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Volume 9, Issue 3, March 1997

Conserved quantities in Stokes flow with free surfaces
View Description Hide DescriptionWe derive an infinite set of invariants for twodimensional Stokes flow with a free surface, driven by a point sink, in the case that surface tension effects are negligible. The complex variable methods used are closely related to those used for certain Hele–Shaw flows, which likewise have infinitely many conserved quantities.

Experimental measurement of dispersion processes at short times using a pulsed field gradient NMR technique
View Description Hide DescriptionDispersion at short times is studied using a PFGNMR (pulsed field gradient NMR) technique inside a fixed bed of nonconsolidated spherical beads saturated with water flowing at a constant velocity. This allows measurement of the probability distribution of the displacement of water molecules along the magnetic field gradient during a preset measurement time : the mean displacement of the water molecules is varied between 0.1 and 7.3 times the bead diameter by varying between 20 and 100 ms and the bead diameter between 800 and 81 m. At short times, the displacement of the molecules is small enough so that the local displacement is proportional to the local velocity component along the magnetic field gradient. At mean displacements larger than 5 bead diameters, the displacement distribution is Gaussian and centered about the mean displacement; the width of the distribution corresponds to the macroscopic dispersion coefficient as measured by other techniques. At intermediate displacements, this distribution displays two peaks corresponding to a combination of the two processes. The main features of this transition can be reproduced by a simple MonteCarlo simulation modeling the porous medium as a set of finite length tubes with random orientations.

Sedimentation of homogeneous suspensions of nonBrownian spheres
View Description Hide DescriptionDynamical simulations of bulk sedimentation have been carried out, using up to 32 000 solid particles. There is no evidence that the longrange hydrodynamic interactions are screened by changes in the pair correlation function at large distances. Instead the velocity fluctuations and diffusion coefficients diverge linearly with the width of the container, consistent with the random longrange microstructures observed in the simulations. Our data suggest that other mechanisms must be uncovered to account for experimental observations.

Experimental investigation of the instabilities in a fluidized bed origin of the pressure fluctuations
View Description Hide DescriptionThis paper presents an experimental investigation of the hydrodynamicinstabilities inside a selfexcited fluidized bed. Pressure fluctuation and bed height measurements have been reported. In the slugging regime (above a critical gas flow rate), the bulk is found to exhibit a regular and periodic macroscopic pattern (large numbers of particles moving collectively in a seemingly organized manner). This state is characterized by nearly signoidal pressure fluctuations and regular bed height oscillations. As the gas flow increases the bulk motion tends toward large oscillations with intermittent smaller ones. The pressure fluctuations above the bed surface and the bed height oscillations are found to be intimately correlated. We report experimental evidence that the pressure fluctuations observed in a fluidized bed under and above its surface are caused by the oscillations of the bed height. The pressure wave initiates at the surface of the bed and propagates both upward and downward.

On the problem of natural convection in liquid phase thermotransport coefficients measurements
View Description Hide DescriptionWe focus in this paper on the effect of natural convection in thermodiffusion coefficients measurements in liquid metal alloys both for normal and microgravity conditions. Our previous experimental results are briefly recalled, with a special emphasis on the data recently obtained from the EURECA space mission. With respect to the ground based values, it is seen that the solutal separation is always significantly higher in microgravity, even in systems where solutal stabilization of the flow has an effect. Simple scalinganalysis arguments show that the error induced by additional convective transport scales with the square of the fluid velocity. Such a result compares favorably with existing three dimensional (3D) numerical data. The theory also accounts qualitatively for the reduced separation observed experimentally in ground based setups. We conclude that it is in principle possible to perform accurate measurements in space, but that the size of the capillaries used in the experiments should always be limited to roughly two millimeters. On Earth on the other hand, the risk of convective interference cannot be avoided.

Numerical simulations of the translational and shape oscillations of a liquid drop in an acoustic field
View Description Hide DescriptionIn this work, the boundary element method combined with the fourth order Runge–Kutta scheme as time integrator is used to simulate the dynamics of an acoustically levitated axisymmetric liquid drop. For a given set of dimensionless parameters—wavenumber, Bond number, and acoustic Bond number—the drop dynamics in an acoustic field is studied in terms of the shape oscillation and the translational motion of the drop. The shape oscillation of the drop is characterized by using the equatorial radius and its rate of change as two phase variables. Fixed points on this phase plane represent the static equilibrium shapes. The translational motion is characterized by using the position and the velocity of the drop centroid as two phase variables. The fixed points on this phase plane represent the equilibrium positions of the drop in the acoustic field. It is found that fixed points corresponding to both translational and shape oscillations undergo saddlenode bifurcations with the acoustic Bond number as a parameter. These saddlenode bifurcations define an upper and a lower limit on the acoustic Bond number that can be used in acoustic levitation. We also investigate the coupling effect between the translational oscillation and the shape oscillation. It is found that owing to the orderofmagnitude difference between the period of translational oscillation and that of shape oscillation the coupling effect is only significant at the boundary of the trapping region.

