Volume 23, Issue 6, June 2011

This paper presents two and threedimensional direct numerical simulations of the flow around a circular cylinder placed symmetrically in a plane channel. Results are presented in the Reynolds number range (based on the cylinder diameter and centerline velocity) of 10 to 390 for a blockage ratio (ratio of the cylinder diameter to the channel height) of 0.2. The aim of this work was to investigate in detail the confinement effect due to the channel’s stationary walls on the force coefficients and the associated Strouhal numbers, as well as on the generated flow regimes. Present results suggest a transition from a 2D to a 3D shedding flow regime between Re = 180 and Re = 210. This transition was found to be dominated by mode A and mode B three dimensional instabilities, similar to those observed in the case of an unconfined circular cylinder. This is the first time that the existence of the two modes, and of naturally occurring vortex dislocations, has been confirmed via full 3D simulations for the case of a confined circular cylinder in a channel. A discontinuity in the variation of the Strouhal number St, and of the base pressure coefficient C _{ pb }, with Re was also observed. This was found to be associated with the onset of mode A instability and the development of vortex dislocations, and parallels what occurs in the unconfined case, but previous studies could not confirm its existence in the confined case. Furthermore, by analyzing the mechanisms affecting the shape and evolution of these instabilities, it is demonstrated that they are significantly affected by the confinement only in the far wake.
 LETTERS


Inner/outer layer interactions in turbulent boundary layers: A refined measure for the largescale amplitude modulation mechanism
View Description Hide DescriptionThe amplitude modulation (AM) imparted by the outer layer largescale motions on the nearwall turbulence is studied through direct numerical simulation of compressible boundary layer flow at moderate Reynolds number. Mathis et al. [J. Fluid Mech. 628, 311 (2009)] introduced an amplitude modulation coefficient to quantify this effect, whereby carrier and modulated signals are decoupled through a procedure based on the Hilbert transform of the streamwise velocity signals. However, Schlatter and Örlü [Phys. Fluids 22, 051704 (2010)] have recently shown that a nonzero amplitude modulation coefficient is closely associated to a nonzero value of the velocity skewness, and therefore, it does not necessarily reflect genuine physics. In this paper, the analysis is extended through systematic use of the twopoint amplitude modulation correlation, which is shown to be a refined measure of the topdown influence of largescale outer events on the inner part of the boundary layer.

Selfsustained processes in the logarithmic layer of turbulent channel flows
View Description Hide DescriptionIt has recently been shown that largescale and verylargescale motions can selfsustain in turbulent channel flows even in the absence of input from motions at smaller scales. Here we show that also motions at intermediate scales, mainly located in the logarithmic layer, survive when motions at smaller scales are artificially quenched. These elementary selfsustained motions involve the bursting and regeneration of sinuous streaks. This is a further indication that a full range of autonomous selfsustained processes exists in turbulent channel flows with scales ranging from those of the buffer layer streaks to those of the large scale motions in the outer layer.

Collisions of ageostrophic modons and formation of new types of coherent structures in rotating shallow water model
View Description Hide DescriptionWe study collisions of recently discovered ageostrophic modons in rotating shallow water model at different values of impact parameter and find that two new types of coherent vortex structures may be formed during this process: “nonlinear” modons, i.e., coherent dipoles with essentially nonlinear scatter plot and coherent tripoles. Both are known for incompressible 2D Euler equations, but were not reported in the “compressible” shallow water model. Inelastic scattering with strong filamentation and shearing is also possible. Surprisingly, the strongly nonlinear process of coherent structure formation leads to almost no emission of inertiagravity waves.

The interaction between strainrate and rotation in shear flow turbulence from inertial range to dissipative length scales
View Description Hide DescriptionDirect numerical simulation data from the self similar region of a planar mixing layer is filtered at four different length scales, from the Taylor microscale to the dissipative scales, and is used to examine the scale dependence of the strainrotation interaction in shear flowturbulence. The interaction is examined by exploring the alignment between the extensive strainrate eigenvector and the vorticity vector. Results show that the mechanism for enstrophy amplification (propensity of which increases when the two vectors are parallel) is scale dependent with the probability of the two vectors being parallel higher for larger length scales. However, the mechanism for enstrophy attenuation, i.e., the probability of the two vectors being perpendicular to each other, appears to be scale independent.

