Volume 27, Issue 4, April 2015
Index of content:

The structure and stability of mode2 internal solitarylike waves is investigated experimentally. A rankordered train of mode2 internal solitary waves is generated using a lock release configuration. The pycnocline is centred either on the middepth of the water column (the 0% offset case) or it is offset in the positive vertical direction by a fraction of 5%, 10%, or 20% of the total fluid depth. It is found that offsetting the pycnocline has little effect on the basic wave properties (e.g., wave speed, wave amplitude, and wavelength) but it does significantly affect wave stability. Instability takes the form of small KHlike billows in the rear of the wave and small scale overturning in the core of the wave. In the 0% offset case, instability occurs on both the upper and lower interfaces of the pycnocline and is similar in extent and vigour over the two interfaces. As the offset percentage is increased, however, instability is more pronounced on the lower interface with little or no evidence of instability being observed on the upper interface. In the 20% offset case, a mode1 tail is associated with the wave and the wave characteristics resemble qualitatively the recent field observations of Shroyer et al. [“Mode 2 waves on the continental shelf: Ephemeral components of the nonlinear internal wavefield,” J. Geophys. Res. 115, C07001, doi:10.1029/2009JC005605 (2010)].
 LETTERS


Kolmogorov’s hypotheses and global energy spectrum of turbulence
View Description Hide DescriptionWe relate the justification of Kolmogorov’s hypotheses on the local isotropy and smallscale universality in real turbulent flows to an observed universality of basis independence for the global energy spectrum and energy flux of smallscale turbulence. To readily examine the smallscale universality, an approach is suggested that investigates the global energy spectrum in a general spectral space for which the nonlinear interscale interaction may not be Fouriertriadic. Specific verifications are performed based on direct numerical simulations of turbulence in a spherical geometry and reexaminations of several existing results for turbulent channel flows.

Turbulent bands in planePoiseuille flow at moderate Reynolds numbers
View Description Hide DescriptionIn this letter, we show via numerical simulations that the typical flow structures appearing in transitional channel flows at moderate Reynolds numbers are not spots but isolated turbulent bands, which have much longer lifetimes than the spots. Localized perturbations can evolve into isolated turbulent bands by continuously growing obliquely when the Reynolds number is larger than 660. However, interactions with other bands and local perturbations cause band breaking and decay. The competition between the band extension and breaking does not lead to a sustained turbulence until Re becomes larger than about 1000. Above this critical value, the bands split, providing an effective mechanism for turbulence spreading.

Multiarmed jets: A subset of the blooming jets
View Description Hide DescriptionThis study focuses on excited circular jets obtained through axial and helical excitations superimposed on an inlet velocity profile. Various forcing frequency ratios (axial to helical) were analysed in the range of 1.6 ≤ fa /fh ≤ 3.2, with occurrence of bifurcating, trifurcating, and blooming jets reported in the literature. Analytical investigations of a spatiotemporal behaviour of forcing show that, apart from the above mentioned jets observed for particular frequency ratios, different types of multiarmed jets are likely to occur. This was reflected by direct numerical simulation performed in the paper. The simulations showed the existence of very strong and noticeable splitting of the jet into a fivefold jet and weaker splitting into 12 and 13armed jets.

Experimental and theoretical models of waveinduced flexure of a sea ice floe
View Description Hide DescriptionAn experimental model is used to validate a theoretical model of a sea ice floe’s flexural motion, induced by ocean waves. A thin plastic plate models the ice floe in the experiments. Rigid and compliant plastics and two different thicknesses are tested. Regular incident waves are used, with wavelengths less than, equal to, and greater than the floe length, and steepnesses ranging from gently sloping to stormlike. Results show the models agree well, despite the overwash phenomenon occurring in the experiments, which the theoretical model neglects.

Superdiffusion in sheared suspensions
View Description Hide DescriptionWe investigate the dispersion of a layer of dye initially applied at the outer wall of a cylindrical Couettecell into a sheared suspension of nonBrownian spherical particles. The process is directly visualized and quantified at the particle scale. A “rollingcoating” mechanism is found to convectively transport the dye at a constant rate directly from the wall towards the bulk. The fluid velocity fluctuations, u′, measured with particle image velocimetry, and the imposed shearrate, , are used to define a diffusion coefficient, , which is found to increase linearly with the distance from the wall. A solution of the transport equation accounting for this inhomogeneous stirring field describes quantitatively the concentration profiles measured experimentally. It exhibits a superdiffusive character, a consequence of the increase of the stirring strength with distance from the wall. Movies are available with the online version of the paper.

