Volume 18, Issue 12, December 2006
Index of content:
 ARTICLES

 Interfacial Flows

Wave impact loads: The role of the flipthrough
View Description Hide DescriptionThe impact of waves upon a vertical, rigid wall during sloshing is analyzed with specific focus on the modes that lead to the generation of a flipthrough [M. J. Cooker and D. H. Peregrine, “A model for breaking wave impact pressures,” in Proceedings of the 22nd International Conference on Coastal Engineering (ASCE, Delft, 1990), Vol. 2, pp. 1473–1486]. Experimental data, based on a timeresolved particle image velocimetry technique and on a novel freesurface tracking method [M. Miozzi, “Particle image velocimetry using feature tracking and Delaunay tessellation,” in Proceedings of the 12th International Symposium on Applications of Laser Techniques to Fluid Mechanics (2004)], are used to characterize the details of the flipthrough dynamics while wave loads are computed by integrating the experimental pressure distributions. Three different flipthrough modes are observed and studied in dependence on the amount and modes of air trapping. No air entrapment characterizes a “mode (a) flipthrough,” engulfment of a single, wellformed air bubble is typical of a “mode (b)” event, while the generation of a finescale airwater mixing occurs for a “mode (c)” event. Upward accelerations of the flipthrough jet exceeding have been measured and the generation/collapse process of a small air cavity is described in conjunction with the available pressure time histories. Predictions of the vertical pressure distributions made with the pressureimpulse model of Cooker and Peregrine [M. J. Cooker and D. H. Peregrine, “Pressureimpulse theory for liquid impact problems,” J. Fluid Mech.297, 193 (1995)] show good agreement with the experimental data.
 Viscous and NonNewtonian Flows

Crossstreamline migration in confined flowing polymer solutions: Theory and simulation
View Description Hide DescriptionTheory and Brownian dynamics (BD) simulations are used to study crossstream migration in confined dilute flowing polymer solutions, using beadspring chain and dumbbell models for the polymer molecules. Different degrees of confinement are explored, from a chain above a single wall to slits whose widths are much bigger than the polymer contour length and radius of gyration , much bigger than the radius of gyration but comparable with the contour length , and comparable with the polymer radius of gyration . The results show that except in the latter case, polymer chains migrate in shear flow away from the confining surfaces due to the hydrodynamic interactions between chains and walls. In contrast, when , the chain migration in flow is toward the walls. This is a steric effect, caused by extension of the chain in the flow direction and corresponding shrinkage of the chains in the confined direction; here the hydrodynamic effects of each wall cancel one another out. Considering the polymer chain as a Stokesletdoublet (pointforcedipole) as in a previously developed kinetic theory captures the correct farfield (relative to the walls) behavior. Once a finitesize dipole is used, the theory improves its nearwall predictions. In the regime , the results are significantly affected by the level of discretization of the polymer chain, i.e., number of springs, because the spatial distribution of the forces exerted by the chain on the fluid acts on the scale of the channel geometry.

Small amplitude oscillations of a flexible thin blade in a viscous fluid: Exact analytical solution
View Description Hide DescriptionThe oscillation of a thin blade immersed in a viscous fluid has received considerable attention recently due to its importance in technological applications such as the atomic force microscope and microelectromechanical systems. In this article, we consider the general case of a flexible thin blade executing spatially varying small amplitude oscillations in a viscous fluid. Exact analytical solutions for the threedimensional flow field and hydrodynamic load are derived for both normal and torsional oscillations of arbitrary wave number. This contrasts previous investigations that focus exclusively on the complementary rigidblade problem, which is twodimensional, and rely on computational techniques.

