Volume 17, Issue 8, August 2005
 LETTERS


Modeling subgridscale effects on particles by approximate deconvolution
View Description Hide DescriptionThe approximate deconvolution is implemented to reconstruct the instantaneous velocities from the filtered velocities before using them in the momentum equations of particles in the largeeddy simulation(LES) of particleladen turbulent flows. It is shown that the various statistics of particles obtained through deconvolution are in good agreement with those obtained by the direct numerical simulation (DNS) by conducting a priori and a posteriori tests in a particleladen homogeneous shear turbulent flow. On the other hand, the neglect of the effects of subgrid scales on the particles results in discrepancies between DNS and LES results.

Energy transfer and intermittency in fourdimensional turbulence
View Description Hide DescriptionThe energy transfer and small scale intermittency in decaying turbulence in four dimensions are studied by direct numerical simulation and by spectral theory. It is found that (1) a law, , in four dimensional (4D) for the longitudinal thirdorder structure function holds, (2) the energy transfer in 4D is more efficient than in three dimensional (3D), (3) the Kolmogorov constant in 4D is which is smaller than in 3D, (4) the velocity gradient intermittency is stronger than in 3D, while (5) the totalenergy dissipation rate in 4D is less intermittent than in 3D. The conflicting trends in (4) and (5) are explained by the changes in the balance between the convective and pressure terms as the dimension increases.

HighReynoldsnumber simulation of turbulent mixing
View Description Hide DescriptionA brief report is given of a new direct numerical simulation of the mixing of passive scalars with uniform mean gradients in forced, stationary isotropic turbulence. The Taylorscale Reynolds number is close to 700 and Schmidt numbers of 1 and are considered. The data provide the most convincing evidence to date for the inertialconvective scaling. Significant departures from smallscale isotropy are sustained in conventional measures. Subject to some stringent resolution requirements, the data suggest that commonly observed differences between the intermittency of energy and scalar dissipation rates may in part be a finiteReynoldsnumber effect.

Enstrophy dissipation in freely evolving twodimensional turbulence
View Description Hide DescriptionFreely decaying twodimensional NavierStokes turbulence is studied. The conservation of vorticity by advective nonlinearities renders a class of Casimirs that decays under viscous effects. A rigorous constraint on the palinstrophy production by nonlinear transfer is derived, and an upper bound for the enstrophy dissipation is obtained. This bound depends only on the decaying Casimirs, thus allowing the enstrophy dissipation to be bounded from above in terms of initial data of the flows. An upper bound for the enstrophy dissipation wave number is derived and the new result is compared with the classical dissipation wave number.

Rayleigh–Taylor turbulent mixing of immiscible, miscible and stratified fluids
View Description Hide DescriptionWe propose a simple empirical model to describe the Rayleigh–Taylor (RT) turbulent mixing of immiscible, miscible and stratified fluids. For immiscible fluids, the rate of momentum loss is an invariant of the flow, whereas the energy dissipation rate increases linearly with time. Turbulent diffusion, accounted for through temperature fluctuations, does not terminate mixing but slows it down significantly. A stratified density distribution can stabilize RT mixing.

“Locally homogeneous turbulence”: Is it an inconsistent framework?
View Description Hide DescriptionIn his first 1941 paper Kolmogorov assumed that the velocity has increments that are homogeneous and independent of the velocity at a suitable reference point. This assumption of local homogeneity is consistent with the nonlinear dynamics only in an asymptotic sense when the reference point is far away. This inconsistency is illustrated numerically using the Burgers equation. Kolmogorov’s derivation of the fourfifths law for the thirdorder structure function and its anisotropic generalization are actually valid only for homogeneous turbulence, but a local version due to Duchon and Robert still holds. A KolmogorovLandau approach is proposed to handle the effect of fluctuations in the largescale velocity on smallscale statistical properties; it is only a mild extension of the 1941 theory and does not incorporate intermittency effects.
 Top

