Volume 26, Issue 5, May 2014
Index of content:

Rotating Rayleigh–Bénard convection in water is studied in direct numerical simulations, where the temperature dependence of the viscosity, the thermal conductivity, and the density within the buoyancy term is taken into account. In all simulations, the arithmetic mean of the lowest and highest temperature in the system equals 40 °C, corresponding to a Prandtl number of Pr = 4.38. In the nonrotational case, the Rayleigh number Ra ranges from 10^{7} to 1.16 × 10^{9} and temperature differences Δ up to 70 K are considered, whereas in the rotational case the inverse Rossby number range from 0.07 ⩽ 1/Ro ⩽ 14.1 is studied for Δ = 40 K with the focus on Ra = 10^{8}. The nonOberbeck–Boussinesq (NOB) effects in water are reflected in an up to 5.5 K enhancement of the center temperature and in an up to 5% reduction of the Nusselt number. The top thermal and viscous boundary layer thicknesses increase and the bottom ones decrease, while the sum of the corresponding top and bottom thicknesses remains as in the classical Oberbeck–Boussinesq (OB) case. Rotation applied to NOB thermal convection reduces the central temperature enhancement. Under NOB conditions the top (bottom) thermal and viscous boundary layers become equal for a slightly larger (smaller) inverse Rossby number than in the OB case. Furthermore, for rapid rotation the thermal bottom boundary layers become thicker than the top ones. The Nusselt number normalized by that in the nonrotating case depends similarly on 1/Ro in both, the NOB and the OB cases. The deviation between the Nusselt number under OB and NOB conditions is minimal when the thermal and viscous boundary layers are equal.
 LETTERS


A loworder decomposition of turbulent channel flow via resolvent analysis and convex optimization
View Description Hide DescriptionWe combine resolventmode decomposition with techniques from convex optimization to optimally approximate velocity spectra in a turbulent channel. The velocity is expressed as a weighted sum of resolvent modes that are dynamically significant, nonempirical, and scalable with Reynolds number. To optimally represent direct numerical simulations (DNS) data at friction Reynolds number 2003, we determine the weights of resolvent modes as the solution of a convex optimization problem. Using only 12 modes per wallparallel wavenumber pair and temporal frequency, we obtain close agreement with DNSspectra, reducing the wallnormal and temporal resolutions used in the simulation by three orders of magnitude.

The deviation from parallel shear flow as an indicator of linear eddyviscosity model inaccuracy
View Description Hide DescriptionA marker function designed to indicate in which regions of a generic flow field the results from linear eddyviscosity turbulence models are plausibly inaccurate is introduced. The marker is defined to identify regions that deviate from parallel shear flow. For two different flow fields it is shown that these regions largely coincide with regions where the prediction of the Reynolds stress divergence is inaccurate. The marker therefore offers a guideline for interpreting results obtained from Reynoldsaveraged NavierStokes simulations and provides a basis for the further development of turbulence modelform uncertainty quantification methods.

The sidewalllocalized mode in a resonant precessing cylinder
View Description Hide DescriptionWe investigate, via direct numerical simulation using a finiteelement method, the precessionally driven flow of a homogeneous fluid confined in a fluidfilled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. Our numerical simulation, after validating with the asymptotic analytical solution for a weakly precessing cylinder and with the constructed exact solution for the strongly nonlinear problem, focuses on the strongly precessing flow at asymptotically small Ekman numbers. An unusual form of the resonant precessing flow is found when the precessing rate is sufficiently large and the corresponding nonlinearity is sufficiently strong. The nonlinear precessing flow is marked by a sidewalllocalized nonaxisymmetric traveling wave and a walllocalized axisymmetric shear together with an overwhelmingly dominant interior rigidbody rotation whose direction and magnitude substantially reduce the angular momentum of the rotating fluid system.

