Volume 25, Issue 11, November 2013
Index of content:

An integrated kineticsbased Large Eddy Simulation (LES) approach for soot evolution in turbulent reacting flows is applied to the simulation of a Pratt & Whitney aircraft gas turbine combustor, and the results are analyzed to provide insights into the complex interactions of the hydrodynamics, mixing, chemistry, and soot. The integrated approach includes detailed models for soot, combustion, and the unresolved interactions between soot, chemistry, and turbulence. The soot model is based on the Hybrid Method of Moments and detailed descriptions of soot aggregates and the various physical and chemical processes governing their evolution. The detailed kinetics of jet fuel oxidation and soot precursor formation is described with the Radiation Flamelet/Progress Variable model, which has been modified to account for the removal of soot precursors from the gasphase. The unclosed filtered quantities in the soot and combustion models, such as source terms, are closed with a novel presumed subfilter PDF approach that accounts for the high subfilter spatial intermittency of soot. For the combustor simulation, the integrated approach is combined with a Lagrangian parcel method for the liquid spray and stateoftheart unstructured LES technology for complex geometries. Two overall fueltoair ratios are simulated to evaluate the ability of the model to make not only absolute predictions but also quantitative predictions of trends. The Pratt & Whitney combustor is a RichQuenchLean combustor in which combustion first occurs in a fuelrich primary zone characterized by a large recirculation zone. Dilution air is then added downstream of the recirculation zone, and combustion continues in a fuellean secondary zone. The simulations show that large quantities of soot are formed in the fuelrich recirculation zone, and, furthermore, the overall fueltoair ratio dictates both the dominant soot growth process and the location of maximum soot volume fraction. At the higher fueltoair ratio, the maximum soot volume fraction is found inside the recirculation zone; at the lower fueltoair ratio, turbulent fluctuations in the mixture fraction promote the oxidation of soot inside the recirculation zone and suppress the accumulation of a large soot volume fraction. Downstream, soot exits the combustor in intermittent fuelrich pockets that are not mixed during the injection of dilution air and subsequent secondary fuellean combustion. At the higher fueltoair ratio, the frequency of these fuelrich pockets is increased, leading to higher soot emissions from the combustor. Quantitatively, the soot emissions from the combustor are overpredicted by about 50%, which is a substantial improvement over previous works utilizing RANS to predict such emissions. In addition, the ratio between the two fueltoair ratios predicted by LES compares very favorably with the experimental measurements. Furthermore, soot growth is dominated by an acetylenebased pathway rather than an aromaticbased pathway, which is usually the dominant mechanism in nonpremixed flames. This finding is the result of the interactions between the hydrodynamics, mixing, chemistry, and soot in the recirculation zone and the resulting residence times of soot at various mixture fractions (compositions), which are not the same in this complex recirculating flow as in nonpremixed jet flames.
 ARTICLES


Micro and Nanofluid Mechanics

Maxwell boundary condition and velocity dependent accommodation coefficient
View Description Hide DescriptionA modification of Maxwell's boundary condition for the Boltzmann equation is developed that allows to incorporate velocity dependent accommodation coefficients into the microscopic description. As a first example, it is suggested to consider the wallparticle interaction as a thermally activated process with three parameters. A simplified averaging procedure leads to jump and slip boundary conditions for hydrodynamics. Coefficients for velocity slip, temperature jump, and thermal transpiration flow are identified and compared with those resulting from the original Maxwell model and the CercignaniLampis model. An extension of the model leads to temperature dependent slip and jump coefficients.