 ARTICLES

 Micro and Nanofluid Mechanics

An analytical and numerical study on the startup flow of slightly rarefied gases in a parallelplate channel and a pipe
View Description Hide DescriptionThe paper presents results of an investigation of the response of an incompressible fluid in a circular micropipe and a parallelplate microchannel to a sudden timeindependent pressure drop. Solutions of the problem were obtained analytically using the Laplace transform technique and numerically using the lattice Boltzmann method. The unsteady velocity profiles in the pipe and in the channel were obtained with the help of the infinite series solutions validated against numerical simulations. In line with the expectations, the flow asymptotically tends to the fully developed pattern, which is attained quicker for smaller Knudsen numbers. The solution enabled also obtaining relations to estimate the hydraulic resistance coefficient.
 Interfacial Flows

Eigenspectra and mode coalescence of temporal instability in twophase channel flow
View Description Hide DescriptionThe stability of two immiscible fluids with different densities and viscosities is examined for channel flow. A multidomain Chebyshev collocation spectral method is used for solving the coupled OrrSommerfeld stability equations for the entire spectrum of eigenvalues and associated eigenfunctions. Numerical solution of the eigenvalue problem is obtained with the QZ eigenvalue solver and is validated with analytical results derived in the long and short wave limits. A parametric study is carried out to investigate the spectral characteristics and eigenfunction structures related to the shear and interfacial modes of instability. The interactions and mode switching between the two instability modes are investigated. Under certain conditions, the two modes are found to become unstable simultaneously. Mode coalescence can occur in either the stable or the unstable part of the spectrum. In general, the eigenfunction of the most dangerous mode is observed to vary sharply in the neighborhood of the interface and the critical points.

Coupling of the interfacial and bulk flow in knifeedge viscometers
View Description Hide DescriptionThe operation of the knifeedge viscometer requires knowledge of the interfacial velocity profile in order to determine the viscous traction between the surface film and the knife edge and hence measure the surface shear viscosity of the film. The interfacial velocity profile can be obtained analytically in two limiting regimes. One is the limit of the surface shear viscosity going to infinity, in which case the interfacial velocity profile is independent of the bulk flow and a simple analytic expression is available. The other limit corresponds to vanishing bulk flow inertia, allowing one to reduce the Navier–Stokes equations to the Stokes equation, and the resulting linear system can be solved analytically. For finite inertia and finite surface shear viscosity, the knifeedge viscometer hydrodynamics is governed by the coupled nonlinear set of equations. Here, we study these numerically, explore the coupling between the interfacial and bulk flow, and delineate the ranges of surface shear viscosity and knifeedge rotation rates where the analytic approximations are appropriate. We also examine a variant of the knifeedge viscometer, known as the doublewall ring viscometer, which is essentially the same geometry but with the addition of a stationary inner cylinder so that the bulk fluid is contained in an annular channel rather than a cylinder.

Does the topography’s specific shape matter in general for the stability of film flows?
View Description Hide DescriptionIn our experimental study on the linear stability of gravitydriven films flowing over inclined topographies, we consider a fundamental question: does the topography’s specific shape matter in general for the stability of film flows? In order to understand this complex problem, we used five topographies of different shapes. For each topography, we characterized the basic flow by measuring the flow field and the free surface contour. Experiments on the flow’s linear stability followed. We obtained astonishing results on how the topography’s shape can manipulate both the basic flow and the linear stability of gravitydriven films.
 Viscous and NonNewtonian Flows

Sticking and splashing in yieldstress fluid drop impacts on coated surfaces
View Description Hide DescriptionYieldstress fluids, including gels and pastes, are effectively fluid at high stress and solid at low stress. In liquidsolid impacts, the fluid motion can be halted by the yield stress at different points of the event, and these fluids can therefore stick and accumulate where they impact, motivating several applications of these rheologically complex materials. Here, we use highspeed imaging to experimentally study liquidsolid impact of yieldstress fluids on precoated horizontal surfaces. With a precoating of the same material, we can observe large longlifetime ejection sheets with redirected momentum which extend away from the impact location. Under critical splash conditions, sheet breakup occurs and ejected droplets can be nonspherical and threadlike due to the inability of capillary stresses to deform material above a certain lengthscale. By varying the droplet size, impact velocity, surface coating thickness, and rheological material properties, we develop appropriate dimensionless parameters and present a lowdimensional regime map of impact behaviors.

