Volume 17, Issue 2, February 2005
 LETTERS


Particle trapping in threedimensional fully developed turbulence
View Description Hide DescriptionThe statistical properties of fluid particles transported by a threedimensional fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. Single trajectory statistics are investigated in a time range spanning more than three decades, from less than a tenth of the Kolmogorov time scale, , up to one largeeddy turnover time. Our analysis reveals the existence of relatively rare trapping events in vortex filaments which give rise to enhanced intermittency on Lagrangian observables up to . Lagrangian velocity structure function attain scaling properties in agreement with the multifractal prediction only for time lags larger than those affected by trapping.

The shearlayer instability of a circular cylinder wake
View Description Hide DescriptionA reinterpretation is made of previously published data concerning the frequency of the instability waves in the separated shear layer from a circular cylinder for Reynolds numbers in the range . An accurate fit to the observed variation can be achieved using a piecewise fit based on theoretical and empirical arguments. A logical conclusion is that the ratio of the frequency of the instability waves to the Kármán vortex shedding frequency is indeed determined by the boundarylayer properties at separation, as suggested by Bloor.

Mixing induced by Rayleigh–Taylor instability in a vortex
View Description Hide DescriptionThe direct numerical simulation (DNS) of a twodimensional Lamb–Oseen vortex with a heavy internal core has been performed. Linear stability theory predicts the existence of Rayleigh–Taylor (RT) instabilities due to the destabilizing effect of the centrifugal force on the radial flow nonhomogeneities. The DNS first exhibits wavy azimuthal perturbations which are nonlinearly distorted into bubblelike patterns, characteristic of the standard development of the RT instabilities, i.e., instabilities obtained in a planar nonhomogeneous flow in the presence of gravity. Nevertheless, important differences may be observed in the late stage development of the instability: contrary to the standard case, the bubbles are then stretched in the azimuthal direction leading to a strong radial filamentation of the flow.
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 ARTICLES

 Interfacial Flows

Effects of static contact angles on inviscid gravitycapillary waves
View Description Hide DescriptionThe effects of a static contact angle on the natural frequencies of inviscid gravitycapillary waves depend on the contactline conditions. The cases of freeend and pinnedend edge conditions are analyzed in the limit of large Bond number . The ratio (where and are the natural frequencies without and with meniscus, respectively) is proportional to or in the case of fixed contact angle or fixed contact line, respectively, and then the effect of the meniscus is greater for pinnedend edges than for freeend ones. When the contact angle is fixed (and for circular cylinders of aspect ratio not small) the natural frequency increases with increasing contact angle. When the contact line is fixed the natural frequency is smaller than in both cases of hydrophilic and hydrophobicsurfaces. It is shown that the reduction in frequency due to the meniscus can exceed the reduction associated with viscous effects. The results of this work compare better with experimental measurements, than those reported in the literature previously.

Dynamics of a gas bubble rising in an inclined channel at finite Reynolds number
View Description Hide DescriptionThe dynamics of a gas bubble rising in an inclined channel are investigated. The solution of this free boundary problem is determined numerically by using a level set method coupled with a finite difference solution of the Navier–Stokes equations. Results are presented as a function of Reynolds number, Bond number, and angle of inclination. Steady solutions for small values of both Reynolds and Bond number are found. In an inclined channel, we find that as these parameters are increased, the bubble will either periodically bounce off of the upper wall or rupture. In a vertical channel, with increasing Bond number, the bubble first begins to oscillate periodically, and then ruptures. In a vertical channel with increasing Reynolds number, the steady solution will bifurcate to a time periodic symmetric oscillation as the bubble rises up the channel, but further increase in Reynolds number allows for solutions that are no longer symmetric and oscillate back and forth between the channel walls. Our results parallel experimental work which shows that there is a critical angle of inclination at which the dynamics changes from bouncing bubbles to steady rising bubbles.

Convective oscillations in multilayer systems under the combined action of buoyancy and thermocapillary effect
View Description Hide DescriptionThe joint action of buoyancy and thermocapillary effect on the convection in a multilayer system is investigated. It is found that the competition between buoyancy and thermocapillarity may lead to the appearance of a specific type of oscillations. Predictions of the linear stability theory are justified by nonlinear simulations. The nonlinear development of the oscillatory instability is studied.

