Volume 22, Issue 1, January 2010

We investigate experimentally the causes of jet plume instability and enhanced mixing observed in the exhaust of shockcontaining convergentdivergent nozzles. Key features of the internal flow are the separation shock, separation shear layers, and pattern of alternating expansion and compression waves downstream of the shock. We focus on two possible reasons for this instability—the motion of the separation shock and the wave pattern downstream of the shock. The nozzle flow was generated in a planar facility with variable area ratio and pressure ratio, and the motion of the shock was tracked using timeresolved wall pressure measurements. The isolated effect of the wave pattern was investigated in a separate facility wherein a sonic shear layer, simulating the nozzle separation shear layer, was disturbed with compression and expansion waves emanating from a wavy wall. In both instances, the instability of the shear layer was characterized by timeresolved measurements of the total pressure. In the nozzle flow, the amplitude of shock motion increases with shock strength. Correlation of shock motion with shear layer total pressure is virtually absent for weak shocks but becomes significant for strong shocks. However, impingement of stationary waves on the shear layer had no impact on its growth rate. We conclude that the enhanced shear layer instability is strongly coupled to shock motion, and that the wave pattern by itself is not a cause of enhanced mixing. The occurrence of asymmetric separation at large shock strengths is a further contributor to the enhancement of instability.
 ANNOUNCEMENTS


Announcement: The 2009 François Naftali Frenkiel Award for Fluid Mechanics
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 LETTERS


On nonlinear multiarmed spiral waves in slowly rotating fluid systems
View Description Hide DescriptionStable nonlinear equilibria of convection in the form of quasistationary, multiarmed spiral waves, up to a maximum of six spiral arms, are found in a slowly rotating fluid confined within a thin spherical shell governed by the threedimensional Navier–Stokes equation, driven by a radial unstable temperature gradient and affected by a weak Coriolis force. It is shown that three essential ingredients are generally required for the formation of the multiarmed spirals: the influence of slow rotation, largeaspectratio geometry and the effect of weak nonlinearity.
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 ARTICLES

 Biofluid Mechanics

The effect of viscoelasticity on the stability of a pulmonary airway liquid layer
View Description Hide DescriptionThe lungs consist of a network of bifurcating airways that are lined with a thin liquid film. This film is a bilayer consisting of a mucus layer on top of a periciliary fluid layer. Mucus is a nonNewtonian fluid possessing viscoelastic characteristics. Surface tension induces flows within the layer, which may cause the lung’s airways to close due to liquid plug formation if the liquid film is sufficiently thick. The stability of the liquid layer is also influenced by the viscoelastic nature of the liquid, which is modeled using the OldroydB constitutive equation or as a Jeffreys fluid. To examine the role of mucus alone, a single layer of a viscoelastic fluid is considered. A system of nonlinear evolution equations is derived using lubrication theory for the film thickness and the film flow rate. A uniform film is initially perturbed and a normal mode analysis is carried out that shows that the growth rate for a viscoelastic layer is larger than for a Newtonian fluid with the same viscosity. Closure occurs if the minimum core radius, , reaches zero within one breath. Solutions of the nonlinear evolution equations reveal that normally decreases to zero faster with increasing relaxation time parameter, the Weissenberg number . For small values of the dimensionless film thickness parameter , the closure time, , increases slightly with , while for moderate values of , ranging from 14% to 18% of the tube radius, decreases rapidly with provided the solventviscosity is sufficiently small. Viscoelasticity was found to have little effect for , indicating the strong influence of surface tension. The film thickness parameter and the Weissenberg number also have a significant effect on the maximum shear stress on tube wall, , and thus, potentially, an impact on cell damage. increases with for fixed , and it decreases with increasing for small provided the solventviscosity parameter is sufficiently small. For large , there is no significant difference between the Newtonian flow case and the large cases.
 Micro and Nanofluid Mechanics

A macromodel for squeezefilm air damping in the freemolecule regime
View Description Hide DescriptionA threedimensional Monte Carlo(MC) simulation approach is developed for the accurate prediction of the squeezefilm air damping on microresonators in the freemolecule gas regime. Based on the MC simulations and the analytical travelingtime distribution, a macromodel, which relates air damping directly with device dimensions and operation parameters, is constructed. This model provides an efficient tool for the design of highperformance microresonators. The accuracy of the macromodel is validated through the modeling of the quality factors of several microresonators. It has been found that the relative errors of the quality factors of two resonators, as compared with experimental data, are 3.9% and 5.7%, respectively. The agreements between the macromodel results and MC simulation results, on the other hand, are excellent in all cases considered.
 Interfacial Flows