Slow viscous gravitydriven interaction between a bubble and a free surface with unequal surface tensions
View Description Hide DescriptionThe axisymmetric gravitydriven dynamics of a bubble rising toward a free surface is addressed for gasliquid interfaces having unequal surface tensions. The liquid flow is governed by the Stokes equations which are here solved using a boundary element method in axisymmetric configuration. Within this framework, two dimensionless numbers arise: the Bond number Bo1 based on the surface tension of the bubble interface and the surface tension ratio comparing the free surface and bubble surface tensions. Under a careful and discussed selection of the code key settings (number of boundary elements, initial bubble location, and distance beyond which the free surface is truncated), it has been possible to numerically and accurately track in time the bubble and free surface shapes for several values of . The longtime shapes are found to deeply depend upon both Bo1 and and also to compare well with the shapes predicted in Princen and Mason [“Shape of a fluid drop at a fluidliquid interface. II. Theory for threephase systems,” J. Colloid. Sci. 20, 246–266 (1965)] using a hydrostatic model in which both surfaces are touching. Similarly, the drainage dynamics of the liquid film thickness between the bubble and the free surface depends on . The longtime film thickness exponentially decays in time and a socalled thinning rate α for which the numerical behaviors and a simple model reveal two basic behaviors: (i) at small Bond number, α behaves as 1/Bo1 and (ii) at large Bond number, α is nearly constant. In addition, it is found that in the entire range of the quantity , the thinning rate α is well approximated by the function 1/(18χ) + α ∞ with α ∞ ≈ 0.158. Such a result also permits one to estimate the typical drainage time versus the initial bubble radius a, the liquid density ρ and viscosity μ, the gravity and the free surface, and bubble surface tensions.
 Particulate, Multiphase, and Granular Flows

Freely cooling granular gases with shortranged attractive potentials
View Description Hide DescriptionWe treat the case of an undriven gas of inelastic hardspheres with shortranged attractive potentials via an extension of the pseudoLiouville operator formalism. New evolution equations for the granular temperature and coordination number are obtained. The granular temperature exhibits deviation from both Haff’s law and the case of longranged potentials. We verify this departure using softsphere discrete element method simulations. Excellent agreement is found for the duration of the simulation even beyond where exclusively binary collisions are expected. Simulations show the emergence of strong spatialvelocity correlations on the length scale of the last peak in the paircorrelation function but do not show strong correlations beyond this length scale. We argue that molecular chaos may remain an adequate approximation if the system is modelled as a Smoluchowski type equation with aggregation and breakup processes.

Topology of pairsphere trajectories in finite inertia suspension shear flow and its effects on microstructure and rheology
View Description Hide DescriptionThe relative motion of pairs of spheres in finite inertia shear flow is examined. The pair relative trajectory is studied, using latticeBoltzmann simulation, both for pairs which are isolated and for pairs in suspension of solid volume fraction of ϕ ≤ 0.3. For the suspension, the average trajectory and aspects of its dispersion are considered. For neutrally buoyant particles in simpleshear flow, particlescale inertia is represented by a Reynolds number , where is the shear rate, a is the particle radius, and ρ and μ are the density and viscosity of the Newtonian suspending fluid. The pair trajectories in a dilute inertial suspension have the same basic features as the streamlines around an isolated particle at similar Re: reversing, inplane and offplane spiraling, and open but foreaft asymmetric trajectories are observed. The origin of the offplane spirals is examined in detail in this work. The average pair trajectory space in a suspension of finite volume fraction is found to be qualitatively similar to the dilute suspension pair trajectories, as the spiraling and reversing zones are retained; the influence of ϕ and Re on the extension of different zones is described. The role of the average pair trajectory in setting the microstructure of the suspension is analyzed through a convection equation description of the pair distribution function, showing extreme accumulation at contact consistent with sampling from simulation. The influence of the microstructure on the bulk rheology is considered, with a focus on the combined effects of ϕ and Re on the near contact structural effects on rheology.
 Laminar Flows