 ARTICLES

 Interfacial Flows

Nonisothermal flow of a thin film of fluid with temperaturedependent viscosity on a stationary horizontal cylinder
View Description Hide DescriptionA comprehensive description is obtained of the twodimensional steady gravitydriven flow with prescribed volume flux of a thin film of Newtonian fluid with temperaturedependentviscosity on a stationary horizontal cylinder. When the cylinder is uniformly hotter than the surrounding atmosphere (positive thermoviscosity), the effect of increasing the heat transfer to the surrounding atmosphere at the free surface is to increase the average viscosity and hence reduce the average velocity within the film, with the net effect that the film thickness (and hence the total fluid load on the cylinder) is increased to maintain the fixed volume flux of fluid. When the cylinder is uniformly colder than the surrounding atmosphere (negative thermoviscosity), the opposite occurs. Increasing the heat transfer at the free surface from weak to strong changes the film thickness everywhere (and hence the load, but not the temperature or the velocity) by a constant factor which depends only on the specific viscosity model considered. The effect of increasing the thermoviscosity is always to increase the film thickness and hence the load. In the limit of strong positive thermoviscosity, the velocity is small and uniform outside a narrow boundary layer near the cylinder leading to a large film thickness, while in the limit of strong negative thermoviscosity, the velocity increases from zero at the cylinder to a large value at the free surface leading to a small film thickness.

Fourthorder coupled nonlinear Schrödinger equations for gravity waves on deep water
View Description Hide DescriptionWe derive a set of two fourthorder coupled nonlinear Schrödinger equations describing the evolution of two twodimensional systems of deepwater gravity waves with different wavenumbers or directions of propagation. It is shown that the coupled equations can be formulated as a Hamiltonian system and that they conserve the total wave action and momentum of the combined wave field. The modulational instability of two interacting uniform wave trains is considered.

A continuum model of colloidstabilized interfaces
View Description Hide DescriptionColloids that are partially wetted by two immiscible fluids can become confined to fluidfluidinterfaces. At sufficiently high volume fractions, the colloids may jam and the interface may crystallize. Examples include bicontinuous interfacially jammed emulsion gels (bijels), which were proposed in this study by Stratford et al. [Science 309, 2198 (2005)] as a hypothetical new class of soft materials in which interpenetrating, continuous domains of two immiscible viscous fluids are maintained in a rigid state by a jammed layer of colloidal particles at their interface. We develop a continuum model for such a system that is capable of simulating the longtime evolution. A NavierStokesCahnHilliard model for the macroscopic twophase flow system is combined with a surface phasefieldcrystal model for the microscopic colloidal system along the interface. The presence of colloids introduces elastic forces at the interface between the two immiscible fluid phases. An adaptive finite element method is used to solve the model numerically. Using a variety of flow configurations in two dimensions, we demonstrate that as colloids jam on the interface and the interfacecrystallizes, the elastic force may be strong enough to make the interface sufficiently rigid to resist external forces, such as an applied shear flow, as well as surface tension induced coarsening in bicontinuous structures.

Air cushioning in droplet impacts with liquid layers and other droplets
View Description Hide DescriptionAir cushioning of a highspeed liquiddropletimpact with a finitedepth liquid layer sitting upon a rigid impermeable base is investigated. The evolution of the droplet and liquidlayer freesurfaces is studied alongside the pressure in the gas film dividing the two. The model predicts gas bubbles are trapped between the liquid freesurfaces as the droplet approaches impact. The key balance in the model occurs when the depth of the liquid layer equals the horizontal extent of interactions between the droplet and the gas film. For liquid layer depths significantly less than this a shallow liquid limit is investigated, which ultimately tends towards the aircushioning behavior seen in dropletimpact with a solid surface. Conversely, for liquid layer depths much deeper than this, the rigid base does not affect the aircushioning of the droplet. The influence of compressibility is discussed and the relevant parameter regime for an incompressible model is identified. The size of the trapped gas bubble as a function of the liquid layer depth is investigated. The deep water model is extended to consider binary droplet collisions. Again, the model predicts gas bubbles will be trapped as the result of air cushioning in highspeed binary dropletimpacts.