Interfacial instability induced by lateral vapor pressure fluctuation in bounded thin liquidvapor layers
View Description Hide DescriptionWe study an instability of thin liquidvapor layers bounded by rigid parallel walls from both below and above. In this system, the interfacial instability is induced by lateral vapor pressure fluctuation, which is in turn attributed to the effect of phase change:evaporation occurs at a hotter portion of the interface and condensation at a colder one. The high vapor pressure pushes the interface downward and the low one pulls it upward. A set of equations describing the temporal evolution of the interface of the liquidvapor layers is derived by applying longwave approximation to both layers. This model neglects the effect of mass loss or gain at the interface and guarantees the mass conservation of the liquid layer. The result of linear stability analysis of the model shows that the presence of the pressure dependence of the local saturation temperature mitigates the growth of longwave disturbances. The thinner vapor layer enhances the vapor pressure effect. We find the stability criterion, which suggests that only slight temperature gradients are sufficient to overcome the gravitational effect for a water/vapor system. The same holds for the Rayleigh–Taylor unstable case, with a possibility that the vapor pressure effect may be weakened if the accommodation coefficient is below a certain critical value.

Spinning droplets on superhydrophobic surfaces
View Description Hide DescriptionThe study of liquiddroplets on solid surfaces is a well established research area. Static measurements include contact angle determinations which allow surface energy measurement. Dynamic measurements generally are reported on vibrating drops on mechanically or sonic driven surfaces. The general analysis of the physics of vibrating drops is complicated due to the internal degrees of freedom of the liquid and the stick or slip conditions at the liquidsolid contact line. Here we propose a simple and straightforward experimental method to measure the physical properties of droplets on highly hydrophobic surfaces. The first dynamic experiments of droplets on superhydrophobicsurfaces are also reported.

On spread extent of sessile droplet into porous medium: Numerical solution and comparisons with experiments
View Description Hide DescriptionThe spread of a wetting liquid sessile droplet into porous medium is solved numerically using the capillary network model with a microforce balance boundary condition at the liquid/gas free interface in the porous medium. The spread starts as the porous medium imbibes the sessile liquid, followed by liquid additionally being spread inside the porous medium itself. After there is no remaining sessile liquid, the net flow across the porous medium boundaries is equal to zero. Either spread, with or without sessile liquid present at the porous mediumsurface, is rendered by local differences in capillary pressure. These local differences are accounted for by implementing the numerical solution over a heterogeneous capillary network, consisting of pores connected by throats. Both pores and throats follow predefined distribution functions. Once there is no sessile liquid present on the porous mediumsurface, it is found from a numerical solution that the liquid front can extend significantly in time, wetting very large volumes of the porous medium. This is also measured in experiments, in which over time, an increase in wetted volume of more than 16 times is observed compared to the wetted volume right after the disappearance of sessile liquid on the porous mediumsurface. The numerical and experimental results for the time changes of (i) volume of liquid remaining at the porous mediumsurface, (ii) porous mediumsurface wetted area of the droplet imprint, and (iii) liquid protrusion depth into porous medium are compared, with very good qualitative and quantitative agreement found.

Steady freesurface flow at the stern of a ship
View Description Hide DescriptionNew solutions for steady twodimensional freesurface flow past a curved plate are considered here. They can be interpreted as approximations to the flow locally at the stern of a ship. Weakly nonlinear solutions are derived analytically and nonlinear solutions are computed by boundary integralequation methods. Analysis in the phase plane provides a way to determine the geometries of hulls that give rise to wavefree stern flows. These waveless flows are desirable as they reduce shipdrag.

