Volume 27, Issue 6, June 2015
Index of content:
 LETTERS


Wake angle for surface gravity waves on a finite depth fluid
View Description Hide DescriptionLinear water wave theory suggests that wave patterns caused by a steadily moving disturbance are contained within a wedge whose halfangle depends on the depthbased Froude number FH . For the problem of flow past an axisymmetric pressure distribution in a finitedepth channel, we report on the apparent angle of the wake, which is the angle of maximum peaks. For moderately deep channels, the dependence of the apparent wake angle on the Froude number is very different to the wedge angle and varies smoothly as FH passes through the critical value FH = 1. For shallow water, the two angles tend to follow each other more closely, which leads to very large apparent wake angles for certain regimes.

 ARTICLES

 Micro and Nanofluid Mechanics

Thermally excited fluid flow in a microsized liquid crystal channel with a free surface
View Description Hide DescriptionThe temperatureinduced reorientation dynamics in microsized liquid crystal (LC) channel with a free LC/vacuum interface has been investigated theoretically based on the hydrodynamic theory including the director motion, the thermally excited fluid flow v, and the temperature T redistribution, produced by induced heating in the interior of the LC sample. Analysis of the numerical results shows that due to interaction between ∇T and the gradient of the director field in the LC channel bounded by the free LC/vacuum interface, a thermally excited vortical fluid flow is maintained in the vicinity of the heat source. Calculations also show that in the case of the fast heating, the LC sample settles down to threevortical flow regime, whereas in the case of the slow heating, the LC material settles down to bivortical flow regime. As for nematogenic material, we have considered the LC channel to be occupied by 4npentyl4′cyanobiphenyl and investigated the effect of both and ∇T on the magnitude and direction of v, as well as on the height of the LC film on the solid surface, for a number of heating and hydrodynamic regimes.

Electrokinetic instability of liquid micro and nanofilms with a mobile charge
View Description Hide DescriptionThe instability of ultrathin films of an electrolyte bordering a dielectric gas in an external tangential electric field is scrutinized. The solid wall is assumed to be either a conducting or charged dielectric surface. The problem has a steady onedimensional solution. The theoretical results for a pluglike velocity profile are successfully compared with available experimental data. The linear stability of the steadystate flow is investigated analytically and numerically. Asymptotic longwave expansion has a triplezero singularity for a dielectric wall and a quadruplezero singularity for a conducting wall, and four (for a conducting wall) or three (for a charged dielectric wall) different eigenfunctions. For infinitely small wave numbers, these eigenfunctions have a clear physical meaning: perturbations of the film thickness, of the surface charge, of the bulk conductivity, and of the bulk charge. The numerical analysis provides an important result: the appearance of a strong shortwave instability. At increasing Debye numbers, the shortwave instability region becomes isolated and eventually disappears. For infinitely large Weber numbers, the longwave instability disappears, while the shortwave instability persists. The linear stability analysis is complemented by a nonlinear direct numerical simulation. The perturbations evolve into coherent structures; for a relatively small external electric field, these are largeamplitude surface solitary pulses, while for a sufficiently strong electric field, these are shortwave inner coherent structures, which do not disturb the surface.
 Viscous and NonNewtonian Flows

Numerical simulation of vortexinduced vibration of a circular cylinder in a spanwise shear flow
View Description Hide DescriptionVortexinduced vibration of a circular cylinder with a lengthtodiameter ratio of 19.2 in a spanwise shear flow is investigated numerically. The Reynolds numbers based on the velocity at the centre of the cylinder and the mass ratio are 500 and 2, respectively. The responses of the cylinder in shear flows with shear factors of 0.05 and 0.1 are compared with that in the uniform flow. Although the oscillation of the lift force for a stationary cylinder in a sheared flow is very weak, it is found that if the cylinder is allowed to vibrate, the lockin regime and the maximum response amplitude are comparable with their counterparts for a cylinder in a uniform flow. The maximum response amplitude for a shear factor of 0.05 is found slightly greater than that for a uniform flow. In the lockin regime, the vortex shedding and the oscillation of the sectional lift coefficient are found to synchronize (have a same frequency) along the cylinder span, leading to strong vibration of the cylinder. The sectional lift coefficient changes from being in phase to being out of phase with the response displacement at a location on the cylinder span, and the location where the lift coefficient changes its phase depends on the reduced velocity. The phase change of the lift coefficient corresponds to the change in the vortex shedding mode. The role of the sectional lift coefficient in the vibration varies along the cylinder span. For a small reduced velocity in the lockin regime, the sectional lift forces near the highvelocity end of the cylinder excite the vibration, while those at the rest of the cylinder span damp the vibration. With increasing reduced velocity, the location where the sectional lift forces excite the vibration moves towards the lowvelocity end of the cylinder.
 Particulate, Multiphase, and Granular Flows