Interfacial thin films rupture and selfsimilarity
View Description Hide DescriptionTwo superposed thin layers of fluids are prone to interfacial instabilities due to Londonvan der Waals forces. Evolution equations for the film thicknesses are derived using lubrication theory. Using the intrinsic scales, for a single layer, results in a system with parametric dependence of four ratios of the two layers: surface tension, Hamaker constant, viscosity, and film thickness. In contrast to the single layer case, the bilayer system has two unstable eigenmodes: squeezing and bending. For some particular parameter regimes, the system exhibits the avoided crossing behavior, where the two eigenmodes are interchanged. Based on numerical analysis, the system evolves into four different rupture states: basal layer rupture, upper layer rupture, double layer rupture, and mixed layer rupture. The ratio of Hamaker constants and the relative film thickness of the two layers control the system dynamics. Remarkably, the line of avoided crossing demarks the transition region of mode mixing and energy transfer, affecting the scaling of the dynamical regime map consequentially. Asymptotic and numerical analyses are used to examine the selfsimilar ruptures and to extract the power law scalings for both the basal layer rupture and the upper layer rupture. The scaling laws for the basal layer rupture are the same as those of the single layer on top of a substrate. The scaling laws for the upper layer rupture are different: the lateral length scale decreases according to and the film thickness decreases according to .

Effect of central slotted screen with a high solidity ratio on the secondary resonance phenomenon for liquid sloshing in a rectangular tank
View Description Hide DescriptionMounting a screen with a high solidity ratio (0.5 ≲ Sn < 1) at the center of a rectangular tank qualitatively changes the secondary resonance phenomenon for liquid sloshing. In contrast to the clean tank, the steadystate sloshing due to lateral excitation is then characterized by multipeak response curves in a neighborhood of the primary resonance frequency. The present paper revises the adaptive nonlinear multimodal method to study the secondary resonance phenomenon for the screenaffected resonant sloshing with a finite liquid depth and, thereby, clarify earlier experimental results of the authors.

Side wall effects in thin gravitydriven film flow – steady and draining flow
View Description Hide DescriptionThe increasing demand for thinner films in scientific and technological applications requires a better knowledge of the effect of side walls on such flows, especially as far as the formation of a capillary boundary layer is concerned. In this paper gravitydriven thin film flow in an open channel is investigated, highlighting the competing effects of the noslip condition and velocity overshoot due to capillary elevation at the bounding side walls. Their influence on the flow rate, the velocity field, the Reynolds number, and the free surface shape is studied for two flow types: (i) the case with vanishingly small capillary elevation at the side walls compared to the film thickness at the center of the channel; (ii) the situation when capillary elevation at the side walls dominates over the film thickness at the center of the channel. For both, large deviations from the twodimensional reference system occur. The theoretical predictions are compared to experimental observations, for the case of side walls with different wetting properties defined in terms of the static contact angle there.
 Viscous and NonNewtonian Flows

Shear flow over a porous layer: Velocity in the real proximity of the interface via rheological tests
View Description Hide DescriptionIn this paper, we have experimentally investigated the velocity profile of a fluid undergoing simple shear above a porous medium. To this end, we used for the first time rheological tests performed with a constant stress rheometer equipped with parallel plate geometry with a real porous medium glued on the lower plate. The velocity at the interface between the porous layer and the free fluid was inferred by extrapolating the linear velocity profile in the free fluid to the interface. These data were nicely compared with predictions obtained integrating the Brinkman extension of Darcy law in the porous medium together with Stokes equations in the free fluid coupled at the interface by the continuity of velocity and by the momentum balance suggested by OchoaTapia and Whitaker [Int. J. Heat Mass Transfer 38(14), 2635 (1995)]. In the literature, the physical origin of the stress jump imposed by OchoaTapia and Whitaker at the interface has been attributed to a perturbation of the velocity profile in the vicinity of the interface, both in the porous medium and in the free fluid. For the first time, the disturbance in the free fluid has been measured and quantified resulting in a satisfactory agreement with theoretical predictions.