The atomic detail of an evaporating meniscus
View Description Hide DescriptionAtomistic simulations of a simple LennardJones fluid are used to investigate the very nearwall dynamics and thermodynamics of evaporating menisci. The specific configuration considered is a twodimensional (in the mean) liquid drop centered on a cold spot on an atomically smooth solid wall with evaporating menisci extending from it onto hotter regions of the wall. In the four cases simulated, the interaction energy between the solid atoms, which make up the wall, and the fluid atoms, which are equilibrated in liquid and vapor phases, is varied by a factor of about 5. Results are interpreted in the context of a recently proposed continuum model [V. S. Ajaev and G. M. Homsy, “Steady vapor bubbles in rectangular microchannels,” J. Colloid. Interface Sci.240, 259 (2001)], which is based on a lowcapillarynumber asymptotic analysis of the flow and heat equations. In this model, the nonlocal influence of the wall is modeled by a disjoining pressure, a common linearized nonequilibrium model is assumed for evaporation kinetics, and the interface curvature impacts thermodynamics through its effect on the local pressure. However, this model and others like it neglect both the atomic granularity of the fluid and any scale associated changes in its properties in the thinnest regions of the evaporating meniscus, which are the subject of this study. Quantitative agreement for meniscus shape and evaporative mass flux is found for a weakly wetting case, but the model must be modified in a straightforward way for more strongly wetting cases to account for a layer of nearly fixed fluid atoms on the wall. A finite solidliquidinterface thermal (Kapitza) resistance is found to be important, and the continuum model is reformulated accordingly. With an appropriate Kapitza resistance value the reformulation yields accurate predictions using the actual wall temperature as a boundary condition, rather than the fluid’s temperature at the wall.

Capture and inception of bubbles near line vortices
View Description Hide DescriptionMotivated by the need to predict vortexcavitation inception, a study has been conducted to investigate bubble capture by a concentrated line vortex of core size and circulation under noncavitating and cavitating conditions. Direct numerical simulations that solve simultaneously for the two phase flow field, as well as a simpler oneway coupled pointparticletracking model (PTM) were used to investigate the capture process. The capture times were compared to experimental observations. It was found that the pointparticletracking model can successfully predict the capture of noncavitating small nuclei by a line vortex released far from the vortex axis. The nucleus grows very slowly during capture until the late stages of the process, where bubble/vortex interaction and bubble deformation become important. Consequently, PTM can be used to study the capture of cavitating nuclei by dividing the process into the noncavitating capture of the nucleus, and then the growth of the nucleus in the lowpressure core region. Bubble growth and deformation act to speed up the capture process.
 Viscous and NonNewtonian Flows

Stokes’ first problem for an OldroydB fluid in a porous half space
View Description Hide DescriptionBased on a modified Darcy’s law for a viscoelastic fluid, Stokes’ first problem was extended to that for an OldroydB fluid in a porous half space. By using Fourier sine transform, an exact solution was obtained. In contrast to the classical Stokes’ first problem for a clear fluid, there is a dependent steady state solution for an OldroydB fluid in the porous half space, which is a damping exponential function with respect to the distance from the flat plate. The thickness of the boundary layer, which tends to be a limited value, is also different from that of a clear fluid. The effect of viscoelasticity on the unsteady flow in porous media is investigated. It was found if , oscillations in velocity occur obviously and the system exhibits viscoelastic behaviors, where and are nondimensional relaxation and retardation times, respectively, Re is Reynold number in porous media. Some previous solutions of Stokes’ first problem corresponding to Maxwell fluid and Newtonian fluid in porous or nonporous half space can be easily obtained from our results in different limiting cases.
 Laminar Flows