 ARTICLES

 Interfacial Flows

Waveturbulence interaction of a lowspeed plane liquid walljet investigated by particle image velocimetry
View Description Hide DescriptionThe surfacewave amplitude (freesurface level) and the turbulent velocity field of the liquid phase of a plane walljet flow have been simultaneously measured by means of particle imagevelocimetry, which allows for the investigation of surface waves and waveturbulence interaction. The Reynolds number, Weber number, and Ohnesorge number of the tested flow, based on the bulk velocity, height of the closed channel, and physical properties of water, were , , and , respectively. Based on the measured datasets of the velocity field and freesurface level, the characteristics of the waveturbulence interaction as well as the statistics of surface waves and turbulent velocity field were studied. The characteristics of the turbulent velocity field near the wavy free surface obtained in this study were different from those obtained previously for openchannel flows with no shear at the interface or negligible deformation of the free surface. It was found that reverse vortex motion was predominant beneath the wave crest to a depth of about under the tested flow condition at the near field of the wall jet, which was conjectured to be the main feature before the onset of breakup of the free surface.

The steady propagation of a surfactantladen liquid plug in a twodimensional channel
View Description Hide DescriptionIn this study, we investigate the steady propagation of a liquid plug in a twodimensional channel lined by a uniform, thin liquid film. The liquid contains soluble surfactant that can exist both in the bulk fluid and on the airliquid interface. The NavierStokes equations with freesurface boundary conditions and the surfactant transport equations are solved using a finite volume numerical scheme. The adsorption/desorption process of the surfactant is modeled based on pulmonary surfactant properties. As the plug propagates, the front meniscus sweeps preexisting interfacial surfactant from the precursor film, and the surfactant accumulates on the front meniscus interface. As the front meniscus converges on the precursor film from the region where the interfacial surfactant concentration is maximized, the Marangoni stress opposes the flow. In this region, the Marangoni stress results in nearly zero surface velocity, which causes the precursor film thickness near the meniscus to be thicker than the leading film thickness. Since the peaks of wall pressure and wall shear stress occur due to narrowing of the film thickness, the observed increase of the minimum film thickness weakens these stresses. In the thicker film region, however, the drag forces increase due to an increase in the surfactant concentration. This causes the overall pressure drop across the plug to increase as a result of the increasing surfactant concentration. A recirculation flow forms inside the plug core and is skewed toward the rear meniscus as the Reynolds number increases. When no surfactant exists, the recirculation flow is in contact with both the front and the rear interfaces. As the surfactant concentration increases, the Marangoni stress begins to rigidify the front interface and forces the recirculation flow away from the front interface. Subsequently, the recirculation flow is directed away from the rear interface in a manner similar to that for the front interface. When the plug length is shorter, this change in recirculation pattern occurs at a smaller surfactant concentration.

Numerical study of liquidhydrogen droplet generation from a vibrating orifice
View Description Hide DescriptionAtomic hydrogen propellant feed systems for farfuture spacecraft may utilize solidhydrogen particle carriers for atomic species that undergo recombination to create hot rocket exhaust. Such technology will require the development of particle generation techniques. One such technique could involve the production of hydrogen droplets from a vibrating orifice that would then freeze in cryogenic helium vapor. Among other quantities, the shape and size of the droplet are of particular interest. The present paper addresses this problem within the framework of the incompressible Navier–Stokes equations for multiphase flows, in order to unravel the basic mechanisms of droplet formation with a view to control them. Surface tension, one of the most important mechanisms to determine droplet shape, is modeled as the source term in the momentum equation. Droplet shape is tracked using a volumeoffluid approach. A dynamic meshing technique is employed to accommodate the vibration of the generator orifice. Numerically predicted droplet shapes show satisfactory agreement with photographs of droplets generated in experiments. A parametric study is carried out to understand the influence of injection velocity, nozzle vibrational frequency, and amplitude on the droplet shape and size. The computational model provides a definitive qualitative picture of the evolution of droplet shape as a function of the operating parameters. It is observed that, primarily, the orifice vibrational frequency affects the shape, the vibrational amplitude affects the time until droplet detachment from the orifice, and the injection velocity affects the size. However, it does not mean that, for example, there is no secondary effect of amplitude on shape or size.