Study of phase transition in homogeneous, rigid extended nematics and magnetic suspensions using an orderreduction method
View Description Hide DescriptionWe study the phase transition in rigid extended nematics and magnetic suspensions by solving the Smoluchowski equation for magnetically polarized rigid nematic polymers and suspensions in equilibrium, in which the molecular interaction is modeled by a dipolar and excluded volume potential. The equilibrium solution (or the probability distribution of the molecular distribution) is given by a Boltzmann distribution parametrized by the (firstorder) polarity vector and the (secondorder) nematic order tensor along with material parameters. We show that the polarity vector coincides with one of the principal axes of the nematic order tensor so that the equilibrium distribution can be reduced to a Boltzmann distribution parametrized by three scalar order parameters, i.e., a polar order parameter and two nematic order parameters, governed by three nonlinear algebraicintegral equations. This reduction in the degree of freedom from 8 (3 in the polarity vector and 5 in the nematic order tensor) to 3 significantly simplifies the solution procedure and allows one to conduct a complete analysis on bifurcation diagrams of the order parameters with respect to the material parameters. The stability of the equilibria is inferred from the second variation of the free energy density.
 Particulate, Multiphase, and Granular Flows

Measurements of the fluid and particle mobilities in strong electric fields
View Description Hide DescriptionWe present a method for measuring both the fluid and particle velocities in strong electric fields and carefully analyze the repeatability and reproducibility of the measurements. The experiments were conducted in capillaries containing dilute aqueous suspensions of polystyrene spheres subjected to dc as well as ac fields of strengths up to and , respectively. These measurements indicate that the predictions of classical linear theories for electrokinetic phenomena apply well beyond the range of relatively weak electric fields for which these theories were developed. The results of our studies are critical for the quantification of microanalytical systems which make use of electrokinetic phenomena for the transport, control, and manipulation of fluids and particles.

Experimental study of twodimensional, monodisperse, frictionalcollisional granular flows down an inclined chute
View Description Hide DescriptionIn this study, positions, velocities, and rotations of monodisperse disks confined twodimensionally in a glasswalled chute are measured using a highspeed camera. Steady, fully developed granular flows (SFD) down bumpy inclines are systematically investigated in the frictionalcollisional (dense, rapid) regime. Three bottoms with different effective roughness heights and roughness distributions are studied to evaluate the influence of the bottom condition. The granular flows are shallow, having a typical depth of ten disk diameters. In the range of flow rates and inclination angles where SFD flows occur, the mean discharge velocity is approximately proportional to the flow depth. The surfacesolid fractions slightly decrease from the bottom to the free surface. The streamwise velocity profiles are close to the linear profile at small inclination angles, whereas at large inclination angles, they are best approximated by the Bagnold profile. The mean angular velocity is equal to the half shear rate everywhere in the flow except near the free surface and the bottom. At large inclination angles, relatively deep SFD flows exhibit an Sshaped granular temperature profile, but in the core, the temperature is far from scaling linearly with the square shear rate. The streamwise and crosswise translational temperatures are slightly different from each other, whereas the rotational temperature is only half of the crosswise translational temperature. The rough bottoms have complex influences on the granular flows as revealed by the velocity and temperature profiles.

The effect of inlet conditions on concentrated suspension flows in abrupt expansions
View Description Hide DescriptionThis work explores the influence of inlet conditions on the particle concentration distribution and flow field of a concentrated suspension undergoing steady flow in an abrupt, axisymmetric 1:4 expansion. Specifically, we consider the impact of inlet conditions in the upstream narrow tube on the resulting downstream profiles. Particle concentration and velocity profiles were measured by using nuclear magnetic resonance imaging. Experiments were conducted with two contrasting inlet tube lengths of and , where is the diameter of the narrow inlet tube. In the short inlet case, none of the particle concentration fields were fully developed upon entering the expansion, whereas several of the suspensions had fully developed profiles in the long inlet case. Results indicate that concentration differences between the core and annular regions were greater for the cases with a long inlet tube than in cases with a short inlet tube, likely owing to the higher degree of particle migration occurring in the long inlet tube before entry into the expansion. Also, the lengths of recirculating regions were greater for suspensions in the long inlet than in the short inlet geometry, while both systems followed the same trend of increasing recirculation length with bulk particle volume fraction and correlation with the ratio of tube center to wall concentration values.
 Instability and Transition