Microscopic steady streaming eddies created around short cylinders in a channel: Flow visualization and Stokes layer scaling
View Description Hide DescriptionMicroscale steady streaming eddies created using lowintensity fluid oscillations offer appealing options for controlling fluids in microfluidic systems. We describe the threedimensional (3D) steady streaming flow formed in a small channel containing single fixed cylinders when the channel fluid is oscillated at low intensity. Experiments include three cylinder sizes (length ; radii , 250, and ) within identical channels (height ; width ) over a range of oscillation frequencies . The size of key flow features is measured from steady particle pathline images recorded within three flow symmetry planes. The resulting 3D streaming exhibits two distinct recirculating flows that are governed by the Stokes layer thickness and geometric length scales. Four symmetric recirculating eddies are created adjacent to the cylinder far from channel walls, and their size is governed by as described by steady streaming theory for a 2D geometry. The cylinder/wall boundary layer junction drives a 3D recirculating flow with size that is directly proportional to and is not affected by a threefold variation of the cylinder radius. The flowimages and scaling describe an organized 3D steady streaming flow that may be tuned to control fluid and its contents in microfluidic devices.
 Instability and Transition

Linear stability and energy growth of viscosity stratified flows
View Description Hide DescriptionThe nonnormality of the Orr–Sommerfeld equation leads to the possibility of disturbance growth even though all eigenvalues are stable. In singlefluid flow the disturbance growth converges to a limit once the number of modes exceeds a minimum number. In the case of a twofluid flow, however, convergence is not found. The problem of nonconvergence is due to the presence of the interface and the corresponding interfacial mode. The interface is replaced with a miscible layer of variable viscosity. When the thickness of the miscible layer is approximately equal to the thickness of the critical layer, the flow resembles twofluid flow and one of the modes starts behaving like the interfacial mode.

Linear stability of a Berman flow in a channel partially filled with a porous medium
View Description Hide DescriptionThe temporal stability of similarity solutions for an incompressible fluid moving in a channel partially filled with a porous medium is analyzed. A constant wall suction acting on the bottom surface of the porous medium drives the fluid; the upper wall of the channel is impermeable. This work extends the work of King and Cox [“Asymptotic analysis of the steadystate and timedependent Berman problem,” J. Eng. Math.39, 87 (2001)] to a wider class of similarity solutions where coupled flow, both above and through a porous medium, is considered. In this work, a similarity transform is proposed which satisfies both the Navier–Stokes equation in the clear fluid portion of the domain and the Brinkman extended Darcy law relationship in the porous medium. The boundary conditions between the clear fluid and porous regions are those outlined by OchoaTapia and Whitaker [“Momentum transfer at the boundary between a porous medium and a homogeneous fluid I: theoretical development,” Int. J. Heat Mass Transfer38, 2635 (1995)]. The solutions of the steady flow are approximated analytically, in the limit of small wall suction, and numerically. Multiple steadystate solutions were found. The temporal stability of the solutions indicates turningpoint bifurcations and instability only occurred with reverse flows.

Thermal convection in a cylindrical enclosure
View Description Hide DescriptionThe paper highlights the onset of convection in a fluid layer partially filled in an axisymmetric container. The equilibrium of the fluid is disturbed with the deformation of the interface due to residual acceleration. The general problem of deformable interface involves a dimensionless parameter, the Bond number. An analytical expression for the natural frequencies of the deformable surface is derived in terms of the Bond number, which determines the time period required for the stable location of the fluid for the propellant management of the spacecraft.

Rayleigh–Taylor mixing rates for compressible flow
View Description Hide DescriptionWe study Rayleigh–Taylor instability in both the moderately compressible and weakly compressible regimes. For the twodimensional single mode case, we find that the dimensionless terminal velocities (and associated Froude numbers) are nearly constant over most of this region of parameter space, as the thermodynamic parameters describing the equation of state are varied. The phenomenological drag coefficient which occurs in the single mode buoyancydrag equation is directly related to the terminal velocities and has a similar behavior. Pressure differences and interface shape, however, display significant dependence on the equation of state parameters even for the weakly compressible flows. For threedimensional multimode mixing, we expect accordingly that density stratification rather than drag will provide the leading compressibility effect. We develop an analytical model to account for density stratification effects in multimode selfsimilar mixing. Our theory is consistent with and extends numerically based conclusions developed earlier which also identify density stratification as the dominant compressibility effect for multimode threedimensional mixing.
 Turbulent Flows