Theory of breakup of core fluids surrounded by a wetting annulus in sinusoidally constricted capillary channels
View Description Hide DescriptionAnalysis of coreannular dynamics in the presence of base flow for arbitrary fluid viscosities leads to an equation describing the temporal evolution of the fluid/fluid interface. The equation follows from the conservation of mass in the “smallslope” approximation. Its useful applications occur, for example, in chemical engineering and petroleum recovery. The nonlinear equation allows inexpensive numerical analysis. For sinusoidally constricted pores, a purely geometric criterion exists that enables or prohibits the corefluid breakup in the necks of the constrictions. The geometrically favoring condition sets up capillarypressure gradients that ensure a continuous outflow of the core fluid from the necks into the “crests” of the profile. Such behavior is indeed observed in the numerical solutions of the evolution equation. For relatively large slopes of the initial configuration, setting up larger pressure gradients, the interface shape remains “smooth,” the evolution times are relatively fast, and the breakup is typically achieved by the growing filmfluid collar touching the axis of the channel at a single point. No satellite droplets are produced. Decreasing the slope lengthens the evolution times, allowing the formation and growth of “wavy” disturbances on the initial interface profile, which may touch the axis of the capillary in several places forming satellite drops. Thinner initial annuli also slow down the evolution process. Instability develops for the cases of the core both more and less viscous than the film. Finally, if the geometry prohibits the snapoff altogether, the initial interface configurations decay into steadystate solutions, and no breakup takes place. The solutions of the evolution equation validate well against two computationalfluiddynamics codes.

Effect of channel width on the primary instability of inclined film flow
View Description Hide DescriptionA procedure is developed to detect the onset of interfacialinstability in inclined filmflows (with estimated accuracy better than 5%) and is used to show that the finite width of experimental channels stabilizes the undisturbed liquid film. Deviation from the classical prediction scales inversely with the product of channel width and sine of inclination angle, and for small inclinations and/or narrow channels is of the order of 100%. The effect is tentatively attributed to the influence of sidewalls on the traveling disturbances, which results in curved crestlines and transverse variation of wave characteristics.
 Viscous and NonNewtonian Flows

Chaotic mixing in a Jouleheated glass melt
View Description Hide DescriptionA numerical study of twodimensional thermal convection of a highly viscous fluid driven by volumetric heating has been carried out to investigate the effect of timedependent thermal forcing on the degree of mixing. The problem is relevant to electrically heated glass melting furnaces which are traditionally operated with timeindependent heating. The numerical computations carried out are for two model fluids representing semitransparent and opaque melts, respectively. Lagrangian motion of passive tracers is numerically simulated and degree of mixing in the glass melt is quantified in terms of mixing entropy. The computed flow patterns indicate that timedependent streamlines intersect transversely and, thereby, result in wellmixed regions. The stirring mechanism involves the rotational stretching and folding produced by two oscillating vortices with varying size and circulation. The model predictions indicate that the chaotic mixing is strongly dependent on the period of electrode firing cycle. The present investigation demonstrates that timedependent thermal forcing improves the mixing characteristics of both semitransparent and opaque glass melts significantly.

Surfactantinduced migration of a spherical drop in Stokes flow
View Description Hide DescriptionIn Stokes flows, symmetry considerations dictate that a neutrally buoyant spherical particle will not migrate laterally with respect to the local flow direction. We show that a loss of symmetry due to flowinduced surfactant redistribution leads to crossstream drift of a spherical drop in Poiseuille flow. We derive analytical expressions for the migration velocity in the limit of small nonuniformities in the surfactant distribution, corresponding to weakflow conditions or a highviscosity drop. The analysis predicts migration toward the flow centerline.

Temporal large eddy simulations of turbulent viscoelastic drag reduction flows
View Description Hide DescriptionWe report on temporal large eddy simulations (TLES) of the turbulent channel flow of a dilute polymer solution modeled with the FENEP (finitely extensible nonlinear elastic in the Peterlin approximation) constitutive equation. The large eddy simulations are based upon an approximate temporal deconvolution method [Pruett et al., Phys. of Fluids, 18, 028104–1, (2006)] for residual Newtonian stress modeling and secondary regularization for unresolved subfilter Newtonian stress. The filtered conformation tensorequation involves deconvolution for stretching and for the nonlinear spring force, as well as secondary regularization. Results are shown at a friction Reynolds number 180 for Weissenberg numbers and molecular extensibilities spanning the moderate to high drag reducing regimes. Excellent agreement is obtained between TLES and direct numerical simulations (DNS) in terms of percent drag reduction prediction. TLES is also able to reproduce the high level of anisotropy of turbulence, which confirms recent findings by Frohnapfel et al. [J. Fluid Mech.577, 457 (2007)] who present high anisotropy as a general mechanism to obtain significant drag reduction. The TLES model proves itself stable and its overall computational workload some 60 times less than DNS.
 Particulate, Multiphase, and Granular Flows