Incipient breaking of unsteady waves on sheared currents
View Description Hide DescriptionIncipient breaking of unsteady waves on sheared currents is experimentally investigated. A new wavegeneration technique, based on the iterative frequencyfocusing concept with the consideration of effects of Doppler shift and current shear, is developed. The surface displacement, the wavelength, and the phase speed of waves at the breaking onset on shear currents are measured. It is found that the steepness of unsteady, incipient breaking waves is altered by the sign and magnitude of current shear (or vorticity). A current with a positive shear, as would be the case in a winddriven current, reduces the steepness of an unsteady incipient breaking wave. A negatively sheared current, such as the jetlike ebb current at a tide inlet, leads to steeper incipient breaking waves. The magnitude of reduction/increase in wave steepness is proportional to the strength of a current shear. In particular, a negative shear can alter the wave steepness more significantly in comparison to a positive shear of the same magnitude. Interestingly, the trend of cresttrough asymmetry with respect to the change of a current shear is in contrast to the limiting wave steepness. A positively sheared current can dramatically increase cresttrough asymmetry for unsteady waves. Dimensionless amplitudes of unsteady waves at incipient breaking are well correlated with surface current drifts. Positive/negative surface drifts lead to the reduced/increased dimensionless wave amplitudes. However, the change in dimensionless wave amplitude of unsteady waves is much smaller than that of steady waves.

Numerical simulation of binary liquid droplet collision
View Description Hide DescriptionA numerical investigation of binary droplet collision has been conducted. The complete process of the collision of two liquid droplets is dynamically simulated by solving the incompressible NavierStokes equations coupled with the convective equation of the level set function that captures the interface between the liquid and the gas phases. The simulations cover four major regimes of binary collision: bouncing, coalescence, reflexive separation, and stretching separation. For waterdroplets in air, the numerical results are compared with the experiments by and Ashgriz and Poo [J. Fluid Mech.221, 183 (1990)] on collision consequences. For hydrocarbon droplets in nitrogen gas, the simulated results are compared in detail with the timeresolved photographic images of the collision processes obtained by Qian and Law [J. Fluid Mech.331, 59 (1997)] in every collision regime. The present numerical results suggest that the mechanism of a bouncing collision is governed by the macroscopic dynamics. However, the fact that the present macroscopic numerical model is unable to capture the collision regime of coalescence after minor deformation supports the speculation that its mechanism is related to the microscopic dynamics. Furthermore, the transition from bouncing to coalescence collisions has been predicted and agrees well with the analytical model. The mechanism of satellitedropletformation for headon collision and stretching separation collision is also studied based on the detailed timeresolveddynamic simulation results. It is then confirmed that end pinching is the main cause of satelliteformation in headon collisions whereas the capillarywave instability becomes dominant in large impact parameter cases. In the case of an intermediate impact parameter, the effects of twisting and stretching due to the angular momentum and the inertia of the colliding droplets are significant for the satelliteformation.

Existence of receding and advancing contact lines
View Description Hide DescriptionWe study a solid plate plunging into or being withdrawn from a liquid bath to highlight the fundamental difference between the local behavior of an advancing or a receding contact line, respectively. It is assumed that the liquid partially wets the solid, making a finite contact angle in equilibrium. In our hydrodynamic description, which neglects the presence of the outer gas atmosphere, an advancing dynamic wetting line persists to arbitrarily high speeds. The receding wetting line, on the other hand, vanishes at a critical speed set by the competition between viscous and surface tension forces. In the advancing case, we apply existing matching techniques to the plunging plate geometry to significantly improve on existing theories. For the receding contact line, we demonstrate for the first time how the local contact line solution can be matched to the farfield meniscus. In doing so, we confirm our very recent criterion for the vanishing of the receding contact line, leading to the formation of a film covering the solid. The results of both the advancing and the receding cases are tested against simulations of the full model equations.

Stretching of material lines in shockaccelerated gaseous flows
View Description Hide DescriptionA Mach 1.2 planar shock wave impulsively accelerates one of five different configurations of heavygas cylinders surrounded by lighter gas (air), producing one or more pairs of interactingvortex columns. The interaction of the columns is investigated with planar laserinduced fluorescence in the plane normal to the axes of the cylinders. For the first time, we experimentally measure the early time stretching rate (in the first after shock interaction before the development of secondary instabilities) of material lines in shockaccelerated gaseous flows resulting from the RichtmyerMeshkov instability at Reynolds number and Schmidt number . The early time specific stretching rate exponent associated with the stretching of material lines is measured in these five configurations and compared with the numerical computations of Yang et al. [AIAA J.31, 854 (1993)] in some similar configurations and time range. The stretching rate is found to depend on the configuration and orientation of the gaseous cylinders, as these affect the refraction of the shock and thus vorticity deposition. Integral scale measurements fail to discriminate between the various configurations over the same time range, however, suggesting that integral measures are insufficient to characterize early time mixing in these flows.
 Viscous and NonNewtonian Flows

