Volume 20, Issue 10, October 2008
Index of content:
Recent theoretical, numerical, and experimental investigations performed at the Department of Mechanics, KTH Stockholm, and the Department of Mechanical Engineering, Eindhoven University of Technology, are reviewed, and new material is presented to clarify the role of the boundary-layer streaks and their instability with respect to turbulentbreakdown in bypass transition in a boundary layer subject to free-stream turbulence. The importance of the streak secondary-instability process for the generation of turbulent spots is clearly shown. The secondary instability manifests itself as a growing wave packet located on the low-speed streak, increasing in amplitude as it is dispersing in the streamwise direction. In particular, qualitative and quantitative data pertaining to temporal sinuous secondary instability of a steady streak, impulse responses both on a parallel and a spatially developing streak, a model problem of bypass transition, and full simulations and experiments of bypass transition itself are collected and compared. In all the flow cases considered, similar characteristics in terms of not only growth rates, group velocity, and wavelengths but also three-dimensional visualizations of the streak breakdown have been found. The wavelength of the instability is about an order of magnitude larger than the local boundary-layer displacement thickness , the group velocity about 0.8 of the free-stream velocity , and the growth rate on the order of a few percent of . The characteristic structures at the breakdown are quasistreamwise vortices, located on the flanks of the low-speed region arranged in a staggered pattern.
- Interfacial Flows
20(2008); http://dx.doi.org/10.1063/1.2998845View Description Hide Description
The dewetting of a nanoscale water film under the action of an electric field is studied with molecular dynamics simulation. Results show that the onset of film rupture is induced by a spontaneous instability mechanism. After the rupture, the rim of the film recedes with a dynamic contact angle. The transient streamlines at a typical moment show that the liquid molecule near the rim moves almost vertically upwards, driven by the repulsive force from the solid surface. The oscillatory behavior of the density profile, resulting from the interaction between attractive and repulsive potentials, is observed near the solid surface. The analyses of the dewetting process demonstrate that the applied electric field will increase the wettability of graphite walls, thus suppressing the rupture, reducing the dynamic contact angle, and raising the liquid density adjacent to both the solid and liquid-vacuumsurfaces. Owing to the polarity of water, the positive voltage produces stronger influences than the negative one.
20(2008); http://dx.doi.org/10.1063/1.3000425View Description Hide Description
The penetration of a wettingliquid in the narrow gap between two vertical plates making a small angle is analyzed in the framework of the lubrication approximation. At the beginning of the process, the liquid rises independently at different distances from the line of intersection of the plates except in a small region around this line where the effect of the gravity is negligible. The maximum height of the liquid initially increases as the cubic root of time and is attained at a point that reaches the line of intersection only after a certain time. At later times, the motion of the liquid is confined to a thin layer around the line of intersection whose height increases as the cubic root of time and whose thickness decreases as the inverse of the cubic root of time. The evolution of the liquid surface is computed numerically and compared with the results of a simple experiment.
20(2008); http://dx.doi.org/10.1063/1.3005453View Description Hide Description
Experiments with glycerol-water thin films flowing down an inclined plane reveal a localized instability that is primarily three dimensional. These transient structures, referred to as “dimples,” appear initially as nearly isotropic depressions on the interface. A linear stability analysis of a binary mixture model in which barodiffusive effects dominate over thermophoresis (i.e., the Soret effect) reveals unstable modes when the components of the mixture have different bulk densities and surface tensions. This instability occurs when Fickian diffusion and Taylor dispersion effects are small, and is driven by solutalcapillary stresses arising from gradients in concentration of one component, across the depth of the film. Qualitative comparison between the experiments and the linear stability results over a wide range of parameters is presented.
- Viscous and Non-Newtonian Flows
20(2008); http://dx.doi.org/10.1063/1.2990751View Description Hide Description
The work in this paper concerns a mathematical model of the contact melting process of a rectangular material in contact with a hot plate. The problem is described by a coupled system of heat equations in the solid and melt layer, fluid flow in the melt, a Stefan condition at the melt interface, and a force balance between the weight of the solid and the fluid pressure. Since the melt layer remains thin throughout the process, we use the lubrication approximation to the fluid equations and assume that the heat flow in the fluid is dominated by conduction across the thin film. In the solid we employ a heat balance integral method. Results show that the film height has initial and final rapid increases, whereas for intermediate times the height slowly increases. The quasisteady state of previous models is never attained: This is shown to be an effect of neglecting the change in mass and conduction in the solid. The previously observed initial infinite velocity of the melt is shown to be a result of the perfect thermal contact assumption. For a water-ice system the melting rate is shown to be approximately linear, this allows us to reduce the problem to solving a single first order differential equation for the liquid layer thickness. The main analysis is carried out in two dimensions, but we briefly highlight the extension to three dimensions. The method is verified by comparison with previously published experimental results on the melting of -octadecane.