Curvature suppresses the RayleighTaylor instability
View Description Hide DescriptionThe dynamics of a thin liquid film on the underside of a curved cylindrical substrate is studied. The evolution of the liquid layer is investigated as the film thickness and the radius of curvature of the substrate are varied. A dimensionless parameter (a modified Bond number) that incorporates both geometric parameters, gravity, and surface tension is identified, and allows the observations to be classified according to three different flow regimes: stable films, films with transient growth of perturbations followed by decay, and unstable films. Experiments and linear stability theory confirm that below a critical value of the Bond number curvature of the substrate suppresses the RayleighTaylor instability.

Extreme waves induced by strong depth transitions: Fully nonlinear results
View Description Hide DescriptionRecent studies on freesurface gravity waves over uneven bathymetries have shown that “rogue” waves can be triggered by strong depth variations. This phenomenon is here studied by means of spectral simulations of the freesurface Euler equations. We focus on the case of a random, onedirectional wave field with prescribed statistics propagating over a submerged step, and consider different depth variations, up to an almost deeptoshallow transition (k p H ≈ 1.8 − 0.78, where k p is the characteristic wavenumber and H the water depth). Strongly nonGaussian statistics are observed in a region localized around the depth transition, beyond which they settle rapidly on the steady statistical state of finitedepth random wave fields. Extreme fluctuations are enhanced by stronger depth variations. Freakwave formation is interpreted as a general signature of outofequilibrium dynamics, associated with the spectral settling from the deepwater to the finitedepth equilibrium. We also document that during such a transition the wave spectrum shows remarkable similarity with Phillips ω^{−5}law for the strongest depth variations considered.

 ARTICLES

 Biofluid Mechanics

Propulsive performance of unsteady tandem hydrofoils in an inline configuration
View Description Hide DescriptionExperiments are reported on the behavior of two hydrofoils arranged in an inline configuration as they undergo prescribed pitching motions over a wide range of phase lags and spacings between the foils. It is found that the thrust production and propulsive efficiency of the upstream foil differed from that of an isolated one only for relatively closely spaced foils, and the effects attenuated rapidly with increasing spacing. In contrast, the performance of the downstream foil depends strongly on the streamwise spacing and phase differential between the foils for all cases considered, and the thrust and propulsive efficiency could be as high as 1.5 times or as low as 0.5 times those of an isolated foil. Particle image velocimetry reveals how the wake interactions lead to these variations in propulsive performance, where a coherent mode corresponds to enhanced performance, and a branched mode corresponds to diminished performance.
 Micro and Nanofluid Mechanics

Capturing nonequilibrium phenomena in rarefied polyatomic gases: A highorder macroscopic model
View Description Hide DescriptionA highorder macroscopic model for the accurate description of rarefied polyatomic gas flows is introduced based on a kinetic equation of BhatnagarGrossKrook (BGK)type, where the different energy exchange processes are accounted for by two collision terms. The order of magnitude method is applied to the primary moment equations to acquire the optimized moment definitions and the final scaled set of Grad's 36 moment equations for polyatomic gases. The two Knudsen numbers of the system are used for model reduction in terms of their powers, which yields a wide range of different reduced systems, a total of 13 different orders. These include, at lower order, a modification of the NavierStokesFourier (NSF) equations which shows considerable extended range of validity in comparison to the classical NSF equations. The highest order of accuracy considered gives a set of 18 regularized partial differential equations (PDEs) (R18). Attenuation and speed of linear waves are studied as the first application of the many sets of equations. For frequencies where the internal degrees of freedom are effectively frozen, the equations reproduce the behavior of monatomic gases.

Knudsen heat capacity
View Description Hide DescriptionWe present a “Knudsen heat capacity” as a more appropriate and useful fluid property in micro/nanoscale gas systems than the constant pressure heat capacity. At these scales, different fluid processes come to the fore that are not normally observed at the macroscale. For thermodynamic analyses that include these Knudsen processes, using the Knudsen heat capacity can be more effective and physical. We calculate this heat capacity theoretically for nonideal monatomic and diatomic gases, in particular, helium, nitrogen, and hydrogen. The quantum modification for para and ortho hydrogen is also considered. We numerically model the Knudsen heat capacity using molecular dynamics simulations for the considered gases, and compare these results with the theoretical ones.