Rotation of a spheroid in a simple shear at small Reynolds number
View Description Hide DescriptionWe derive an effective equation of motion for the orientational dynamics of a neutrally buoyant spheroid suspended in a simple shear flow, valid for arbitrary particle aspect ratios and to linear order in the shear Reynolds number. We show how inertial effects lift the degeneracy of the Jeffery orbits and determine the stabilities of the logrolling and tumbling orbits at infinitesimal shear Reynolds numbers. For prolate spheroids, we find stable tumbling in the shear plane and logrolling is unstable. For oblate spheroids, by contrast, logrolling is stable and tumbling is unstable provided that the particle is not too disklike (moderate asphericity). For very flat oblate spheroids, both logrolling and tumbling are stable, separated by an unstable limit cycle.
 Laminar Flows

On the symmetry of proper orthogonal decomposition modes of a lowaspectratio plate
View Description Hide DescriptionIn this paper, the symmetry property and corresponding virtual force contribution of the proper orthogonal decomposition (POD) modes are numerically investigated for the lowReynolds number flows passing over a lowaspectratio pitchingplunging plate. It is found that the flow and its POD modes have the same reflectional symmetry about the spanwise central plane. However, about the crossflow central plane, the spatiotemporal flow symmetry results in a change of symmetry pattern every two POD modes, which corresponds to odd or even multiples of the vortex shedding frequency. Based on a wake survey method for virtual forces, the POD modes are further classified into two groups, thrust and liftproducing modes, respectively. Results have also shown that the distinct symmetry properties of these modes can be used to identify the correlation between the wake structure and the hydrodynamic force production.
 Turbulent Flows

Secondsound studies of coflow and counterflow of superfluid ^{4}He in channels
View Description Hide DescriptionWe report a comprehensive study of turbulent superfluid ^{4}He flow through a channel of square cross section. We study for the first time two distinct flow configurations with the same apparatus: coflow (normal and superfluid components move in the same direction), and counterflow (normal and superfluid components move in opposite directions). We realise also a variation of counterflow with the same relative velocity, but where the superfluid component moves while there is no net flow of the normal component through the channel, i.e., pure superflow . We use the secondsound attenuation technique to measure the density of quantised vortex lines in the temperature range 1.2 K ≲ T ≲ Tλ ≈ 2.18 K and for flow velocities from about 1 mm/s up to almost 1 m/s in fully developed turbulence. We find that both the steadystate and temporal decay of the turbulence significantly differ in the three flow configurations, yielding an interesting insight into twofluid hydrodynamics. In both pure superflow and counterflow, the same scaling of vortex line density with counterflow velocity is observed, , with a pronounced temperature dependence; in coflow instead, the vortex line density scales with velocity as L ∝ V ^{3/2} and is temperature independent; we provide theoretical explanations for these observations. Further, we develop a new promising technique to use different secondsound resonant modes to probe the spatial distribution of quantised vortices in the direction perpendicular to the flow. Preliminary measurements indicate that coflow is less homogeneous than counterflow/superflow, with a denser concentration of vortices between the centre of the channel and its walls.

Direct detection of linearized bursts in turbulence
View Description Hide DescriptionThe relevance of linear transitional mechanisms in fully turbulent shear flows, and in particular of the Orrlike inviscid transient amplification of disturbances, is explored in the context of the prediction of bursting behavior. Although the logarithmic layer of wallbounded turbulence is used as the primary example, most conclusions should apply to other flows with linearly stable mean profiles that are dominated by largescale streamwisevelocity streaks and intermittent bursts of the crossshear velocity. When the linearised problem is solved in the limit of small viscosity, it has previously been shown that statistical properties, such as the bursting time and lengthscales, the energy fluxes between components, and the mean inclination angles, are consistent in linear and nonlinear systems. The question addressed here is whether the individual structures predicted by the linearised solution can be detected in fully nonlinear simulations, and whether the linearized approximation can be used to predict their evolution. It is found that strong bursting of the largest scales is well described linearly, comprising about 65%–70% of the total time, but that weaker fluctuations are not. It is also found that adding an eddy viscosity does not substantially improve predictions.