Vertical dispersion of light inertial particles in stably stratified turbulence: The influence of the Basset force
View Description Hide DescriptionThe dispersion of light inertial particles in statistically stationary stably stratified turbulence is studied by means of direct numerical simulations. The light particle dispersion behavior is found to be comparable to that of heavy particles when displayed as a function of the Stokes number. Deviations from fluid particle dispersion are found already for small Stokes numbers; the length of the typical plateau for vertical dispersion is shorter for the light inertial particles. All the forces in the Maxey–Riley equation are taken into account and they are found to be of similar magnitude as the Stokes drag for particles with . However, not all forces directly influence the particle dispersion. It is shown that especially the often neglected Basset force plays a considerable role in the vertical dispersion of light particles in stratified turbulence. Neglecting this force results in an overprediction of the vertical dispersion by about 15%–20%.
 Laminar Flows

Mixing kinematics of moderate Reynolds number flows in a Tchannel
View Description Hide DescriptionAn experimental study of water flow in a Tshaped channel with rectangular cross section ( inlet ID and outlet ID) has been conducted for a Reynolds number Re range based on inlet geometry of . Dynamical conditions and Tchannel geometry of the current study are applicable to the microscale. This study supports a large body of numerical work, and resolution and the interrogation region are extended beyond previous experimental studies. Laser induced fluorescence(LIF) permits a detailed look at the flow fields that evolve in the outlet channel over the broad range of Re where realistic Tchannels operate. Scalar structures previously unresolved in the literature are presented. Unsteady flow regimes numerically predicted to occur at higher Re are characterized, and simultaneous planar and discretepoint LIF measurements relate the development of oscillatory behavior in the outlet channel to flow structure in the junction. Further, the development of an unsteady symmetric topology at higher Re, which negatively affects mixing, is presented, and mechanisms behind the wide range of mixing qualities predicted for this regime are explained. Characteristics of steady and unsteady flows are tracked with Re to elucidate mixing behavior on a fundamental level. Practical conclusions for experimental mixing in a Tchannel are extracted.
 Instability and Transition

Dynamic stability analysis for capillary channel flow: Onedimensional and threedimensional computations and the equivalent steady state technique
View Description Hide DescriptionSpacecraft technology provides a series of applications for capillary channel flow. It can serve as a reliable means for positioning and transport of liquids under low gravity conditions. Basically, capillary channels provide liquid paths with one or more free surfaces. A problem may be flow instabilities leading to a collapse of the liquid surfaces. A result is undesired gas ingestion and a two phase flow which can in consequence cause several technical problems. The presented capillary channel consists of parallel plates with two free liquid surfaces. The flow rate is established by a pump at the channel outlet, creating a lower pressure within the channel. Owing to the pressure difference between the liquid phase and the ambient gas phase the free surfaces bend inwards and remain stable as long as they are able to resist the steady and unsteady pressure effects. For the numerical prediction of the flow stability two very different models are used. The onedimensional unsteady model is mainly based on the Bernoulliequation, the continuity equation, and the Gauss–Laplace equation. For threedimensional evaluations an open source computational fluid dynamics(CFD) tool is applied. For verifications the numerical results are compared with quasisteady and unsteady data of a sounding rocket experiment. Contrary to previous experiments this one results in a significantly longer observation sequence. Furthermore, the critical point of the steady flow instability could be approached by a quasisteady technique. As in previous experiments the comparison to the numerical model evaluation shows a very good agreement for the movement of the liquid surfaces and for the predicted flow instability. The theoretical prediction of the flow instability is related to the speed index, based on characteristic velocities of the capillary channel flow. Stable flow regimes are defined by stability criteria for steady and unsteady flow. The onedimensional computation of the speed index is based on the technique of the equivalent steady system, which is published for the first time in the present paper. This approach assumes that for every unsteady state an equivalent steady state with a special boundary condition can be formulated. The equivalent steady state technique enables a reformulation of the equation system and an efficient and reliable speed index computation. Furthermore, the existence of the numerical singularity at the critical point of the steady flow instability, postulated in previous publication, is demonstrated in detail. The numerical singularity is related to the stability criterion for steady flow and represents the numerical consequence of the liquid surface collapse. The evaluation and generation of the pressure diagram is demonstrated in detail with a series of numerical dynamic flow studies. The stability diagram, based on onedimensional computation, gives a detailed overview of the stable and instable flow regimes. This prediction is in good agreement with the experimentally observed critical flow conditions and results of threedimensional CFD computations.