Scale effects in gas nano flows
View Description Hide DescriptionMost previous studies on gas transport in nanoscale confinements assume dynamic similarity with rarefied gas flows, and employ kinetic theory based models. This approach is incomplete, since it neglects the van der Waals forces imposed on gas molecules by the surfaces. Using threedimensional molecular dynamics (MD) simulations of force driven gas flows, we show the significance of wall force field in nanoscale confinements by defining a new dimensionless parameter (B) as the ratio of the wall forcepenetration length to the channel height. Investigation of gas transport in different nanochannels at various Knudsen numbers show the importance of wall force field for finite B values, where the dynamic similarity between the rarefied and nanoscale gas flows break down. Comparison of MD results employing molecularly structured threedimensional walls versus reflection of gas molecules from a twodimensional planar surface with Maxwell distribution show that the nanoconfinement effects cannot be resolved by the latter approach, frequently used in kinetic theory calculations. Molecularly structured walls determine the bulk flow physics by setting a proper tangential momentum accommodation coefficient, and they also determine the transport in the near wall region. Gas nanoflows with finite B exhibit significant differences in the local density and velocity profiles, affecting the mass flow rate and the formation of Knudsen's minimum in nanochannels.

Flow rate through microfilters: Influence of the pore size distribution, hydrodynamic interactions, wall slip, and inertia
View Description Hide DescriptionWe examine the fluid mechanics of viscous flow through filters consisting of perforated thin plates. We classify the effects that contribute to the hydraulic resistance of the filter. Classical analyses assume a single pore size and account only for filter thickness. We extend these results to obtain an analytical formula for the pressure drop across the microfilter versus the flow rate that accounts for the nonuniform distribution of pore sizes, the hydrodynamic interactions between the pores given their layout pattern, and wall slip. Further, we discuss inertial effects and their order of scaling.
 Interfacial Flows

Tear film dynamics with evaporation, wetting, and timedependent flux boundary condition on an eyeshaped domain
View Description Hide DescriptionWe study tear film dynamics with evaporation on a wettable eyeshaped ocular surface using a lubrication model. The mathematical model has a timedependent flux boundary condition that models the cycles of tear fluid supply and drainage; it mimics blinks on a stationary eyeshaped domain. We generate computational grids and solve the nonlinear governing equations using the OVERTURE computational framework. In vivo experimental results using fluorescent imaging are used to visualize the influx and redistribution of tears for an open eye. Results from the numerical simulations are compared with the experiment. The model captures the flow around the meniscus and other dynamic features of human tear film observed in vivo.

Numerical simulation of drop and bubble dynamics with soluble surfactant
View Description Hide DescriptionNumerical computations are presented to study the effect of soluble surfactant on the deformation and breakup of an axisymmetric drop or bubble stretched by an imposed linear strain flow in a viscous fluid. At the high values of bulk Peclet number Pe in typical fluidsurfactant systems, there is a thin transition layer near the interface in which the surfactant concentration varies rapidly. The large surfactant gradients are resolved using a fast and accurate “hybrid” numerical method that incorporates a separate, singular perturbation analysis of the dynamics in the transition layer into a full numerical solution of the free boundary problem. The method is used to investigate the dependence of drop deformation on parameters that characterize surfactant solubility. We also compute resolved examples of tipstreaming, and investigate its dependence on parameters such as flow rate and bulk surfactant concentration.

A study of dynamic contact angles of shearthickening powerlaw fluids
View Description Hide DescriptionWe study the dynamic wetting of shearthickening powerlaw fluids in a liquidsolidgas contact system. In the previous model based on hydrodynamic analysis, microscopic effects near the contact line are neglected. In this work, we adopt two different physical models, slip boundary model and molecular force model, to incorporate microscopic effects and relieve the stress singularity at the moving contact line in hydrodynamics analysis. The two models, which are mathematically equivalent for Newtonian fluids, lead to different results on the dependence of the liquid's dynamic contact angle on its moving speed in both complete wetting and partial wetting cases. By comparing with experiments, we find that the slip boundary model matches the experiments better than the previous model and the molecular force model.