A theoretical bridge between linear and nonlinear microrheology
View Description Hide DescriptionPassive microrheology exploits the fluctuationdissipation theorem to relate thermal fluctuations of a colloidal probe to the nearequilibrium linear response behavior of the material through an assumed generalized Stokes Einstein relation (GSER). Active and nonlinear microrheology, on the other hand, measures the nonlinear response of a strongly driven probe, for which fluctuationdissipation does not hold. This leaves no clear method for recovering the macroscopic rheological properties from such measurements. Although the two techniques share much in common, there has been little attempt to relate the understanding of one to the other. In passive microrheology, the GSER is generally assumed to hold, without the need for explicit calculation of the microstructural deformation and stress, whereas in nonlinear microrheology, the microstructure must be explicitly determined to obtain the drag force. Here we seek to bridge the gap in understanding between these two techniques, by using a single model system to explicitly explore the gentleforcing limit, where passive () and active () microrheology are identical. Specifically, we explicitly calculate the microstructural deformations and stresses as a microrheological probe moves within a dilute colloidalsuspension. In the gentleforcing limit, we find the microstructural stresses in the bulk material to be directly proportional to the local strain tensor, independent of the detailed flow, with a prefactor related to the effective shear modulus. A direct consequence is that the probe resistance due to the bulk stresses in passive (linear response) microrheology quantitatively recovers the results of macroscopic oscillatory shear rheology. Direct probebath interactions, however, lead to quantitative discrepancies that are unrelated to macroscopic shear rheology. We then examine the microstructural equations for nonlinear microrheology, whose limit reduces to the limit in passive microrheology. Guided by the results from passive microrheology, we show that direct probematerial interactions are unrelated to the macroscopic shear rheology. Moreover, we show that the bulk microstructural deformations (which quantitatively recover macroscopic shear rheology in the linear limit) now obey a governing equation that differs qualitatively from macroscopic rheology, due to the spatially dependent, Lagrangian unsteady mixture of shear and extensional flows. This inherently complicates any quantitative interpretation of nonlinear microrheology.

Thin viscous sheets with inhomogeneous viscosity
View Description Hide DescriptionWe derive the equations governing the dynamics of thin viscoussheets having nonhomogeneous viscosity, via asymptotic expansion methods. We consider distributions of viscosity that are inhomogeneous in the longitudinal and transverse directions and arbitrary (bulk and surface) external forces. Two specific problems are solved as an illustration. In a first example, we study the effects of purely inplane variations of viscosity, which lead to thickness modulations when the sheet is stretched or compressed. In a second example, we study a stretched viscoussheet whose viscosity varies both across thickness and inplane; in that case, we find that inplane strain leads to outofplane displacement as the inplane forces become coupled to transverse ones.
 Particulate, Multiphase, and Granular Flows

Fluid velocity fluctuations in a collision of a sphere with a wall
View Description Hide DescriptionWe report on the results of a combined experimental and numerical study on the fluid motion generated by the controlled approach and arrest of a solid sphere moving towards a solid wall at moderate Reynolds number. The experiments are performed in a small tank filled with water for a range of Reynolds numbers for which the flow remains axisymmetric. The fluid agitation of the fluid related to the kinetic energy is obtained as function of time in the experiment in a volume located around the impact point. The same quantities are obtained from the numerical simulations for the same volume of integration as in the experiments and also for the entire volume of the container. As shown in previous studies, this flow is characterized by a vortex ring, initially in the wake of the sphere, that spreads radially along the wall, generating secondary vorticity of opposite sign at the sphere surface and wall. It is also observed that before the impact, the kinetic energy increases sharply for a small period of time and then decreases gradually as the fluid motion dies out. The measure of the relative agitation of the collision is found to increase weakly with the Reynolds number. The close agreement between the numerics and experiments is indicative of the robustness of the results. These results may be useful in light of a potential modelling of particleladen flows. Movies illustrating the spatiotemporal dynamics are provided with the online version of this paper.

Breakup of suspension drops settling under gravity in a viscous fluid close to a vertical wall
View Description Hide DescriptionThe evolution of suspensiondrops sedimenting under gravity in a viscous fluid close to a vertical wall was studied experimentally and numerically with the use of the pointforce model, in the Stokes flow regime. The fluid inside and outside the drop was identical. The initial distribution of the suspended solid heavy particles was uniform inside a spherical volume. In the experiments and in the simulations, the suspensiondrops evolved qualitatively in the same way as in an unbounded fluid. However, it was observed, both experimentally and numerically, that, on the average, the destabilization time T and the distance L traveled by the drop until breakup were smaller for a closer distance h of the drop center from the wall, with approximately linear dependence of T and L on D/h, for h larger or comparable to the drop diameter D. Destabilization times and lengths of individual drops with different random configurations of the particles were shown to differ significantly from each other, owing to the chaotic nature of the particle dynamics.
 Laminar Flows