Linear stability of a nonorthogonal axisymmetric stagnation flow on a rotating cylinder
View Description Hide DescriptionThe present analysis deals with the onset of instability in an axisymmetric stagnation flow obliquely impinging on a uniformly rotating circular cylinder. The basic flow is described by an exact solution of the NavierStokes equations, discovered by Weidmann and Putkaradze [Eur. J. Mech. B/Fluids22, 123 (2003)]. An eigenvalue problem for the linear stability is formulated, regardless of the free stream obliqueness, and then solved numerically by means of a collocation method using Laguerre’s polynomials. It is established that the basic stagnation flow is stable for sufficiently high Reynolds numbers. This is in conformity with the unconditional linear stability of twodimensional Hiemenz stagnation flow.Instability occurs for Reynolds numbers smaller than some threshold value that increases with the rotation rate of the cylinder. At criticality, the flow undergoes a Hopf bifurcation, leading then to an oscillatory secondary motion.

Convective instabilities in a rotating vertical HeleShaw cell
View Description Hide DescriptionConvective instabilities driven by vertical buoyancy in a Boussinesq fluid in a rotating vertical HeleShaw cell, a long channel with rectangular cross section of finite height and small width with , are investigated both analytically and numerically. The problem is characterized by the Taylor number , the Rayleigh number , and the aspect ratio . Explicit asymptotic solutions describing convective instabilities are derived for , where is assumed to be large compared to unity. Comparison between the asymptotic and fully numerical solutions shows a satisfactory quantitative agreement. It is found that an overall condition for convective instabilities becomes optimal when . Direct threedimensional simulations for strongly nonlinear convection are also carried out in the regime , where denotes the critical Rayleigh number. As a consequence of both the geometric and dynamic constraints imposed by the narrow channel (geometric) and rapid rotation (dynamic), the nonlinear flow remains temporally stationary and spatially simple and is comprised primarily of vertically long thin convection cells that transport heat all the way from the bottom to the top of the channel.

Influence of Prandtl number on stability of mixed convective flow in a vertical channel filled with a porous medium
View Description Hide DescriptionBuoyancy opposed mixed convection is considered in a vertical channel filled with an isotropic, porous medium, in which the motion of an incompressible fluid is induced by external pressure gradients and buoyancy forces. The BrinkmanWoodingextended Darcy model has been used to study the instability mechanisms of the basic flow and its dependence on the Prandtl number (Pr) of the fluid. The stability analysis indicated that for the same Reynolds number (Re), the fully developed base flow was highly unstable for a fluid with high Pr. For a porous medium with a Darcy number (Da) of and , two different types of instability, RayleighTaylor (RT) and buoyant instability, are observed. The RT instability mode is observed for relatively small values of Re. Further, the results show that for and , the spectrum of the energy profile is abrupt and sudden, whereas the same is smooth when . In the case of RT instability, the critical value of Ra at low Re is given by . Though the RT mode of instability is independent of Pr, the range of Re that sustains the RT mode is a function of Pr. It has been found that enhancement of Pr reduces the Re range mentioned above. In contrast to the case of a purely viscous fluid, where the effect of Pr is not significant, in isotropic porous media Pr plays a significant role in characterizing the flow stability. The instability characteristics of zero temperature flux perturbation (BCI) and zero heat flux perturbation (BCII) on the boundaries differ significantly in the case of the RT stability mode. However, both conditions lead to similar results for buoyant stability, except at small values of Re.

Experimental study of the suppression of RayleighBénard instability in a cylinder by combined rotating and static magnetic fields
View Description Hide DescriptionWe consider experimentally transitions in a liquid metal cylinder heated from below and subject to superimposed rotating and static magnetic fields. The applied static magnetic fields are too weak to influence the characteristic velocity of the rotating field driven basic flow. Being itself turbulent, a strong enough magnetically driven flow suppresses considerably the temperature fluctuations due to the thermogravitational convection. The remaining background fluctuations are caused by unsteady Taylor vortices generated near the sidewall. Our experiment shows that the superimposed static “cusp” magnetic field reduces the amplitude of these remaining temperature fluctuations by a factor of 4, compared to the case with a superimposed uniform axial field. The observed behavior agrees well with the static field effect on the amplitude of the additional unstable Taylor vortextype solutions. These solutions bifurcate subcritically and represent the governing structures in the background turbulence. Thus, the observations are consistent with the description of the background turbulence as an irregular phase trajectory around the skeleton of the subcritical flow states. If this “skeleton” is compressed by an external influence (the cusp static field in our case), then also the amplitude of turbulent fluctuations decreases by the same factor. Another effect of the cusp field is to sharpen the transition between buoyancy and magnetic forcing dominated regimes. This allowed us to obtain an empirical expression for the conditions of this transition. We conclude that the rotating magneticfielddriven flow suppresses the buoyant flow at a much lower angular velocity than a rigidbody mechanical rotation.