Surface wave dynamics in orbital shaken cylindrical containers
View Description Hide DescriptionBe it to aerate a glass of wine before tasting, to accelerate a chemical reaction, or to cultivate cells in suspension, the “swirling” (or orbital shaking) of a container ensures good mixing and gas exchange in an efficient and simple way. Despite being used in a large range of applications this intuitive motion is far from being understood and presents a richness of patterns and behaviors which has not yet been reported. The present research charts the evolution of the waves with the operating parameters identifying a large variety of patterns, ranging from single and multiple crested waves to breaking waves. Free surface and velocity fields measurements are compared to a potential sloshing model, highlighting the existence of various flow regimes. Our research assesses the importance of the modal response of the shaken liquids, laying the foundations for a rigorous mixing optimization of the orbital agitation in its applications.

Origin of ejecta in the water impact problem
View Description Hide DescriptionThis work presents the analysis of the flow resulting from the flat plate impact on the surface of an incompressible viscous liquid at zero deadrise angle—of particular interest is the flow structure near the plate edge, r → 0, evolving at early times, t → 0. The deduced mathematical formulation proves to be of a singular perturbation type with the underlying governing equations having a linear structure. The key goals here are to elucidate the effects of viscosity and surface tension, which turn out to contribute to the solution at the leading order, and to resolve both t → 0 and r → 0 limit singularities in the classical pressureimpulse theory. In the course of construction of the solution, first the standard assumptions behind the existence of the inviscid approximation are revisited, which leads to correcting the previously known interpretation of the selfsimilarity of the classical inviscid solution near the plate edge. Second, new scalings of the solution structure near the plate edge are determined, with which the viscous solution near the edge is constructed analytically and matched to the inviscid one. Finally, the analysis of both the Stokes and inviscid limits of this uniformly valid solution allows us to uncover the scalings for the early timeevolution of ejecta—a jet forming during the impact—as well as to clarify the applicability of the KuttaJoukowsky condition used in previous studies.
 Viscous and NonNewtonian Flows

Compact bubble clusters in Newtonian and nonNewtonian liquids
View Description Hide DescriptionWe studied the terminal velocity of a packed array of bubbles, a bubble cluster, rising in different fluids: a Newtonian fluid, an elastic fluid with nearly constant viscosity (Boger fluid), and a viscoelastic fluid with a shear dependent viscosity, for small but finite Reynolds numbers (1 × 10^{−4} < Re < 4). In all three cases, the cluster velocity increased with the total volume, following the same trend as single bubbles. For the case of clusters in elastic fluids, interestingly, the socalled velocity discontinuity was not observed, unlike the single bubble case. In addition to the absence of jump velocity, the clusters did not show the typical teardrop shape of large bubbles in viscoelastic fluids and the strength of the negative wake is much weaker than the one observed behind single bubbles. Dimensional analysis of the volumevelocity plots allowed us to show that, while the equivalent diameter (obtained from the total cluster volume) is the appropriate length to determine buoyancy forces and characteristic shear rates, the individual bubble size is the appropriate scale to account for surface forces.

A macroscopic model for slightly compressible gas slipflow in homogeneous porous media
View Description Hide DescriptionThe study of gas slipflow in porous media is relevant in many applications ranging from nanotechnology to enhanced oil recovery and in any situation involving lowpressure gastransport through structures having sufficiently small pores. In this paper, we use the method of volume averaging for deriving effectivemedium equations in the framework of a slightly compressible gas flow. The result of the upscaling process is an effectivemedium model subjected to time and lengthscale constraints, which are clearly identified in our derivation. At the first order in the Knudsen number, the macroscopic momentum transport equation corresponds to a Darcylike model involving the classical intrinsic permeability tensor and a slipflow correction tensor that is also intrinsic. It generalizes the DarcyKlinkenberg equation for ideal gas flow, and exhibits a more complex form for dense gas. The component values of the two intrinsic tensors were computed by solving the associated closure problems on two and threedimensional periodic unit cells. Furthermore, the dependence of the slipflow correction with the porosity was also verified to agree with approximate analytical results. Our predictions show a powerlaw relationship between the permeability and the slipflow correction that is consistent with other works. Nevertheless, the generalization of such a relationship to any configuration requires more analysis.
 Particulate, Multiphase, and Granular Flows