20(2008); http://dx.doi.org/10.1063/1.2994750View Description Hide Description
Two-dimensional Stokes flow in a rectangular driven cavity is studied. Flow in very tall or shallow cavities has many counter-rotating eddies lying along the cavity centerline. This structure is investigated by constructing asymptotic approximations to the flow based on the assumptions and , where is the cavity’s aspect ratio. We show that the number of eddies increases as tends to infinity or zero and derive asymptotic formulas for the values of the aspect ratio at which the streamline topology bifurcates and a new eddy appears. We concentrate particularly on flow driven by translation of the top and bottom cavity walls with equal and opposite velocities. For this benchmark problem our asymptotics are able to connect existing computations (performed in the approximate range of ) with Moffatt’s 1964 theory for an infinite channel (i.e., or ). We show that tall cavities have sequences of eddies which match the infinite channel flow, as observed in the previous computations. In contrast, shallow cavities have only half the number of eddies and a more complicated streamline topology. In both cases our asymptotic approximations give analytic formulas for the number of eddies and the shape of the streamlines. Other flows, corresponding to different driving at the boundaries, are also discussed and can be treated by the asymptotic methods we derive.
- Particulate, Multiphase, and Granular Flows
20(2008); http://dx.doi.org/10.1063/1.2996134View Description Hide Description
It has long been known that shaking a granular bed can produce circulating convection. This is the first in a series of papers that explores convection in beds a hundred particles or more deep. In such deep beds, the pressures are high enough that the particles remain in contact with their neighbors throughout much of the vibrational period. As such, they interact elastically and convection becomes dependent on the elastic properties of the bed. Changing the particle stiffness can dramatically alter the convection, and for very soft or very hard particles, eliminate it completely. In this paper, the effect of stiffness, bed geometry, and frictional properties on the global convection properties are assessed.
Convection in deep vertically shaken particle beds. II. The relationship between convection and internal wave propagation20(2008); http://dx.doi.org/10.1063/1.2996135View Description Hide Description
The convective motion in deep vertically shaken beds is not continuous but occurs only during a brief portion of a cycle that roughly repeats over three periods of vibration. The convection is coordinated by a series of waves that propagate through the bed, a compression wave formed as the flask’s bottom pushes upward against the bottom of the bed, and two expansion waves: a Type 1 expansion wave that is the reflection of the compression wave and a Type 2 expansion wave that forms as the flask’s bottom moves away from the bottom of the bed. Convection only is observed after Type 1 expansion wave has passed the convective zone, relaxing the stresses in the bed and leaving the particles free to move. However the convective motion is confined to the region above Type 2 wave and convection disappears as a Type 2 wave passes.
20(2008); http://dx.doi.org/10.1063/1.2996136View Description Hide Description
Convection in a deep vertically vibrated two-dimensional cell of granular material occurs in the form of counter-rotating cells that move material from the walls to the center of the channel and back again. At least for deep beds, where for much of the cycle, particles are in long duration contact with their neighbors, convection only appears for a short potion of every third vibrational period. That period is delimited by the interaction of three types of internal waves, a compression wave, and two types of expansion waves. Four mechanisms are identified that drive the four basic motions of convection: (1) particles move upward at the center as the result of compression wave, (2) downward at the wall as a combined effect of frictional holdback by the walls and the downward pull of gravity, (3) from the center to the walls along the free surface due to the heaping of the bed generated by the compression wave, and (4) toward the center in the interior of the box to form the bottom of convection rolls due to the relaxation of compressive stresses caused by an expansion wave.Convection only occurs when the conditions are right for all four mechanisms to be active simultaneously.