Stability of a dragged viscous thread: Onset of “stitching” in a fluidmechanical “sewing machine”
View Description Hide DescriptionA thin thread of viscous fluid that falls on a moving belt acts like a fluidmechanical “sewing machine,” exhibiting a rich variety of “stitch” patterns including meanders, translated coiling, slanted loops, braiding, figuresofeight, Wpatterns, side kicks, and perioddoubled patterns. Using a numerical linear stability analysis, we determine the critical belt speed and oscillation frequency of the first bifurcation, at which a steady dragged viscous thread becomes unstable to transverse oscillations or “meandering.” The predictions of the stability analysis agree closely with the experimental measurements of ChiuWebster and Lister [J. Fluid Mech.569, 89 (2006)]. Moreover, the critical belt speed and onset frequency for meandering are nearly identical to the contactpoint migration speed and angular frequency, respectively, of steady coiling of a viscous thread on a stationary surface, implying a remarkable degree of dynamical similarity between the two phenomena.

Stability diagram and effect of initial density stratification for a twolayer system in a supercritical fluid
View Description Hide DescriptionA numerical study of the stability in a twolayer system filled with a single pure supercritical fluid subjected to an initial temperature difference is performed. The very large compressibility and the very low heat diffusivity of nearcritical fluids lead to a RayleighTaylorlike gravitational instability of the heat diffusion layer. This instability is similar to the one of two misciblefluids where molecular species diffusion coefficient is replaced by the heat diffusion coefficient. Our numerical results are consistent with respect to the dispersion relation derived by Duff et al. [Phys. Fluids5, 417 (1962)] for a system of two incompressible misciblefluids (argonbromine mixture falling into helium or air). It has also been shown that when the thickness of the lower layer becomes smaller than the heat diffusion length based on the maximum growth rate, the system is stable [Phys. Fluids17, 054102 (2005)]. A linear stability diagram has been established as a function of three parameters: the thickness of the lower layer, the density difference between the two layers and the distance to the critical point. When the critical point is approached, the high initial stratification (due to the high compressibility) of this RayleighTaylorlike configuration has seen the effect of stabilizing the system.

Magnetorotational instability in electrically driven flow of liquid metal: Spectral analysis of global modes
View Description Hide DescriptionThe spectral magnetohydrodynamics stability of liquid metal differentially rotating in transverse magnetic field is studied numerically by solving the eigenvalue problem with rigidwall boundary conditions. The equilibrium velocity profile used in calculations corresponds to the electrically driven flow in circular channel with the rotation law . This type of flow profile is planned to be used in new experimental devices to test the magnetorotational instability(MRI) in the laboratory. Our analysis includes calculations of the eigenfrequency spectra for both axisymmetric (with azimuthal wavenumber ) and nonaxisymmetric modes. It is shown that for certain parameters the flow is unstable with respect to MRI with the fastest growth rate corresponding to the axisymmetric mode. For other parameters, the axisymmetric MRI modes can be suppressed and the instability develops only for modes with .
 Turbulent Flows