Coupled continuum and molecular model of flow through fibrous filter
View Description Hide DescriptionA coupled approach combining the continuum boundary singularity method (BSM) and the molecular direct simulation Monte Carlo (DSMC) is developed and validated using TaylorCouette flow and the flow about a single fiber confined between two parallel walls. In the proposed approach, the DSMC is applied to an annular region enclosing the fiber and the BSM is employed in the entire flow domain. The parameters used in the DSMC and the coupling procedure, such as the number of simulated particles, the cell size, and the size of the coupling zone are determined by inspecting the accuracy of pressure drop obtained for the range of Knudsen numbers between zero and unity. The developed approach is used to study flowfield of fibrous filtration flows. It is observed that in the partialslip flow regime, Kn ⩽ 0.25, the results obtained by the proposed coupled BSMDSMC method match the solution by BSM combined with the heuristic partialslip boundary conditions. For transition moleculartocontinuum Knudsen numbers, 0.25 < Kn ⩽ 1, the difference in pressure drop and velocity between these two approaches is significant. This difference increases with the Knudsen number that confirms the usefulness of coupled continuum and molecular methods in numerical modeling of transition low Reynolds number flows in fibrous filters.

Interfacial Flows

The influence of inertia and charge relaxation on electrohydrodynamic drop deformation
View Description Hide DescriptionWe quantify the transient deformation of a droplet immersed in a weakly conductive (leaky dielectric) fluid upon exposure to a uniform DC electric field. Capillary forces are assumed to be sufficiently large that the drop only slightly deviates from its equilibrium spherical shape. In particular, we account for transient (or linear) fluid inertia via the unsteady Stokes equations, and also account for a finite electrical relaxation time over which the drop interface charges. The temporal droplet deformation is governed by two dimensionless groups: (i) the ratio of capillary to momentum diffusion time scales: an Ohnesorge number Oh and (ii) the ratio of charge relaxation to momentum diffusion time scales, which we denote by Sa. If charge and momentum relaxation occur quickly compared to interface deformation, Sa ≪ 1 and Oh ≫ 1 for the droplet and medium, a monotonic deformation is acquired. In contrast, Sa > 1 and Oh < 1 for either phase can lead to a nonmonotonic development in the deformation. Numerical values for the deformation are calculated by inverting an analytical expression obtained in the Laplace domain, and are corroborated by asymptotic expansions at early and late times. The droplet and medium behave as perfect dielectrics at early times, which always favors an initial prolate (parallel to the applied field) deformation. As a consequence, for a final oblate (normal to the applied field) deformation, there is a shape transition from prolate to oblate at intermediate times. This transition is caused by the accumulation of sufficient charge at the interface to generate electrical and viscous shear stresses. Notably, after the transition, there may be an “overshoot” in the deformation, i.e., the magnitude exceeds its steadystate value, which is proceeded by an algebraic tail describing the arrival towards the final, steady deformation. Our work demonstrates that transient inertia or a nonzero electrical relaxation time can yield nonmonotonic electrohydrodynamic drop deformation.

Dynamic coupling in Cooker's sloshing experiment with baffles
View Description Hide DescriptionThis paper investigates the dynamic coupling between fluid sloshing and the motion of the vessel containing the fluid, for the case when the vessel is partitioned using nonporous baffles. The vessel is modelled using Cooker's sloshing configuration [M. J. Cooker, “Water waves in a suspended container,” Wave Motion20, 385–395 (1994)]. Cooker's configuration is extended to include n − 1 nonporous baffles which divide the vessel into n separate fluid compartments each with a characteristic length scale. The problem is analysed for arbitrary fill depth in each compartment, and it is found that a multitude of resonance situations can occur in the system, from 1 : 1 resonances to (n + 1)−fold 1 : 1: ⋯ : 1 resonances, as well as ℓ: m: ⋯ : n for natural numbers ℓ, m, n, depending upon the system parameter values. The conventional wisdom is that the principle role of baffles is to damp the fluid motion. Our results show that in fact without special consideration, the baffles can lead to enhancement of the fluid motion through resonance.

Comparison of NavierStokes simulations with longwave theory: Study of wetting and dewetting
View Description Hide DescriptionThe classical longwave theory (also known as lubrication approximation) applied to fluid spreading or retracting on a solid substrate is derived under a set of assumptions, typically including small slopes and negligible inertial effects. In this work, we compare the results obtained by using the longwave model and by simulating directly the full twophase NavierStokes equations employing a volumeoffluid method. In order to isolate the influence of the small slope assumption inherent in the longwave theory, we present a quantitative comparison between the two methods in the regime where inertial effects and the influence of gas phase are negligible. The flow geometries that we consider include wetting and dewetting drops within a broad range of equilibrium contact angles in planar and axisymmetric geometries, as well as liquid rings. For perfectly wetting spreading drops we find good quantitative agreement between the models, with both of them following rather closely Tanner's law. For partially wetting drops, while in general we find good agreement between the two models for small equilibrium contact angles, we also uncover differences which are particularly evident in the initial stages of evolution, for retracting drops, and when additional azimuthal curvature is considered. The contracting rings are also found to evolve differently for the two models, with the main difference being that the evolution occurs on the faster time scale when the longwave model is considered, although the ring shapes are very similar between the two models.