The effects of nonnormality and nonlinearity of the Navier–Stokes operator on the dynamics of a large laminar separation bubble
View Description Hide DescriptionThe effects of nonnormality and nonlinearity of the twodimensional Navier–Stokes differential operator on the dynamics of a large laminar separation bubble over a flat plate have been studied in both subcritical and slightly supercritical conditions. The global eigenvalueanalysis and direct numerical simulations have been employed in order to investigate the linear and nonlinear stability of the flow. The steadystate solutions of the Navier–Stokes equations at supercritical and slightly subcritical Reynolds numbers have been computed by means of a continuation procedure. Topological flow changes on the base flow have been found to occur close to transition, supporting the hypothesis of some authors that unsteadiness of separated flows could be due to structural changes within the bubble. The global eigenvalueanalysis and numerical simulations initialized with small amplitude perturbations have shown that the nonnormality of convective modes allows the bubble to act as a strong amplifier of small disturbances. For subcritical conditions, nonlinear effects have been found to induce saturation of such an amplification, originating a wavepacket cycle similar to the one established in supercritical conditions, but which is eventually damped. A transient amplification of finite amplitude perturbations has been observed even in the attached region due to the high sensitivity of the flow to external forcing, as assessed by a linear sensitivity analysis. For supercritical conditions, the nonnormality of the modes has been found to generate lowfrequency oscillations (flapping) at large times. The dependence of such frequencies on the Reynolds number has been investigated and a scaling law based on a physical interpretation of the phenomenon has been provided, which is able to explain the onset of a secondary flapping frequency close to transition.

Sensitivity analysis of a streamwise corner flow
View Description Hide DescriptionThe stability of the flow formed by the intersection of two perpendicular flat plates is revisited through a study of the sensitivity to baseflow variations. After a brief presentation of the asymptotic regime, sensitivity functions underlying corner mode (concentrated close to the intersection) and Tollmien–Schlichting modes, with different obliqueness angles, are computed. Taking this into consideration, associated mechanisms as well as active regions are identified, which further confirm the sensitivity area of the corner mode along the intersection of two flat plates. Furthermore, the concept of pseudospectra indicates that under a small baseflow modification, a certain range of frequencies underlying the corner mode could become unstable. Then, an optimization technique tracking the worstcase scenario, i.e., the deviation leading to a maximum temporal amplification rate, shows that a small variation in the reference field in the area of uncertainty leads to a significant decrease in the critical Reynolds number as observed in experiments. A hypothesis based on the onset of an inflectional mechanism is thus proposed to explain the experimental results.

Simulations and model of the nonlinear Richtmyer–Meshkov instability
View Description Hide DescriptionThe nonlinear evolution of the Richtmyer–Meshkov (RM)instability is investigated using numerical simulations with the FLASH code in two dimensions. The purpose of the simulations is to develop an empirical nonlinear model of the RMinstability that is applicable to inertial confinement fusion(ICF) and ejecta formation, namely, at large Atwood number and scaled initial amplitude of the perturbation. The FLASH code is first validated with a variety of RMexperiments that evolve well into the nonlinear regime. They reveal that bubbles stagnate when they grow by an increment of and that spikes accelerate for due to higher harmonics that focus them. These results are then compared with a variety of nonlinear models that are based on potential flow. We find that the models agree with simulations for moderate values of and , but not for the larger values that characterize ICF and ejecta formation. We thus develop a new nonlinear empirical model that captures the simulation results consistent with potential flow for a broader range of and . Our hope is that such empirical models concisely capture the RM simulations and inspire more rigorous solutions.

Ordered microdroplet formations of thin ferrofluid layer breakups
View Description Hide DescriptionThe ordered breakup pattern of a thin layer of ferrofluiddrop subjected to a uniform perpendicular field is experimentally investigated. The results confirm a universal pattern formation containing numerous breaking droplets of a uniform size, which is independent of the initial area of ferrofluiddrop and the propagating directions of the formation waves. Two quantitative observations regarding the size and number of breaking droplets are concluded. Both the experiments and theoretical analysis show the correlation between the diameter of breaking droplets and magnetization strength can be characterized as . The uniform size of breaking droplets under a constant field strength results in a linear proportionality between the number of breaking droplets and the initial area of ferrofluiddrop as , which is verified by the experiments.