Terminal velocity of a bubble in a vertically vibrated liquid
View Description Hide DescriptionWe rigorously derive a formula for the terminal velocity of a small bubble in a vertically vibrated viscous incompressible liquid starting from the full NavierStokes equations and the exact boundary conditions at the bubble surface. This formula is derived using a perturbation analysis in which the small parameter is the nondimensional amplitude of the pressure oscillation. The analysis does not assume that the bubble remains spherical but does assume that the bubble is axisymmetric. It is shown that the bubble terminal velocity can be computed to second order while computing the full solution only to first order by applying a compatibility condition on the firstorder solution. To second order, the bubble terminal velocity is shown to be the net value from an upward steady term and a rectified term that can be downward or upward. The perturbation formula depends on the vibration frequency nondimensionalized by the bubble radius and the liquid kinematic viscosity. We show that our perturbation formula links two heuristically developed formulas for the rectified component, which we denote the velocityaveraged and forceaveraged formulas. Our perturbation formula reproduces the velocityaveraged formula for low frequencies and the forcedaveraged formula for high frequencies and varies monotonically between these limits for intermediate frequencies. We furthermore develop a highresolution spectral code specifically to simulate this type of bubble motion. Results from this code verify that the perturbation formula is correct for infinitesimal oscillating pressure amplitudes and suggest that it provides an upper bound for finite amplitudes of the pressure oscillation.

Sedimentation of an ellipsoidal particle in narrow tubes
View Description Hide DescriptionSedimentation behaviours of an ellipsoidal particle in narrow and infinitely long tubes are studied by a multirelaxationtime lattice Boltzmann method (LBM). In the present study, both circular and square tubes with 12/13 ⩽ D/A ⩽ 2.5 are considered with the Galileo number (Ga) up to 150, where D and A are the width of the tube and the length of major axis of the ellipsoid, respectively. Besides three modes of motion mentioned in the literature, two novel modes are found for the narrow tubes in the higher Ga regime: the spiral mode and the vertically inclined mode. Near a transitional regime, in terms of average settling velocity, it is found that a lighter ellipsoid may settle faster than a heavier one. The relevant mechanism is revealed. The behaviour of sedimentation inside the square tubes is similar to that in the circular tubes. One significant difference is that the translation and rotation of ellipsoid are finally constrained to a diagonal plane in the square tubes. The other difference is that the anomalous rolling mode occurs in the square tubes. In this mode, the ellipsoid rotates as if it is contacting and rolling up one corner of the square tube when it settles down. Two critical factors that induce this mode are identified: the geometry of the tube and the inertia of the ellipsoid.

How do neighbors affect incipient particle motion in laminar shear flow?
View Description Hide DescriptionWe experimentally study how neighboring particles affect the incipient motion of particles on regular substrates and exposed to a laminar shear flow. To this end, we determine the critical Shields number and determine whether the particle rolls or slides. The substrates consist of a monolayer of fixed spheres of uniform size that are regularly arranged in triangular and quadratic configurations. Neighboring particles influence the incipient motion by shielding to the shear flow and may inhibit continuous motion once they are in direct contact with the particle. At the low particle Reynolds numbers studied, neighboring spheres on the monolayer only affect the incipient particle motion if they are closer than about 3 particle diameters. Direct contact inhibits continuous motion and results in a strong increase of the critical Shields number. For identical beads, we found two different regimes for the onset of continuous motion. Depending on the substrate geometry, the upstream particle may start to roll like a single particle passing the downstream neighbor or it may push its downstream neighbor forward. In the latter case, the downstream sphere rolls while the upstream bead slides in contact with the downstream neighbor. Both regimes yield about the same critical Shields number although the critical Shields number for single particle motion differs by about 50%. If particle contact is avoided by a sudden jump in the Shields number, the critical Shields number for onset of continuous particle motion can be reduced considerably. Finally, the lowest critical Shields numbers for dislodging buried beads in the configurations studied coincides with the critical Shields number for incipient motion of irregular granular beds.