Capillary instability of a viscous liquid thread in a cylindrical tube
View Description Hide DescriptionWe study axisymmetric breakup and drop formation of a liquid thread confined in a cylindrical tube by using a long wave model. The limit with large viscosity contrast is investigated, consisting of a highly viscous core and a less viscous annulus. Satellite drop formation near pinching is found to be dependent on the location of the tube wall, while inertia is found to have little effect. It is also shown that main drops tend to form so called “plug with collar” structures when the tube wall is close to the thread interface so that rings of fluid are trapped inside the plugs. Finally, our numerical simulations show that a double cone structure occurs on the breakup and the cone angles are fixed, even though selfsimilar solutions may not exist.

The stability of a static liquid column pulled out of an infinite pool
View Description Hide DescriptionWe study the stability of a static liquid column rising from an infinite pool, with its top attached to a horizontal plate suspended at a certain height above the pool's surface. Two different models are employed for the column's contact line. Model 1 assumes that the contact angle always equals Young's equilibrium value. Model 2 assumes a functional dependence between the contact angle and the velocity of the contact line, and we argue that, if this dependence involves a hysteresis interval, linear perturbations cannot move the contact line. It is shown that, within the framework of Model 1, all liquid columns are unstable. In Model 2, both stable and unstable columns exist (the former have larger contact angles θ and/or larger heights H). For Model 2, the marginal stability curve on the (θ, H)plane is computed. The mathematical results obtained imply that, if the plate to which a stable liquid column is attached is slowly lifted up, the column's contact line remains pinned while the contact angle is decreasing. Once it reaches the lower boundary of the hysteresis interval, the column breaks down.

Oscillations of a sessile droplet in open air
View Description Hide DescriptionThe open system consisting of a sessile drop, a neutral gas, and a substrate is analyzed by numerical methods. The mode with constant contact angle is considered. The model takes into account evaporation from drop surface, free and forced convection in gas, buoyancy, and Marangoni effect in the liquid phase. It was established that every considered mechanical and thermodynamical disturbance of the system leads to the drop surface oscillations, and thus a drop in an open air oscillates almost inevitably. The displacement of the liquidgas interface due to oscillations is analyzed in terms of its impact on the accuracy of measurement of the surface tension by sessile drop method.

Effect of rising motion on the damped shape oscillations of drops and bubbles
View Description Hide DescriptionThe objective of this work is to determine the effect of the rising motion on the dynamics of inertial shape oscillations of drops and bubbles. We have carried out axisymmetric direct numerical simulations of an ascending drop (or bubble) using a levelset method. The drop is initially elongated in the vertical direction and therefore performs shape oscillations. The analysis is based on the decomposition of the interface into spherical harmonics, the time evolutions of which are processed to obtain the frequency and the damping rate of the oscillations. As the drop accelerates, its shape flattens and oscillations no longer take place around a spherical equilibrium shape. This causes the eigenmode of oscillations to change, which results in the appearance of spherical harmonics of high order that all oscillate at the same frequency. For both drops and bubbles, the frequency, which remains controlled by the potential flow, slightly decreases with the rising velocity. The damping rate of drops, which is controlled by the dissipation within boundary layers at the interface, strongly increases with the rising velocity. At terminal velocity, the damping rate of bubbles, which results from the dissipation by the potential flow associated with the oscillating motion, remains close to that of a nonrising bubble. During the transient, the rate of deformation of the equilibrium shape of bubbles can be comparable to the oscillation frequency, which causes complex evolutions of the shape. These results extend the description of shape oscillations to common situations where gravity plays a role. In particular, the present conclusions are useful to interpret experimental results where the effect of the rising motion is often combined with that of surfactant.