Stalactite growth as a freeboundary problem
View Description Hide DescriptionStalactites, the most familiar structures found hanging from the ceilings of limestone caves, grow by the precipitation of calcium carbonate from within a thin film of fluid flowing down their surfaces. We have recently shown [M. B. Short, J. C. Baygents, J. W. Beck, D. A. Stone, R. S. Toomey III, and R. E. Goldstein, “Stalactitegrowth as a freeboundary problem: A geometric law and its Platonic ideal,” Phys. Rev. Lett.94, 018501 (2005)] that the combination of thinfilmfluid dynamics,calcium carbonate chemistry, and carbon dioxide diffusion and outgassing leads to a local geometric growth law for the surface evolution which quantitatively explains the shapes of natural stalactites. Here we provide details of this freeboundary calculation, exploiting a strong separation of time scales among that for diffusion within the layer, contact of a fluid parcel with the growingsurface, and growth. When the flow rate, the scale of the stalactite, and the chemistry are in the ranges typically found in nature, the local growth rate is proportional to the local thickness of the fluid layer, itself determined by Stokes flow over the surface. Numerical studies of this law establish that a broad class of initial conditions is attracted to an ideal universal shape, whose mathematical form is found analytically. Statistical analysis of stalactite shapes from Kartchner Caverns (Benson, AZ) shows excellent agreement between the average shape of natural stalactites and the ideal shape. Generalizations of these results to nonaxisymmetric speleothems are discussed.

Optimal probes for withdrawal of uncontaminated fluid samples
View Description Hide DescriptionWithdrawal of fluid by a composite probe pushed against the face of a porous halfspace is modeled assuming incompressible Darcy flow. The probe is circular, of radius , with an inner sampling section of radius and a concentric outer guard probe . The porous rock in is saturated with fluid 1, and the region is saturated with fluid 2; the two fluids have the same viscosity. It is assumed that the interface between the two fluids is sharp and remains so as it moves through the rock. The pressure in the probe is lower than that of the pore fluid in the rock, so that the fluid interface is convected with the fluids towards the probe. This idealized axisymmetric problem is solved numerically, and it is shown that an analysis based on farfield spherical flow towards a point sink is a good approximation when the nondimensional depth of fluid 1 is large, i.e., . The inner sampling probe eventually produces pure fluid 2, and this technique has been proposed for sampling pore fluids in rock surrounding an oil well [A. Hrametz, C. Gardner, M. Wais, and M. Proett, U.S. Patent No. 6,301,959 B1 (16 October 2001)]. Fluid 1 is drilling fluid filtrate, which has displaced the original pore fluid (fluid 2), a pure sample of which is required. The time required to collect an uncontaminated sample of original pore fluid can be minimized by a suitable choice of the probe geometry [J. Sherwood, J. Fitzgerald and B. Hill, U.S. Patent No. 6,719,049 B2 (13 April 2004)]. It is shown that the optimal choice of depends on the depth of filtrate invasion and the volume of sample required.

Theory of shearinduced migration in dilute polymer solutions near solid boundaries
View Description Hide DescriptionIn this work, a continuum theory is developed for the behavior of flowing dilute polymer solutions near solid surfaces, using a beadspring dumbbell model of the dissolved polymer chains. Hydrodynamic interactions between the chains and the wall lead to migration away from the wall in shear flow. At steady state, this hydrodynamic effect is balanced by molecular diffusion; an analytical expression for the resulting concentration profile is derived. It is shown that the depletion layer thickness is determined by the normal stresses that develop in flow and can be much larger than the size of the polymer molecule. The transient development of this depletion layer is also studied, as well as the spatial development downstream from an entrance. Numerical and similarity solutions in these cases show that the developing concentration profile generally displays a maximum at an intermediate distance from the wall.
 Particulate, Multiphase, and Granular Flows