Similarity in the far field of a turbulent round jet
View Description Hide DescriptionIn this paper, we test the idea of equilibrium similarity, for which all scales evolve in a similar way in a turbulent round jet, for a prescribed set of initial conditions. Similarity requirements of the mean momentum and turbulent energy equations are reviewed briefly but the main focus is on the velocity structure function equation, which represents an energy budget at any particular scale. For similarity of the structure function equation along the jet axis, it is found that the Taylor microscale is the relevant characteristic length scale. Energy structure functions and spectra, measured at a number of locations along the axis of the jet, support this finding reasonably well, i.e., they collapse over a significant range of scales when normalized by and the mean turbulent energy Since the Taylor microscale Reynolds number is approximately constant along the jet axis, the structure functions and spectra also collapse approximately when the normalization uses either the Kolmogorov or integral length scales. Over the dissipative range, the best collapse occurs when Kolmogorov variables are used. The use of and the integral length scale provides the best collapse at large separations. A measure of the quality of collapse is given.

Large eddy simulation of a plane turbulent wall jet
View Description Hide DescriptionThe meanflow and turbulence properties of a plane wall jet, developing in a stagnant environment, are studied by means of large eddy simulation. The Reynolds number, based on the inlet velocity and the slot height , is , corresponding to recent wellresolved laser Doppler velocimetry and pulsed hot wire measurements of Eriksson et al. The relatively low Reynolds number and the high numerical resolution adopted (8.4 million nodes) allow all scales larger than about 10 Kolmogorov lengths to be captured. Of particular interest are the budgets for turbulence energy and Reynolds stresses, not available from experiments, and their inclusion sheds light on the processes which play a role in the interaction between the nearwall layer and the outer shear layer. Profiles of velocity and turbulentReynolds stresses in the selfsimilar region are presented in inner and outer scaling and compared to experimental data. Included are further results for skin friction, evolution of integral quantities and thirdorder moments. Good agreement is observed, in most respects, between the simulated flow and the corresponding experiment. The budgets demonstrate, among a number of mechanisms, the decisive role played by turbulenttransport (via the third moments) in the interaction region, across which information is transmitted between the nearwall layer and the outer layer.

Direct numerical simulations of turbulent flow over a permeable wall using a direct and a continuum approach
View Description Hide DescriptionA direct numerical simulation (DNS) has been performed of turbulent channel flow over a threedimensional Cartesian grid of cubes in, respectively, the streamwise, spanwise, and wallnormal direction. The grid of cubes mimics a permeable wall with a porosity of 0.875. The flow field is resolved with mesh points. To enforce the noslip and nopenetration conditions on the cubes, an immersed boundary method is used. The results of the DNS are compared with a second DNS in which a continuum approach is used to model the flow through the grid of cubes. The continuum approach is based on the volumeaveraged Navier–Stokes (VANS) equations [S. Whitaker, “The Forchheimer equation: a theoretical development,” Transp. Porous Media25, 27 (1996)] for the volumeaveraged flow field. This method has the advantage that it requires less computational power than the direct simulation of the flow through the grid of cubes. More in general, for complex porous media one is usually forced to use the VANS equations, because a direct simulation would not be possible with presentday computer facilities. A disadvantage of the continuum approach is that in order to solve the VANS equations, closures are needed for the drag force and the subfilterscale stress. For porous media, the latter can often be neglected. In the present work, a relation for the drag force is adopted based on the Irmay [“Modèles théoriques d’écoulement dans les corps poreux,” Bulletin Rilem 29, 37 (1965)] and the Burke–Plummer model [R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena (Wiley, New York, 2002)], with the model coefficients determined from simulations reported by W. P. Breugem, B. J. Boersma, and R. E. Uittenbogaard [“Direct numerical simulation of plane channel flow over a 3D Cartesian grid of cubes,” Proceedings of the Second International Conference on Applications of Porous Media, edited by A. H. Reis and A. F. Miguel (Évora Geophysics Center, Évora, 2004), p. 27]. The results of the DNS with the grid of cubes and the second DNS in which the continuum approach is used, agree very well.