The distribution of “time of flight” in three dimensional stationary chaotic advection
View Description Hide DescriptionThe distributions of “time of flight” (time spent by a single fluid particle between two crossings of the Poincaré section) are investigated for five different three dimensional stationary chaotic mixers. Above all, we study the large tails of those distributions and show that mainly two types of behaviors are encountered. In the case of slipping walls, as expected, we obtain an exponential decay, which, however, does not scale with the Lyapunov exponent. Using a simple model, we suggest that this decay is related to the negative eigenvalues of the fixed points of the flow. When noslip walls are considered, as predicted by the model, the behavior is radically different, with a very large tail following a power law with an exponent close to −3.
 Instability and Transition

Twodimensional instability of the bottom boundary layer under a solitary wave
View Description Hide DescriptionThe objective of this paper is to establish a detailed map for the temporal instability of the bottom boundary layer (BBL) flow driven by a solitonlike wave induced pressure gradient in a Utubeshaped tunnel, which serves as an approximation to the BBL under surface solitary waves. Both linear stability analysis and fully nonlinear twodimensional simulations using highorder numerical methods have been carried out. The process of delineation of the stability regions as a function of boundary layer thicknessbased Reynolds number of the temporally evolving base flow, Re δ , consists of two parts. In the first part, we assess the lower limit of the Re δ range within which the standard, quasisteady, linear stability analysis is applicable when considering individual base flow profiles sampled during its transient evolution. Below this limit, transient linear stability analysis serves as a more accurate predictor of the stability properties of the base flow. In the second step, above the Re δ limit where the BBL is determined to be linearly unstable, the base flow is further classified as unconditionally stable, conditionally unstable, or unconditionally unstable in terms of its sensitivity to the amplitude and the insertion time of perturbations. Two distinct modes of instability exist in this case: post and preflowreversal modes. At a moderate value of Re δ , both modes are first observed in the wave deceleration phase. The post flow reversal mode dominates for relatively low Re and it is the one observed in Sumer et al. [“Coherent structures in wave boundary layers. Part 2. Solitary motion,” J. Fluid Mech. 646, 207 (2010)]. For Re above a threshold value of the base flow in the unconditionally unstable regime, the preflow reversal mode, which is longer in wavelength than its postreversal counterpart, becomes dominant. In the same regime, the threshold Re δ value above which instability is observed in the acceleration phase of the wave is also identified. In this case, the base flow velocity profiles lack any inflection point, suggesting that the origin of such an instability is viscous. Finally, the lower Re δ limit above which quasisteady linear stability analysis is valid may be independently obtained by adapting to the surface solitary wave BBL, an instability criterion which links the average growth rate and wave event timescale, as previously proposed in studies of the instability of the interior of progressive and solitary internal waves.

A study of the geometry and parameter dependence of vortex breakdown
View Description Hide DescriptionThe types of vortex breakdown observed in the torsionally driven cylinder (TDC) flow and in the flow through an openended pipe are compared. The connection between the various breakdown types is specifically addressed, and the differences in manifestation of breakdown are attributed to the different Reynolds number regimes involved. Here, in both cases, the Reynolds number is based on quantities associated with the vortex core immediately upstream of breakdown, rather than the more geometryspecific Reynolds number defined in the previous work. Thus, the relationship between the TDC flow and the flows observed in other, more open geometries, is clarified. The predominantly asymmetric breakdown observed in open high Reynolds number flows is replaced by a closed bubble form with decreasing Reynolds number in the TDC. Threedimensional numerical simulations support this interpretation, showing that the 3D spiral type of breakdown is replaced by a TDCtype axisymmetric breakdown in an open pipe as the Reynolds number is reduced. The stability of the threedimensional solutions indicates that spiral breakdown modes stabilise at lower Reynolds number, leading to an axisymmetric breakdown state as a stable evolved flow solution.