On turbulent kinetic energy production and dissipation in dilute polymer solutions
View Description Hide DescriptionDrag reduction by dilute polymer solutions is the most recognized phenomenon in wallbounded turbulent flows, which is associated with large scales (e.g., velocity scales) in spite of a consensus that polymers act mainly on much smaller scales of velocity derivatives. We demonstrate that drag reduction is only one sort of polymers’ effect on a turbulent flow and show how turbulent velocity and velocity derivatives are altered in the presence of dilute polymers, irrespective of drag reduction phenomena. This is an experimental study on the interaction of dilute polymers with a complex threedimensional turbulent flow with small mean velocity gradients. Lagrangian data (e.g., velocities and velocity gradients) of flow tracers were obtained by using threedimensional particle tracking velocimetry in an observational volume in the turbulent bulk region, far from the boundaries. The focus is on aspects related to the turbulent kinetic energy (TKE) production, ( is the fluctuating velocity, is the Reynolds stresstensor, and is the mean rateofstrain tensor), such as an anisotropy of Reynolds stresses and the alignment of the velocity vector field with respect to the eigenframe of , among others. We base our study on the comparison of turbulent quantities in flows of water and of dilute polymer solution, forced in two distinct ways: frictional forcing by smooth rotating disks and inertial forcing by disks with baffles. The comparison of the results from the water and from the dilute polymer solution flows allows a critical examination of the influence of polymers on the TKE production, viscous dissipation, and the related turbulent properties. We conclude with (i) quantification of the direct effect of polymers on the small scales of velocity derivatives, (ii) evidence of an additional dissipation mechanism by the polymers, which is the main reason for the strong inhibition of the viscous dissipation, , in a turbulent bulk, (iii) verification that TKE production does not change if the energy input to the flow is at the scales that are not affected by polymers (e.g., inertial forcing or a very rough wall), and last, (iv) evidence for qualitative modification of the turbulent structure, which is not exhausted by the additional dissipation mechanism.

Bounds on the attractor dimension for low wallbound magnetohydrodynamic turbulence
View Description Hide DescriptionSteady lowmagnetohydrodynamic(MHD)turbulence is investigated here through estimates of upper bounds for attractor dimension. A flow between two parallel walls with an imposed perpendicular magnetic field is considered. The flow is defined by its maximum velocity and the intensity of the magnetic field. Given the corresponding Reynolds and Hartmann numbers, one can rigorously derive an upper bound for the dimension of the attractor and find out which modes must be chosen to achieve this bound. The properties of these modes yield quantities that we compare to known heuristic estimates for the size of the smallest turbulent vortices and the degree of anisotropy of the turbulence. Our upper bound derivation is based on known bounds of the nonlinear inertial term, while low Lorentz forces—being linear—can be relatively easily dealt with. The simple configuration considered in this paper allows us to identify some boundaries separating different sets of modes in the space of nondimensional parameters, which are reminiscent of three important previously identified transitions observed in the real flow. The first boundary separates classical hydrodynamic sets of modes from MHD sets where anisotropy takes the form of a “Joule cone.” In the second, one can define the boundary separating threedimensional (3D) MHD sets from quasitwodimensional (2D) MHD sets, when all “OrrSommerfeld modes” disappear and only “Squire modes” are left. The third separates sets where all the modes exhibit the same boundary layer thickness or so, and sets where many different “boundary layer modes” coexists in the set. The nondimensional relations defining these boundaries are then compared to the heuristics known for the transition between isotropic and anisotropicMHDturbulence, 3D and quasi2D MHDturbulence, and that between a turbulent and a laminar Hartmann layer. In addition to this 3D approach, we also determine upper bounds for the dimension of forced turbulent flows modeled using a 2D MHDequation, which should become physically relevant in the quasi2D MHD regime. The advantage of this 2D approach is that, while upper bounds are quite loose in three dimensions, optimal upper bounds exist for the 2D nonlinear term. This allows us to derive realistic attractor dimensions for quasi2D MHDflows.

Influence of outlet geometry on strongly swirling turbulent flow through a circular tube
View Description Hide DescriptionThe results are reported for an extensive series of measurements (using laser Doppler anemometry) of the mean and fluctuatingflow fields for swirling turbulent flow downstream of an orifice in a tube. The influence of a concentric outlet contraction is found to be negligible for low “supercritical” swirl. For high “subcritical” swirl, the outlet geometry is found to have a significant influence throughout the flow field and, in the case of an eccentric (i.e., offset) outlet, to lead to an asymmetric flow with a distorted core. In no case was the core found to precess or the flow to be periodic.