Mechanism of bubble detachment from vibrating walls
View Description Hide DescriptionWe discovered a previously unobserved mechanism by which air bubbles detach from vibrating walls in glasses containing water. Chaotic oscillation and subsequent water jets appeared when a wall vibrated at greater than a critical level. Wave forms were developed at waterair interface of the bubble by the wall vibration, and water jets were formed when sufficiently grown wavecurvatures were collapsing. Droplets were pinched off from the tip of jets and fell to the surface of the glass. When the solidair interface at the bubblewall attachment point was completely covered with water, the bubble detached from the wall. The water jets were mainly generated by subharmonic waves and were generated most vigorously when the wall vibrated at the volume resonant frequency of the bubble. Bubbles of specific size can be removed by adjusting the frequency of the wall's vibration.

Vortices catapult droplets in atomization
View Description Hide DescriptionA droplet ejection mechanism in planar twophase mixing layers is examined. Any disturbance on the gasliquid interface grows into a KelvinHelmholtz wave, and the wave crest forms a thin liquid film that flaps as the wave grows downstream. Increasing the gas speed, it is observed that the film breaks up into droplets which are eventually thrown into the gas stream at large angles. In a flow where most of the momentum is in the horizontal direction, it is surprising to observe these large ejection angles. Our experiments and simulations show that a recirculation region grows downstream of the wave and leads to vortex shedding similar to the wake of a backwardfacing step. The ejection mechanism results from the interaction between the liquid film and the vortex shedding sequence: a recirculation zone appears in the wake of the wave and a liquid film emerges from the wave crest; the recirculation region detaches into a vortex and the gas flow over the wave momentarily reattaches due to the departure of the vortex; this reattached flow pushes the liquid film down; by now, a new recirculation vortex is being created in the wake of the wave—just where the liquid film is now located; the liquid film is blown up from below by the newly formed recirculation vortex in a manner similar to a bagbreakup event; the resulting droplets are catapulted by the recirculation vortex.

Oblique wave trapping by porous and flexible structures in a twolayer fluid
View Description Hide DescriptionTrapping of obliquely incident surface waves by permeable flexible barriers placed near a vertical rigid wall in a twolayer fluid having free surface and an interface is studied for both surfacepiercing and bottomstanding partial barriers. For the surfacepiercing permeable flexible barrier, the barrier is assumed to be fixed near the free surface and is free at the submerged end. On the other hand, for the bottom standing permeable flexible barrier, the barrier is assumed to be fixed at the bottom and the other end is free. As special cases of the permeable flexible barrier, the results associated with surfacepiercing and bottomstanding permeable membrane barriers are obtained by assuming that the two ends of the barriers are fixed. Appropriate continuity conditions are used to deal with the interfacepiercing flexible/membrane barriers. The mathematical problem is handled for solution using a generalized orthogonal relation suitable for twolayer fluid along with the least square approximation method. Explicit relations are derived to ensure full reflection by porous flexible barriers of any configuration placed near a vertical rigid wall for waves in surface and internal modes, which are validated through numerical computations in various cases. The effect of critical angle of incidence on wave reflection and trapping by barriers, surface and interface wave elevations, deflection of the flexible barrier under wave action, pressure distribution on the barrier, wave loads on barrier and rigid wall are analyzed. The finding of the present study is likely to play a significant role in the design of marine facilities with less wave force on the infrastructure. The present concept and methodology can be easily extended to similar problems in acousticstructure interactions.