A numerical study of evaporation characteristics of spherical ndodecane droplets in high pressure nitrogen environment
View Description Hide DescriptionEvaporation of stagnant (zero relative velocity) as well as moving spherical droplets of ndodecane in a zerogravity and high pressure nitrogen environment is modeled. The nonideal effects, solubility of ambient gas into the liquidphase, variable thermophysical properties, and gas and liquidphase transients are included in the model. The model is quantitatively validated using published experimental data. Numerical predictions show that, for stagnant droplets at subcritical ambient temperatures, the droplet lifetime continuously increases with pressure, while at critical temperature, the lifetime initially increases and thereafter remains almost constant. At supercritical temperatures, the lifetime decreases continuously with increasing ambient pressure and the average evaporation constant shows a local maximum at a particular ambient pressure. In the case of moving droplets, at supercritical ambient temperature, the rate of increase of average evaporation constant with ambient pressure becomes significant as the initial droplet relative velocity increases. For low initial velocities (<1 m/s), the average evaporation constant gradually increases with ambient pressure and subsequently levelsoff with further increase in ambient pressure. The droplet lifetime decreases with increase in ambient pressure or initial velocity. Penetration distance of the moving droplets decreases with ambient pressure and increases with initial droplet relative velocity. The mechanisms influencing the differences in evaporation under varying conditions of pressure and temperature are discussed.

The shape of an elastic filament in a twodimensional corner flow
View Description Hide DescriptionThe deformation of a flexible filament held fixed at one end in a nonuniform viscousflow with curved streamlines is considered, with a focus on the filament dynamics and steadystate shape. Our motivation arises from recent microfluidic experiments on biofilm formation in a channel with bends, where threadlike structures, or streamers, were observed, attached to the side walls downstream of each corner and connecting consecutive corners while floating in the middle plane of the channel [Rusconi et al., J. R. Soc. Interface 7, 1293 (2010)]. We discuss the time evolution and final shape of the filament in different corner geometries as a function of a nondimensional elasticity parameter that compares viscous and elastic effects. Since the filament develops tension, when the flow has curved streamlines the filament does not align with the flow, as occurs in a rectilinear flow, but rather it crosses the streamlines.
 Instability and Transition

Model reduction for fluids using frequential snapshots
View Description Hide DescriptionThis paper deals with model reduction of highorder linear systems. An alternative method to approximate proper orthogonal decomposition (POD) and balanced truncation is exposed in this paper within the framework of the incompressible NavierStokes equations. The method of snapshots used to obtain lowrank approximations of the system controllability and observability Gramians is carried out in the frequency domain. Model reduction is thus performed using flow states that are longtime harmonic responses of the flow to given forcings, we call them frequential snapshots. In contrast with the recent works using timestepping approach, restricted to stable systems, this one can always be computed for systems without marginal modes while it reduces to the same procedure for stable systems. We show that this method is efficient to perform POD and balanced proper orthogonal decomposition reducedorder models in both globally stable and unstable flows through two numerical examples: the flow over a backwardfacing step and the flow over a square cavity. The first one is a globally stable flow exhibiting strong transient growths as a typical noise amplifier system while the second is a globally unstable flow representative of an oscillator system. In both cases, it is shown that the frequencybased snapshot method yields reducedorder models that efficiently capture the inputoutput behavior of the system. In particular, regarding the unstable cavity flow, our resulting unstable reducedorder models possess the same unstable global modes and stable transfer functions as those of the full system.

The onset of double diffusive reactionconvection in an anisotropic porous layer
View Description Hide DescriptionThe onset of double diffusive reactionconvection in a horizontal anisotropic porous layer saturated with binary mixture, which is heated and salted from below and subjected to chemical equilibrium on the boundaries, is studied analytically using both linear and nonlinear stability analyses. A linear stability analysis is performed to investigate how the dissolution or precipitation of reactive component affects the onset of convection. The Darcy model is employed as momentum equation. The effect of mechanical and thermal anisotropy parameters, reaction rate, Lewis number, solute Rayleigh number, and normalized porosity on the stability of the system is investigated. We find that the chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.