Elevator mode convection in flows with strong magnetic fields
View Description Hide DescriptionInstability modes in the form of axially uniform vertical jets, also called “elevator modes,” are known to be the solutions of thermal convection problems for vertically unbounded systems. Typically, their relevance to the actual flow state is limited by threedimensional breakdown caused by rapid growth of secondary instabilities. We consider a flow of a liquid metal in a vertical duct with a heated wall and strong transverse magnetic field and find elevator modes that are stable and, thus, not just relevant, but a dominant feature of the flow. We then explore the hypothesis suggested by recent experimental data that an analogous instability to modes of slow axial variation develops in finitelength ducts, where it causes largeamplitude fluctuations of temperature. The implications for liquid metal blankets for tokamak fusion reactors that potentially invalidate some of the currently pursued design concepts are discussed.

Interaction between counterpropagating Rossby waves and capillary waves in planar shear flows
View Description Hide DescriptionA counterintuitive destabilizing effect of the surface tension in planar wakes has been observed by Tammisola et al. [“Effect of surface tension on global modes of confined wake flows,” Phys. Fluids 23, 014108 (2011)] and Biancofiore et al. [“Direct numerical simulations of twophase immiscible wakes,” Fluid Dyn. Res. 46, 041409 (2014)] by means of linear global analyses and direct numerical simulations, respectively. In the present study, we approximate the velocity profile of a wake flow through a piecewise brokenline profile and explain the presence of temporal unstable modes using an interfacial wave interaction perspective. With this perspective, we associate to each vorticity discontinuity an individual counterpropagating Rossby wave (RW), while the introduction of a finite amount of surface tension at the interface creates two capillary waves (CWs) which propagate with respect to the interface velocity with the same relative velocity but in opposite directions. The addition of the surface tension generates a new unstable mode, which is a Rossbycapillary mode, since it is due to the interaction between one RW and one CW. Furthermore, we capture the spatiotemporal evolution of the interacting fourwaves system by means of an impulse response analysis. The spreading of the wavepacket, and consequently the absolute nature of the instability, is enhanced by a moderate surface tension, especially if the interface is located close to the faster edge of the brokenline wake profile. This can be explained by the influence of the surface tension on the group velocities of the waves, taken in isolation.

Modal and nonmodal evolution of perturbations for parallel round jets
View Description Hide DescriptionThe present work investigates the local modal and nonmodal stability of round jets for varying aspect ratios α = R/θ, where R is the jet radius and θ the shear layer momentum thickness, for Reynolds numbers ranging from 10 to 10 000. The competition between axisymmetric (azimuthal wavenumber m = 0) and helical (m = 1) perturbations depending on the aspect ratio, α, is quantified at different time horizons. Three different techniques have been used, namely, a classical temporal stability analysis in order to characterize the unstable modes of the jet; an optimal excitation analysis, based on the resolution of the adjoint problem, to quantify the potential for nonmodal perturbation dynamics; and finally an optimal perturbation analysis, focused on the very short time transient dynamics, to complement the adjointbased study. Besides providing with the determination of the critical aspect ratio below which the most unstable perturbations switch from m = 0 to m = 1 depending on the Reynolds number, the study shows that perturbations can undergo a rapid transient growth. It is found that helical perturbations always experience the highest transient growth, although for large values of aspect ratio, this transient domination can be overcome by the eventual emergence of axisymmetric perturbation when more exponentially unstable. Furthermore, the adjoint mode, which excites optimally the most unstable mode of the flow, is found to coincide with the optimal perturbation even for short time horizons, and to drive the transient dynamics for finite times. Therefore, the adjointbased analysis is found to characterize adequately the transient dynamics of jets, showing that a mechanism equivalent to the Orr one takes place for moderate to small wavelengths. However, in the long wavelength limit, a specific mechanism is found to shift the jet as a whole in a way that resembles the classical liftup effect active in wall shear flows.

Temporal and spatial instability of a compound jet in a surrounding gas
View Description Hide DescriptionDroplet generated from the rupture of a compound liquid jet can be used to produce encapsulated droplets which have applications in a wide variety of industrial processes. In this paper, we examine the instability of a two dimensional axisymmetric inviscid compound jet falling vertically downwards in a surrounding gas under the influence of gravity. The steady state equations are derived using an asymptotic method and the linear instability, including temporal and spatial instability, is determined using a multiple scales approach. The results are analysed to investigate how the gastoshell density ratio affects key features of the jet including theoretical breakup lengths.