Dynamics of a prelens tear film after a blink: Model, evolution, and rupture
View Description Hide DescriptionA mathematical model is developed to investigate the dynamics and rupture of a prelens tear film on a contact lens. The contact lens is modeled as a saturated porous medium of constant, finite thickness and is described by the DarcyBrinkman equations with stressjump condition at the interface. The model incorporates the influence of capillarity, gravitational drainage, contact lens properties such as the permeability, the porosity, and the thickness of the contact lens on the evolution and rupture of a prelens tear film, when the eyelid has opened after a blink. Two models are derived for the evolution of a prelens tear film thickness using lubrication theory and are solved numerically; the first uses shearfree surface condition and the second, the tangentially immobile free surface condition. The results reveal that life span of a prelens tear film is longer on a thinner contact lens for all values of permeability and porosity parameter considered. An increase in permeability of contact lens, porosity or stressjump parameter increases the rate of thinning of the film and advances the rupture time. The viscousviscous interaction between the porous contact lens and the prelens tear film increases the resistance offered by the frictional forces to the rate of thinning of prelens tear film. This slows down the thinning process and hence delays the rupture of the film as compared to that predicted by the models of Nong and Anderson [SIAM. J. Appl. Math.70, 2771–2795 (2010)] derived in the framework of Darcy model.

Viscous and NonNewtonian Flows

Temperaturedriven motion of liquid crystals confined in a microvolume
View Description Hide DescriptionThe theoretical description of the reorientational dynamics in microsized liquid crystal (LC) cell, where the nematic sample is confined by two horizontal and two lateral surfaces, under the influence of a temperature gradient ∇T, caused by a laser beam focused on the bounding surface with and without the orientational defect, whereas the rest of the bounding surfaces of the LC cell are kept at constant temperature, has been presented. Our calculations, based on the appropriate nonlinear extension of the classical EricksenLeslie theory, show that due to interaction between ∇T and the gradient of the director field in the LC sample, a thermally excited vortical fluid flow is maintained in the vicinity of the orientational defect, with the motion in the positive sense (clockwise) around the middle part of that defect. In the case of the same LC cell, but without the orientational defect on the lower hotter boundary, the heating regime can also produce the vortical flow in the vicinity of the lower boundary, but with the motion in the negative sense (anticlockwise) around the middle part of that boundary. At that, the second vortex is characterized by a much slower speed than the vortical flow in the first case.

Particulate, Multiphase, and Granular Flows

Segregation and mixture profiles in dense, inclined flows of two types of spheres
View Description Hide DescriptionWe study dry flows of two types of spheres down an inclined, rigid, bumpy bed in the absence of sidewalls. The flow is assumed to be steady and uniform in all but the direction normal to the free surface, collisions between particles are dissipative, and the sizes and masses of the particles are not too different. We restrict our analysis to dense flows and use an extension of kinetic theory to predict the concentration of the mixture and the profile of mixture velocity. A kinetic theory for a binary mixture of nearly elastic spheres that do not differ by much in their size or mass is employed to predict profiles of the concentration fraction of one type of sphere. We also determine the ratio of the radii and of the masses of the two species for which there is no segregation. We compare the predictions of the theory to the results of numerical simulations.

Uniform selfdiffusion in a granular gas
View Description Hide DescriptionA granular gas composed of inelastic hard spheres or disks in the homogeneous cooling state is considered. Some of the particles are labeled and their number density exhibits a timeindependent linear profile along a given direction. As a consequence, there is a uniform flux of labeled particles in that direction. It is shown that the inelastic BoltzmannEnskog kinetic equation has a solution describing this selfdiffusion state. Approximate expressions for the transport equation and the distribution function of labeled particles are derived. The theoretical predictions are compared with simulation results obtained using the direct simulation Monte Carlo method to generate solutions of the kinetic equation. A fairly good agreement is found.