Lattice-Boltzmann simulation of finite Reynolds number buoyancy-driven bubbly flows in periodic and wall-bounded domains20(2008); http://dx.doi.org/10.1063/1.3001728View Description Hide Description
A lattice-Boltzmann method is used to probe the structure and average properties of suspensions of monodisperse, spherical, noncoalescing bubbles rising due to buoyancy with Reynolds numbers based on the bubble terminal velocities of 5.4 and 20. Unbounded suspensions subject to periodic boundary conditions exhibit a microstructure with a strong tendency toward horizontal alignment of bubble pairs even at volume fractions of as high as 0.2. This microstructure leads to a mean rise velocity whose dependence on the bubble volume fraction is not well fitted by a standard power-law function. Simulations with bounding vertical walls exhibit a deficit of bubbles near each wall and a peak of volume fraction approximately one bubble diameter from the wall. We attribute this structure to the effects of a repulsive wall-induced force and a lift force associated with the liquid flow driven by the variation in the buoyancy force with horizontal position. Weaker peaks of bubble volume fraction extend into the bulk of the suspension and these peaks are separated by a distance equal to the peak in the pair distribution function for bubble pairs in an unbounded fluid. This suggests that the layering is a result of hydrodynamic bubble-bubble interactions.
Detailed characteristics of drop-laden mixing layers: Large eddy simulation predictions compared to direct numerical simulation20(2008); http://dx.doi.org/10.1063/1.2990758View Description Hide Description
Results are compared from direct numerical simulation (DNS) and large eddy simulation(LES) of a temporal mixing layer laden with evaporating drops to assess the ability of LES to reproduce detailed characteristics of DNS. The LES used computational drops, each of which represented eight physical drops, and a reduced flow field resolution using a grid spacing four times larger than that of the DNS. The LES also used models for the filtered source terms, which express the coupling of the drops with the flow, and for the unresolved subgrid-scale (SGS) fluxes of species mass, momentum, and enthalpy. The LESs were conducted using one of three different SGS-flux models: dynamic-coefficient gradient (GRD), dynamic-coefficient Smagorinsky (SMD), and constant-coefficient scale similarity (SSC). The comparison of the LES with the filtered-and-coarsened (FC) DNS considered detailed aspects of the flow that are of interest in ignition or full combustion. All LESs captured the largest-scale vortex, the global amount of vapor emanating from the drops, and the overall size distribution of the drops. All LESs tended to underpredict the global amount of irreversible entropy production (dissipation). The SMD model was found unable to capture either the global or local vorticity variation and had minimal small-scale activity in dynamic and thermodynamic variables compared to the FC-DNS. The SMD model was also deficient in predicting the spatial distribution of drops and of the dissipation. In contrast, the GRD and SSC models did mimic the small-scale activity of the FC-DNS and the spatial distribution of drops and of the dissipation. Therefore, the GRD and SSC models are recommended, while the SMD model seems inappropriate for combustion or other problems where the local activity must be predicted.
Experimental study of oscillatory motion of particles and bubbles with applications to Coriolis flow meters20(2008); http://dx.doi.org/10.1063/1.3001725View Description Hide Description
The present experimental study is designed to measure the motion of a spherical particle in a noninertial reference frame when the environment oscillates horizontally at a prescribed frequency and amplitude. Measurements are compared with theoretical equations of motion, for example, Basset [A Treatise on Hydrodynamics (Deighton Hall, London, 1888), Vol. 2], over wide ranges of density ratio , inverse Stokes number , and amplitude ratio , the three most critical nondimensional parameters. The experimental configuration consists of a bubble or solid sphere rising or falling in a bubble column while vibration occurs in the horizontal direction. Motion is measured with a high speed video camera and contemporary image and signal processing techniques are used to evaluate the data. The setup closely resembles multiphase flow in a Coriolisflow meter, a device which measures mass flow rate and density by oscillating two tubes at resonance. Accurate predictions of the motion of the sphere may lead to estimates of measurement errors due to entrained gas or solid particles. Excellent agreement for amplitude and phase shift is found between theory and experiment over the full range of testing, which is defined by small oscillatory Reynolds numbers, finite Strouhal numbers , widely varying density ratios , inverse Stokes numbers , and amplitude ratios .
