system are strikingly similar to those which arise by adding stressreducing additives to a wallbounded shear flow. Both show familiar log regions with essentially the same gradient as the undisturbed (polymer free or no minimum length scale) system but which are now displaced vertically in the traditional diagram.

Derivative moments in turbulent shear flows
View Description Hide DescriptionWe propose a generalized perspective on the behavior of highorder derivative moments in turbulentshear flows by taking account of the roles of smallscale intermittency and mean shear, in addition to the Reynolds number. Two asymptotic regimes are discussed with respect to shear effects. By these means, some existing disagreements on the Reynolds number dependence of derivative moments can be explained. That oddorder moments of transverse velocity derivatives tend not to vanish as expected from elementary scaling considerations does not necessarily imply that smallscale anisotropy persists at all Reynolds numbers.

Predicting continuum breakdown in hypersonic viscous flows
View Description Hide DescriptionThis paper presents a study of the breakdown of the Navier–Stokes equations in hypersonic viscousflows over a sharp cone tip and a hollow cylinder/flare geometry. Investigations are performed through detailed comparisons of the numerical results obtained with continuum and particle techniques. The objective of the study is to predict conditions under which the continuum approach may be expected to fail. A modified breakdown parameter is proposed that can predict the failure of the continuum approach accurately for the simple cone flow and fairly well for the more complex cylinder/flare flow. The study of continuum breakdown is the first step toward development of a hybrid numerical code.

Laminar boundary layer response to rotation of a finite diameter surface patch
View Description Hide DescriptionThe responses of the flat plate laminar boundary layer to perturbations generated by rotating a finite patch of the bounding surface are explored experimentally. The size of the surface patch was of the same order as the boundary layer thickness. The displacement thickness Reynolds number range of the boundary layers explored was 72–527. The rotation rates of the surface patch ranged from 2.14 to 62.8 s^{−1}. Qualitative flow visualizations and quantitative molecular tagging velocimetrymeasurements revealed that rotation of a finite surface patch generates an asymmetric looplike vortex. Significant features of this vortex include that, (i) the sign of the vorticity in the vortex head is opposite that of the boundary layervorticity regardless of the sign of the input rotation, (ii) one leg of the vortex exhibits motion akin to solid body rotation while the other leg is best characterized as a spanwise shear layer, (iii) the vortex leg exhibiting near solid body rotation lifts more rapidly from the surface than the leg more like a shear layer, and (iv) the vortex leg exhibiting near solid body rotation always occurs on the side of the surface patch experiencing downstream motion. These asymmetries switch sides depending on the sign of the input rotation. The present results are interpreted and discussed relative to analytical solutions for infinite geometries. By way of analogy, plausible connections are drawn between the present results and the influences of wall normal vortices in turbulent boundary layer flows.

Detailed features of onedimensional detonations
View Description Hide DescriptionThe oscillation mechanism and reignition process of onedimensional unsteady detonations are numerically studied using a onestep chemical reactionmodel governed by Arrhenius kinetics. A series of simulations, without perturbations from the outflow boundary to the detonation front, are carried out while the degree of overdrive,f, is varied between 1.10 and 1.74 where D is detonation velocity). Shock pressure histories and x–t diagrams are utilized in order to attain precise understanding of the onedimensional unsteady detonations. At higher degrees of overdrive, shock pressure histories agree with those of previous studies. The oscillation mechanism is the same as that of the largedisturbance regime of unsteady shockinduced combustion around a projectile. At lower degrees of overdrive, grid resolution affects the eventual results, because half reaction time in the shock pressure exhibits considerable variation. Four typical kinds of oscillation pattern are discussed and are explained by their x–t diagrams and shock pressure histories. Each oscillation mechanism is essentially the same as that of the largedisturbance regime. The reignition process in the failed regime was numerically investigated at The reignition points tend to converge on a specified point in study of grid refinement, although the oscillation of the shock pressure histories becomes chaotic, suggesting the existence of a unique solution for reignition. All the simulation results for show the failed regime after initial disturbance at the early stage. The failed regime is compared with the solution of the Riemann problem, and analysis consisting of a Rayleigh line for weak leading shock and a partially burnt Hugoniot curve is adopted. Analysis suggests the concept of partial chemical heat release, indicating the possibility of discontinuous change in conditions, and, indeed, simulation indicates occurrence of explosion. The explosion time derived from the analysis agrees well with the results of simulation.