Building proper invariants for eddyviscosity subgridscale models
View Description Hide DescriptionDirect simulations of the incompressible NavierStokes equations are limited to relatively lowReynolds numbers. Hence, dynamically less complex mathematical formulations are necessary for coarsegrain simulations. Eddyviscosity models for largeeddy simulation is probably the most popular example thereof: they rely on differential operators that should properly detect different flow configurations (laminar and 2D flows, nearwall behavior, transitional regime, etc.). Most of them are based on the combination of invariants of a symmetric tensor that depends on the gradient of the resolved velocity field, . In this work, models are presented within a framework consisting of a 5D phase space of invariants. In this way, new models can be constructed by imposing appropriate restrictions in this space. For instance, considering the three invariants P GG^{ T } , Q GG^{ T } , and R GG^{ T } of the tensorGG^{ T }, and imposing the proper cubic nearwall behavior, i.e., , we deduce that the eddyviscosity is given by . Moreover, only R GG^{ T } dependent models, i.e., p > − 5/2, switch off for 2D flows. Finally, the model constant may be related with the Vreman’s model constant via ; this guarantees both numerical stability and that the models have less or equal dissipation than Vreman’s model, i.e., . The performance of the proposed models is successfully tested for decaying isotropic turbulence and a turbulent channel flow. The former testcase has revealed that the model constant, C s3pqr , should be higher than 0.458 to obtain the right amount of subgridscale dissipation, i.e., C s3pq = 0.572 (p = − 5/2), C s3pr = 0.709 (p = − 1), and C s3qr = 0.762 (p = 0).
 Compressible Flows

Dynamics of cylindrical converging shock waves interacting with aerodynamic obstacle arrays
View Description Hide DescriptionCylindrical converging shock waves interacting with an array of aerodynamic obstacles are investigated numerically for diverse shock strengths and for different obstacle configurations in air in standard conditions. The considered number of obstacles N is 4, 6, 8, 16, and 24. Obstacles are lenticular airfoils with thicknesstochord ratios of 0.07, 0.14, and 0.21. The distances of the airfoil leading edge from the shock focus point are 1, 2, and 2.5, where is the dimensionless reference distance from the origin. Considered impinging shock Mach numbers Ms are 2.2, 2.7, and 3.2 at the reference distance from the origin. The reference experimental configuration ( ) was proposed by Kjellander et al. [“Thermal radiation from a converging shock implosion,” Phys. Fluids 22, 046102 (2010)]. Numerical results compare fairly well to available onedimensional models for shock propagation and to available experimental results in the reference configuration. Local reflection types are in good agreement with the classical criteria for planar shock waves. The main shock reshaping patterns are identified and their dependence on the shock strength and obstacle configuration is exposed. In particular, different shock patterns are observed after the leading edge reflection, which results in polygonal shock wave with N, 2N, 3N, and 4N sides. The largest temperature peak at the origin is obtained for the 8 and the 16obstacle configurations and for the smallest thickness to length ratio, 0.07, located at distance from the origin of . In terms of compression efficiency at the origin, the 16obstacle configuration is found to perform slightly better than the reference 8obstacle configuration—with an efficiency increase of about 2%3%, which is well within the model accuracy—thus confirming the goodness of the obstacle arrangement proposed by Kjellander and collaborators.
 Others

An experimental investigation of vortexinduced vibration of a rotating circular cylinder in the crossflow direction
View Description Hide DescriptionVortexinduced vibration of a flexiblymounted circular cylinder free to oscillate in the crossflow direction with imposed rotation around its axis was studied experimentally. The rotation rate, α, defined as the ratio of the surface velocity and free stream velocity, was varied from 0 to 2.6 in small steps. The amplitudes and frequencies of oscillations as well as the flow forces were measured in a Reynolds number range of Re = 350 1000. The maximum amplitude of oscillations was limited to values less than a diameter of the cylinder at high rotation rates. Also, the lockin range became narrower at higher rotation rates and finally the oscillations ceased beyond α = 2.4. Vortex shedding pattern was found to be 2S (two single vortices shed per cycle of oscillations) for rotation rates up to α = 1.4 and transitioned toward an asymmetric P shedding (one pair of vortices shed in a cycle of oscillations) for rotation rates within the range of 1.4 ≤ α ≤ 1.8. Vortex shedding was found to persist up to higher rotation rates than those observed for a nonoscillating cylinder. The phase difference between the flow forces and displacement of the cylinder in the crossflow direction was influenced as the rotation rate was increased: At high reduced velocities, the phase difference decreased from 180° for a nonrotating cylinder to values close to 90° for a rotating cylinder at large rotation rates. Different shedding patterns resulted in flow forces with different frequencies. In the crossflow direction, the dominant frequency of flow forces was found to be close to the system’s natural frequency for all the rotation rates tested with either 2S or P vortex shedding pattern. In the inline direction, however, the change from 2S to P shedding at high rotation rates resulted in a shift of the ratio of the dominant frequency of the inline flow forces to the natural frequency of the system from 2:1 to 1:1.