Inertial migration of spherical particles in circular Poiseuille flow at moderately high Reynolds numbers20(2008); http://dx.doi.org/10.1063/1.3005427View Description Hide Description
The inertial migration of spherical particles in a circular Poiseuille flow is numerically investigated for the tube Reynolds number up to 2200. The periodic boundary condition is imposed in the streamwise direction. The equilibrium positions, the migration velocity, and the angular velocity of a single particle in a tube cell are examined at different Reynolds numbers, particle-tube size ratios, and tube lengths. Inner equilibrium positions are observed as the Reynolds number exceeds a critical value, in qualitatively agreement with the previous experimental observations [J.-P. Matas, J. F. Morris, and E. Guazzelli, J. Fluid Mech.515, 171 (2004)]. Our results indicate that the hydrodynamic interactions between the particles in different periodic cells have significant effects on the migration of the particles at the tube length being even as large as 6.7 particle diameters and they tend to stabilize the particles at the outer Segré–Silberberg equilibrium positions and to suppress the emergence of the inner equilibrium positions. A mirror-symmetric traveling-wave-like structure is observed when the particle Reynolds number is large enough. A pair of counter-rotating streamwise vortices exists at both upstream and downstream of the particle but with different rotating directions. The fluids in the half of the pipe without the particle flow more slowly and most fluids in the other half with the particle move faster with respect to the parabolic profile. The intensity of the structure is influenced by the local particle Reynolds number, the particle motion, and the tube length. In addition, the migration of multiple particles in a periodic tube cell is examined. We attribute the disparity in the critical particle Reynolds number for the occurrence of the inner particle annulus for the experiments and our simulations to the effect of the tube length or the periodic boundary condition in our numerical model.
- Laminar Flows
20(2008); http://dx.doi.org/10.1063/1.2980039View Description Hide Description
Dissipative particle dynamics (DPD) has recently attracted great interest due to its potential to simulate the dynamics of colloidal particles in fluidic devices. In this work, we explore the validity of DPD to reproduce the hydrodynamic interaction between a suspended particle and confining solid walls. We first show that a relatively large Schmidt number of the DPD fluid can be obtained by increasing the ratio between the strength of the dissipative force and the kinetic energy of the particles. We then measure the mobility and diffusion coefficient of the colloidal particles and show good agreement with the predicted results. We then focus on the particle-solid interactions and measure the force on a colloidal particle moving both parallel and perpendicular to two parallel walls. In both cases we found good agreement with the theoretical predictions based on Stokes flows for separations as small as one-tenth of the particle radius.
20(2008); http://dx.doi.org/10.1063/1.2997367View Description Hide Description
Flow in a rectangular basin driven by a surface force is considered. The problem is motivated by flow in geophysical bodies of water driven by wind at the water surface. Results are obtained via numerical computations of the Navier–Stokes equations assuming constant density. The numerical integration is achieved with a splitting method, with Crank–Nicolson for the linear terms, and Adams–Bashforth for the nonlinear terms. Spatial derivatives are treated with finite differences. The forcing has a sinusoidal variation across the top with a sequence of length scales. The results show a symmetric steady stable flow for small Reynolds numbers. As the Reynolds number is increased, the system experiences either a subcritical or supercritical pitchfork bifurcation to an asymmetric steady stable flow, or a local Hopf bifurcation, depending on the aspect ratio of the container and the length scale of the forcing. The asymmetric flow is cellular for forcing length scales commensurate with the depth. For smaller forcing length scales, the asymmetric flow has a basin-filling character at the bottom portion of the basin.
20(2008); http://dx.doi.org/10.1063/1.3003525View Description Hide Description
A transversely oscillating circular cylinder confined in a channel has the potential to promote mixing and heat transfer at moderate Reynolds numberflows. In the present study, simulation results for flow past a circular cylinder subjected to forced cross-flow oscillations in a straight channel with an upstream splitter plate are presented in a wide range of cylinder oscillation frequencies, including the subharmonic, superharmonic, and primary lock-in regimes. Simulations are performed at , with cylinder oscillation amplitude of 0.4 diameters and a blockage ratio of 1/3. A spectral element algorithm based on the arbitrary Lagrangian Eulerian formulation is utilized. The numerical method exhibits spectral accuracy and allows large mesh deformation in the computational domain without mesh refinements. The main objective of this study is systematic investigations of the cylinder oscillation on the vortex shedding mechanism, downstream vortex patterns, and forces exerted on the cylinder.
- Instability and Transition
20(2008); http://dx.doi.org/10.1063/1.2987435View Description Hide Description
The present paper deals with the onset of the two-dimensional Rayleigh–Bénard convection for a plane channel flow of viscoplasticfluid. The influence of the yield stress on the instability and stability conditions characterized by the Rayleigh numbers denoted, respectively, and is investigated in the framework of linear analysis using modal and energetic approaches. The results show that the yield stress, represented by the Bingham number , delays the onset of convection. For low values of the Reynolds number Re, the critical conditions and tend to be equal and the difference increases with increasing Re, highlighting the non-normality of the linear operator. For and large , it is shown that the critical Rayleigh number increases as and the critical wave number evolves according to .