Control of vortex breakdown by a coaxial wire
View Description Hide DescriptionA very small diameter wire is tethered from the apex of a delta wing and nominally aligned with the centerline of the leadingedge vortex. The wire can alter both the onset and structure of vortex breakdown. A technique of highimagedensity particle image velocimetry allows acquisition of patterns of instantaneous and averaged vorticity and velocity, which reveal the relationship between: Advancement of vortex breakdown towards the apex of the wing; and corresponding changes of patterns of vorticity and velocity contours. The diameter of the wire is one percent of the core diameter of the prebreakdown vortex. It is possible to alter the onset of vortex breakdown by as much as approximately one chord length of the wing. A critical parameter is the length of the wire, which is normalized by: The chord of the wing; or the distance to onset of vortex breakdown in absence of the wire. Once a critical length of the wire is attained, further increases in length have no effect on the onset of breakdown. This effect is interpreted in terms of abrupt changes in patterns of vorticity and streamwise gradients of velocity along the central region of the vortex. It is possible to attain a switch in sign of azimuthal vorticity and a wakelike region of the vortex, in absence of a stagnation point.

Linear stability analysis and numerical calculations of the liddriven flow in a toroidally shaped cavity
View Description Hide DescriptionThe liddriven incompressible flow in a toroidally shaped cavity of square crosssection and radius of curvature is studied. For discrete values of the curvature ratio the Reynolds number at which the flow becomes threedimensional is determined by means of linear stability analysis. Present results show that both and the wavelength of the critical mode depend strongly on curvature. For small curvatures steady modes of short wavelength render the flow threedimensional. For the dominant modes are unsteady and of longer wavelength For larger curvatures the first active modes are stationary and of even longer wavelength Numerical solutions of the Navier–Stokes equations for slightly to moderately supercritical conditions show a good agreement with the structure of the critical mode predicted by linear analyses. The flow is unstable for all curvatures studied due to instability of centrifugal type. The slightly supercritical flow presents pairs of Görtlerlike vortices and the perturbation flow draws its energy from regions that are located along the outermost streamlines of the primary vortex. For small curvatures, these regions are located near the upstream wall where the Görtler vortices are stronger and the perturbation flow is similar to that in a straight cavity. For high curvatures, the region of maximum production of kinetic energy occurs instead near the downstream wall, but the pairs of counterrotating Görtlerlike vortices still appear near the upstream wall. The slightly supercritical (global) flow is characterized by a complex structure that is highly threedimensional throughout most of the cavity.

Singularity formation in threedimensional vortex sheets
View Description Hide DescriptionWe study singularity formation of threedimensional (3D) vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integralequation governing the 3D vortex sheet. This leading order equation captures the most singular contributions of the integral equation. By introducing an appropriate change of variables, we show that the leading order vortex sheet equation degenerates to a twodimensional vortex sheet equation in the direction of the tangential velocity jump. This change of variables is guided by a careful analysis based on properties of certain singular integral operators, and is crucial in identifying the leading order singular behavior. Our result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. We also show that the singularity type of the threedimensional problem is similar to that of the twodimensional problem. Moreover, we introduce a model equation for 3D vortex sheets. This model equation captures the leading order singularity structure of the full 3D vortex sheet equation, and it can be computed efficiently using fast Fourier transform. This enables us to perform wellresolved calculations to study the generic type of 3D vortex sheet singularities. We will provide detailed numerical results to support the analytic prediction, and to reveal the generic form of the vortex sheet singularity.

Air bubble entrapment under an impacting droplet
View Description Hide DescriptionWe simulated impact of water, nheptane, and molten nickeldroplets on a solid surface. A numerical code was developed to model the motion of both the liquid in the droplet and the surrounding air. The model used a volumeoffluid method to track the droplet surface and assumed that only one flow field governed the motion of all the fluids present. Predicted droplet shapes during impact agreed well with photographs. When a droplet approached another surface, air in the gap between them was forced out. Increased air pressure below a droplet created a depression in its surface in which air was trapped. The magnitude of pressure rise could be predicted using a simple analysis of fluid between two solid planes moving closer together. The air bubble formed at the solid–liquid interface remained attached to the solid surface in a waterdroplet. In an nheptane droplet the bubble moved away from the surface and broke into two or three smaller bubbles before escaping through the droplet surface. This difference in behavior was attributed to the contact angle of water being much larger than that of nheptane.