Kinetic theory for identical, frictional, nearly elastic disks
View Description Hide DescriptionWe develop kinetic theory for slightly frictional and nearly elastic disks. The tangential interaction is modeled by two parameters: a Coulomb friction coefficient and a tangential restitution coefficient. Assuming Maxwellian velocity distribution functions for both translational and rotational velocities, we derive exact expressions for the rates of dissipation of translational and rotational fluctuation energies per unit area. Setting the rotational dissipation rate to zero, as in a steady, homogeneous shearing flow, we find the ratio of the rotational temperature to the translational. In the case of small friction, this is used to determine an effective coefficient of normal restitution. In this way, the effects of small friction can be incorporated into the theory, thereby dispensing with the need to separately consider the complete balances for the momentum and the energy of the rotational motion.
 Instability and Transition

Experimental study of RichtmyerMeshkov instability induced by cylindrical shock waves
View Description Hide DescriptionThe paper describes the results of holographic interferometric flow visualization of the RichtmyerMeshkov instability induced by cylindrical shock waves propagating across cylindrical interfaces. Experiments were conducted in an annular coaxial vertical diaphragmless shock tube, which can produce converging cylindrical shock waves with minimum disturbances. The shock wave converged and interacted with a cylindrical soap bubble filled with He, Ne, air, Ar, Kr, Xe, or . The soap bubble was placed coaxially in the test section. The effects of density variation on the RichtmyerMeshkov instability for a wide range of Atwood numbers were determined. Pressure histories at different radii during the shock wave implosion and reflection from the center were measured. Doubleexposure holographic interferometry was used and the motion of the converging shock wave and its interaction with the gaseous interface were visualized. The variation of the pressure at the center with interface Atwood number for constant incident shock Mach number was studied. It is found that the dominant mechanism limiting the maximum pressure at the center of convergence is related to the instability of the converging shock wave induced by its interaction with the interface. A short time after the impulsive acceleration, the interface started deforming, and the growth of these perturbations is described. The results show that after diverging shock wave interaction, the reshocked cylindrical interfaces have a higher growth rate of the turbulent mixing zone than that of the reshocked interface in a plane geometry reported by previous works.

Instability and mixing flux in frontal displacement of viscous fluids from porous media
View Description Hide DescriptionThe goal of the present study is to investigate the instability of viscous fluid displacement by a less viscous one in a twodimensional channel, and to the determine the characteristics of displacement quality and entrapment zones. Experiments on miscible displacement of fluids in HeleShaw cells were conducted under microgravity conditions. Extensive direct numerical simulations allowed to investigate the sensitivity of the displacement process to the variation of values of the main governing parameters. Validation of the code was performed by comparing the results of model problems simulations with experiments and with the existing solutions published in literature. Numerical simulations allowed to explain new experimental results on the pear shape of fingers and periodical separation of their tip elements from the main body of displacing fluid. These separated blobs of less viscous fluid move much faster than the mean flow of the displaced viscous fluid. The results of numerical simulations processed on the basis of dimensions analysis allow to introduce parameters characterizing the quality of displacement and the mixing flux induced by instability. Functional forms describing additional mixing induced by displacement instability were developed. The influence of inhomogeneity of porous matrix on displacement instability is being investigated. The modified HeleShaw cell containing regular and randomized obstacles is used to study the effect of inhomogeneity on displacement instability. The result of numerical simulations as well as physical experiment shows that the presence of inhomogeneity of a definite length scale could stabilize unstable displacement and could destabilize a stable one.

On Holmboe’s instability for smooth shear and density profiles
View Description Hide DescriptionThe linear stability of a stratified shear flow for smooth density profiles is studied. This work focuses on the nature of the stability boundaries of flows in which both Kelvin–Helmholtz and Holmboe instabilities are present. For a fixed Richardson number the unstable modes are confined to finite bands between a smallest and a largest marginally unstable wavenumber. The results in this paper indicate that the stability boundary for small wavenumbers is comprised of neutral modes with phase velocity equal to the maximum/minimum wind velocity whereas the other stability boundary, for large wavenumbers, is comprised of singular neutral modes with phase velocity in the range of the velocity shear. We show how these stability boundaries can be evaluated without solving for the growth rate over the entire parameter space as was previously done. The results indicate further that there is a new instability domain that has not been previously noted in the literature. The unstable modes, in this new instability domain, appear for larger values of the Richardson number and are related to the higher harmonics of the internal gravity wave spectrum.