Particle dispersion in a turbulent vortex core
View Description Hide DescriptionA study is performed using direct numerical simulation to examine the effect of turbulence on particle dispersion in a columnar vortex. A procedure is used to generate initial conditions in which turbulence external to the vortex has the wrapped, nearly azimuthal form characteristic of turbulence around a largescale vortex structure. This approach enables control of the initial values of the external turbulence intensity and length scale for a given vortex size and strength. The effect of turbulence on dispersion of particles by the vortexflow is examined for a wide range of different values of the initial turbulence intensity. Cases both with zero mean axial flow and unstable vortices with nonzero mean axial flow are examined. The state of the external turbulence is determined by a balance between dissipation, stretching, and vorticity reorientation and stripping from the largescale vortex core. We examine cases with peak initial turbulence kinetic energy differing by a ratio of up to 400, and for all cases we observe that the external turbulence has a dramatic effect on the particle dispersion. The results of the study have implications for largeeddy simulation of particulate flows for which the particle concentration in the region surrounding the coherent vortex structures will not be accurately predicted if the effect of smallerscale turbulence surrounding these structures on particle dispersion is not resolved or appropriately modeled.

A scaledependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows
View Description Hide DescriptionA scaledependent dynamic subgrid model based on Lagrangian time averaging is proposed and tested in large eddy simulations(LES) of highReynolds number boundary layerflows over homogeneous and heterogeneous rough surfaces. The model is based on the Lagrangian dynamic Smagorinsky model in which required averages are accumulated in time, following fluid trajectories of the resolved velocity field. The model allows for scale dependence of the coefficient by including a second testfiltering operation to determine how the coefficient changes as a function of scale. The model also uses the empirical observation that when scale dependence occurs (such as when the filter scale approaches the limits of the inertial range), the classic dynamic model yields the coefficient value appropriate for the testfilter scale. Validation tests in LES of high Reynolds number, rough wall, boundary layerflow are performed at various resolutions. Results are compared with other eddyviscosity subgridscale models. Unlike the Smagorinsky–Lilly model with walldamping (which is overdissipative) or the scaleinvariant dynamic model (which is underdissipative), the scaledependent Lagrangian dynamic model is shown to have good dissipation characteristics. The model is also tested against detailed atmospheric boundary layer data that include measurements of the response of the flow to abrupt transitions in wall roughness. For such flows over variable surfaces, the planeaveraged version of the dynamic model is not appropriate and the Lagrangian averaging is desirable. The simulated wall stress overshoot and relaxation after a jump in surface roughness and the velocity profiles at several downstream distances from the jump are compared to the experimental data. Results show that the dynamic Smagorinsky coefficient close to the wall is very sensitive to the underlying local surface roughness, thus justifying the use of the Lagrangian formulation. In addition, the Lagrangian formulation reproduces experimental data more accurately than the planaraveraged formulation in simulations over heterogeneous rough walls.

On the vortical structure in a round jet
View Description Hide DescriptionTurbulent vortical structures in a round free jet of water were experimentally visualized by using stereo particle image velocimetry (PIV). A laser light sheet illuminated a crosssectional plane normal to the axis of the jet, and two chargecoupled device cameras captured particle images in the same region of interest but from different directions. The stereoPIV algorithm had been applied to obtain twodimensional, threecomponent (2D3C) velocity distributions on various crosssectional planes along the axis downstream. All nine components of the velocity gradient tensor were reconstructed from timedependent 2D3C velocity data by locally assuming Taylor’s frozen field hypothesis based on the convective velocity evaluated from the mean flow profile. Isosurfaces of the swirling strength revealed that the existence of a group of hairpinlike vortex structures was quite evident around the rim of the shear layer of the jet. The center of curvature of the head of the hairpin was typically observed around , and the azimuthal spacing between the legs of the hairpin was roughly . A similar hairpin structure was estimated by linear stochastic estimation. The typical spacing between the legs of the estimated hairpin was , which is generally constant over the range of .