The mechanism of zero mean absolute vorticity state in rotating channel flow
View Description Hide DescriptionIn the core region of spanwise rotating channel flows, the mean velocity profile is approximately linear with a slope of twice the system rotation rate. The mechanism of this zero mean absolute vorticity state is investigated from the turbulence modeling point of view. The mean velocity profile is calculated using three simple nonlinear eddyviscosity models. It is shown that two models, the curvaturecorrected type of explicit algebraic Reynolds stressmodel and the model with the corotational derivative of the secondorder nonlinear term, reproduce well the zero mean absolute vorticity profile. Both models are derived by taking into account the effect of the advection of the Reynolds stress in the rotating frame. In particular, the latter model reflects the memory effect of the secondorder nonlinear term. This effect means that a nonzero absolute vorticity creates a difference between the normal stresses, leading to a large shear stress in a rapidly rotating system. Since the actual value of the shear stress does not increase with the rotation rate, the absolute vorticity needs to be very small. To confirm the memory effect of the nonlinear term, anisotropic and temporally nonlocal effects of the mean velocity on the Reynolds stress are evaluated using Green’s function for the velocity fluctuation.

Investigation of coherent structures in rotating RayleighBenard convection
View Description Hide DescriptionThe current work involves identification of coherent structures in rotating turbulent RayleighBenard convection (RBC) in a moderately large aspectratio (8:8:1) rectangular enclosure. The enclosure is rotating about a vertical axis passing through its center of gravity. The incompressible NavierStokes and energy equations are solved in a rotating frame of reference and the resulting velocity and thermal fields are analyzed to educe coherent structures. The flowstructures have been investigated at different nondimensional rotation rates ranging from to for a fixed Rayleigh number and Prandtl number , keeping the ratio constant at in order to stringently maintain Boussinesq approximation as well as to attain experimentally realizable rotational Rayleigh numbers. Coherent structures in the flow domain have been sought using different identification techniques, namely large eddy simulation(LES) decomposition using tophat filter, proper orthogonal decomposition (POD) considering larger energy modes, second invariant of the velocity gradient tensor, and regions of negative , the second largest eigenvalue of the tensor. It has been found that the coherent structures educed using POD or LES decomposition at low to moderate rotation ( to ) show the formation of two to three largescale rolls aligned along both horizontal directions. At higherrotation rates corresponding to , there is a breakup of largescale structures into multiple smallscale rolls having random spatial orientation. The thermal structures educed using both POD and largescale LES decomposition at zero rotation show randomly rising and descending plumes that at coalesce to form a large cylindrical thermal plume in the core of the cavity. Further increase of rotation leads again to breakup of the cylindrical plume into multiple random plumes. Isosurfaces of and reveal elongated tubular rolllike structures mainly concentrated near the sidewall regions, though at higher rotationrate the density of tubular structures increases significantly. There is a strong similarity of results obtained by largescale LES decomposition and results obtained using mean plus first mode of POD decomposition. The structures educed using and are different in shape compared to LES or POD educed structures.

Direct numerical simulation of turbulent channel flow under a uniform magnetic field for largescale structures at high Reynolds number
View Description Hide DescriptionA direct numerical simulation (DNS) of turbulent channel flow with high Reynolds number has been carried out to show the effects of the magnetic field. In this study, the Reynolds number for channel flow based on bulk velocity , viscosity, and channel width was set to be constant; . A uniform magnetic field was applied in the direction of the wall normal. The value of the Hartmann number, Ha were 32.5 and 65, where . The turbulent quantities such as the mean flow,turbulent stress, and turbulent statistics were obtained by DNS. Although the influence of the magnetohydrodynamic dissipation terms in the turbulent kinetic energy budget was small, largescale turbulent structures, e.g., vertical structures, lowspeed streaks, ejection, and sweep, were found to decrease at the central region of the channel. Consequently, the difference between production and dissipation in the turbulent kinetic energy decreased with increasing Hartmann number at the central region and largescale structures at this region were reduced.