20(2008); http://dx.doi.org/10.1063/1.2990763View Description Hide Description
This paper develops a one-dimensional formulation to simulate Taconis oscillations in a helium-filled, quarter-wavelength tube in cryogenics within a framework of the boundary-layer theory. Dividing an acoustic field in the tube into a boundary layer on the wall and a main-flow region outside of it, the fluid-dynamical equations are averaged over the whole cross section of the tube, from which the equations averaged over the main-flow region are derived by using the boundary-layer solutions. Nonlinear theory is employed for the main-flow region, whereas the boundary layer is assumed to be described by the linear theory. Resultant equations for the main-flow region are posed in the form of integrodifferential equations due to memory effects by the boundary layer. An initial- and boundary-value problem is solved numerically for the evolution of a small disturbance. It is demonstrated that for temperature distribution of a smooth, step function, a transient behavior leading to emergence of self-excited Taconis oscillations can be simulated numerically. Although the first-order theory in the boundary-layer thickness has been regarded as being incapable of describing Taconis oscillations, it turns out to be applicable to a case with plausible temperature gradient.
20(2008); http://dx.doi.org/10.1063/1.2991431View Description Hide Description
A linear stability analysis of the Rayleigh–Taylor instability(RTI) between two ideal inviscid immiscible compressible fluids in cylindrical geometry is performed. Three-dimensional (3D) cylindrical as well as two-dimensional (2D) axisymmetric and circular unperturbed interfaces are considered and compared to the Cartesian cases with planar interface. Focuses are on the effects of compressibility, geometry, and differences between the convergent (gravity acting inward) and divergent (gravity acting outward) cases on the early instability growth. Compressibility can be characterized by two independent parameters—a static Mach number based on the isothermal sound speed and the ratio of specific heats. For a steady initial unperturbed state, these have opposite influence, stabilization and destabilization, on the instability growth, similar to the Cartesian case [D. Livescu, Phys. Fluids16, 118 (2004)]. The instability is found to grow faster in the 3D cylindrical than in the Cartesian case in the convergent configuration but slower in the divergent configuration. In general, the direction of gravity has a profound influence in the cylindrical cases but marginal for planar interface. For the 3D cylindrical case, instability grows faster in the convergent than in the divergent arrangement. Similar results are obtained for the 2D axisymmetric case. However, as the flow transitions from the 3D cylindrical to the 2D circular case, the results above can be qualitatively different depending on the Atwood number, interface radius, and compressibility parameters. Thus, 2D circular calculations of RTI growth do not seem to be a good model for the fully 3D cylindrical case.
20(2008); http://dx.doi.org/10.1063/1.2996326View Description Hide Description
We investigate the stability of a fluid confined between two cylinders that rotate at same constant angular speed. In the case of infinite cylinders, or endwalls rotating with the cylinders, the flow is in solid-body rotation and hence linearly stable for any rotation speed. However, when the endwalls are stationary, a large-scale circulation is driven by radially inward boundary layer flow on the endwalls. For sufficiently high angular speeds, this circulation becomes unstable to azimuthal waves. As the length-to-gap aspect ratio of the system is increased, a wealth of instabilities is revealed. It is particularly interesting that for all these instabilities the associated energy is localized in the equatorial region, as far from the endwalls as possible. This shows that care must be taken when assuming localized endwall effects in simplified models.
20(2008); http://dx.doi.org/10.1063/1.3000643View Description Hide Description
The instability of Poiseuille flow in a fluid-porous system is investigated. The system consists of a fluid layer overlying porous media and is subjected to a horizontal plane Poiseuille flow. We use Brinkman’s model instead of Darcy’s law to describe the porous layer. The eigenvalue problem is solved by means of a Chebyshev collocation method. We study the influence of the depth ratio and the Darcy number on the instability of the system. We compare systematically the instability of Brinkman’s model with the results of Darcy’s model. Our results show that no satisfactory agreement between Brinkman’s model and Darcy’s model is obtained for the instability of a fluid-porous system. We also examine the instability of Darcy’s model. A particular comparison with early work is made. We find that a multivalued region may present in the plane, which was neglected in previous work. Here is the dimensionless wavenumber and Re is the Reynolds number.