Flow structures and their contribution to turbulent dispersion in a randomly packed porous bed based on particle image velocimetry measurements
View Description Hide DescriptionAn experimental study was undertaken to explore the evolution of flow structures and their characteristics within a randomly packed porous bed with particular attention to evaluating turbulent scalar dispersion. A low aspect ratio bed of 4.67 (bed width to spherical solid phase particle diameter) with fluid phase refractive index matched to that of the solid phase was used in order to obtain time resolved two component particle image velocimetry data. Results are based on detailed velocity vector maps obtained at selected pores near the bed center. Pore, or large scale, regions that are associated with the mean flow were identified based on Reynolds decomposed velocity fields, while smaller scale structures embedded within pore scale regions were identified and quantified by combining large eddy scale decomposition and swirling strength analysis. The velocity maps collected in distinctive pore geometries showed presence of three types of flow regions that display very different mean flow conditions, described as regions with tortuous channel like flow, high fluid momentum jet like regions, and low fluid momentum recirculating regions. The major portion of pore space is categorized as tortuous channel flow. Time series of instantaneous velocity field maps were used to identify mean and turbulent flow structures based on their spatial scales in the different regions. Even though regions exhibit varied Eulerian statistics, they show very similar eddy characteristics such as spinning rate and number density. The integral scale eddy structures show nearly a linear rate of increase in their rotation rate with increasing pore Reynolds number, indicating a linear decrease in their time scales. The convective velocities of these eddies are shown to reach an asymptotic limit at high pore Reynolds numbers, unique for each flow region. Detailed Eulerian statistics for the identified flow regions are presented and are used to predict mechanical dispersion through the use of estimated Lagrangian statistics. Contributions from each of the flow regions are presented and the recirculating regions are shown to contribute most to the overall longitudinal dispersion, whereas the tortuous channel regions contribute most to the transverse dispersion. The overall dispersion estimates agree well with global data in the limit of high Schmitt number.

Laminar Flows

Groove optimization for drag reduction
View Description Hide DescriptionOptimal shapes of laminar, drag reducing longitudinal grooves in a pressure driven flow have been determined. It has been shown that such shapes can be characterized using reduced geometry models involving only a few Fourier modes. Two classes of grooves have been studied, i.e., the equaldepth grooves, which have the same height and depth, and the unequaldepth grooves. It has been shown that the optimal shape in the former case can be approximated by a certain universal trapezoid. There exists an optimum depth in the latter case and this depth, combined with the corresponding groove shape, defines the optimal geometry; this shape is wellapproximated by a Gaussian function. Drag reduction due to the use of the optimal grooves has been determined. The analysis has been extended to kinematically driven flows. It has been shown that in this case the longitudinal grooves always increase the flow resistance.

RayleighBénard convection at high Rayleigh number and infinite Prandtl number: Asymptotics and numerics
View Description Hide DescriptionThe problem of fast viscous steady RayleighBénard convection in a rectangular enclosure is revisited using asymptotic and numerical methods. There are two generic cases: in the first, there is zero shear stress at all boundaries; in the second, there is zero shear stress at the vertical boundaries, but no slip at the horizontal ones. For the first case, we reconcile our new numerical solutions to the full equations with earlier asymptotic results for large Rayleigh number and effectively infinite Prandtl number. For the second case, we first derive the corresponding asymptotic theory and then reconcile it also with the relevant full numerical solutions. However, the latter also indicate behavior which the asymptotic theory does not predict, for Rayleigh numbers in excess of just over 10^{6} and aspect ratios in excess of around 1.1.

Instability and Transition

Incident shock Mach number effects on RichtmyerMeshkov mixing in a heavy gas layer
View Description Hide DescriptionExperiments were performed at the horizontal shock tube facility at Los Alamos National Laboratory to study the effect of incident shock Mach number (M) on the development of RichtmyerMeshkov instability after a shock wave impulsively accelerates a varicoseperturbed, heavygas curtain. Three cases of incident shock strength were experimentally investigated: M = 1.21, 1.36, and 1.50. We discuss the state of the mixing and the mechanisms that drive the mixing at both large and small scales by examining the time evolution of 2D density fields derived from quantitative planar laserinduced fluorescence measurements. Several differences in qualitative flow features are identified as a result of Mach number variation, and differences in vortex interaction, observed using particle image velocimetry, play a critical role in the development of the flow field. Several quantities, including mixing layer width, mixing layer area, interface length, instantaneous mixing rate, the density selfcorrelation parameter, probability density functions of the density field, and mixing progress variables are examined as a function of time. These quantities are also examined versus time scaled with the convection velocity of the mixing layer. A higher incident Mach number yields greater mixing uniformity at a given downstream location, while a lower Mach number produces a greater amount of total mixing between the two gases, suggesting possible implications for optimization in applications with confined geometries.