Linear stability and transient growth in driven contact lines
View Description Hide DescriptionFluid flowing down an inclined plane commonly exhibits a fingering instability in which the contact line corrugates. We show that below a critical inclination angle the base state before the instability is linearly stable. Several recent experiments explore inclination angles below this critical angle, yet all clearly show the fingering instability. We explain this paradox by showing that regardless of the long time linear stability of the front, microscopic scale perturbations at the contact line grow on a transient time scale to a size comparable with the macroscopic structure of the front. This amplification is sufficient to excite nonlinearities and thus initiate finger formation. The amplification is a result of the wellknown singular dependence of the macroscopic profiles on the microscopic length scale near the contact line. Implications for other types of forced contact lines are discussed.

Cavity dynamics in highspeed water entry
View Description Hide DescriptionA method is presented for modeling the cavity formation and collapse induced by highspeed impact and penetration of a rigid projectile into water. The approach proposes that highspeed waterentry is characterized by a cavity that experiences a deep closure prior to closure at the surface. This sequence in the physical events of the induced cavitydynamics is suggested by the most recent highspeed waterentry experimental data, by results from numerical experiments using a hydrocode, and by an understanding of the fundamental physics of the processes that govern surface closure. The analytical model, which specifies the energy transfer for cavity production as equivalent to the energy dissipated by velocitydependent drag on the projectile, provides accurate estimates for variables that are important in characterizing the cavitydynamics, and reveals useful knowledge regarding magnitudes and trends. In particular, it is found that the time of deep closure is essentially constant and independent of the impact velocity for a given projectile size, while the location of deep closure has a weak dependence on impact velocity. Comparison of these analytical results with experimental results from the literature and with results from numerical simulations verifies the analytical solutions.

Reduced dynamical models of nonisothermal transitional groovedchannel flow
View Description Hide DescriptionReduced dynamical models are derived for transitional flow and heat transfer in a periodically grooved channel. The full governing partial differential equations are solved by a spectral element method. Spontaneously oscillatory solutions are computed for Reynolds number Re⩾300 and proper orthogonal decomposition is used to extract the empirical eigenfunctions at Re=430, 750, 1050, and Pr=0.71. In each case, the organized spatiotemporal structures of the thermofluid system are identified, and their dependence on Reynolds number is discussed. Lowdimensional models are obtained for Re=430, 750, and 1050 using the computed empirical eigenfunctions as basis functions and applying Galerkin’s method. At least four eigenmodes for each field variable are required to predict stable, selfsustained oscillations of correct amplitude at “design” conditions. Retaining more than six eigenmodes may reduce the accuracy of the loworder models due to noise introduced by the lowenergy high order eigenmodes. The loworder models successfully describe the dynamical characteristics of the flow for Re close to the design conditions. Far from the design conditions, the reduced models predict quasiperiodic or perioddoubling routes to chaos as Re is increased. The case Pr=7.1 is briefly discussed.

Flow regimes in model viscoelastic fluids in a circular couette system with independently rotating cylinders
View Description Hide DescriptionFlow visualization of two highly elastic, nonshearthinning polyisobutylene/polybutene fluids in the gap between concentric cylinders was performed over a range of shear rates and choices of relative cylinder rotations. The observed secondary flows are discussed in terms of destabilizing elastic and centrifugal forces. In the more viscous, more elastic fluid, instabilities are found to be independent of the choice of rotating cylinder and due entirely to elasticity. At the lowest shear rates examined, the first detectable secondary flows are steady counterrotating vortices forming after a shearing time more than five orders of magnitude greater than the characteristic relaxation time of the fluid. At somewhat higher shear rates, a much more rapidly appearing oscillatory flow is observed to evolve into the steady vortex structure. In the less elastic fluid, the structure first detectable at the lowest shear rates is again steady vortices regardless of the choice of driving cylinder. At all shear rates examined, only elastic stationary vortices are observed in the absence of centrifugal destabilization (outer cylinder rotating). Secondary flows are significantly stronger in the presence of the centrifugal destabilization due to a rotating inner cylinder. Interaction of elasticity and centrifugal forces is found to generate a number of axially translating vortex structures, many of which are described here for the first time. At a shear rate more than five times the critical, another family of instability is observed which closely resembles a purely elasticinstability observed by Baumert and Muller (1995). These experimental results are expected to provide a challenging test of numerical simulations of these viscoelasticflows.

Drag reduction by dc corona discharge along an electrically conductive flat plate for small Reynolds number flow
View Description Hide DescriptionCoronainduceddrag reduction was studied numerically over a finite region of a semiinfinite flat plate having small Ohmic surface conductivity for low Reynolds number flow (<100 000, based on the farthest downstream electrode distance). The model simulates a corona discharge along a surface from two parallel wire electrodes of infinite length immersed flush on the surface and oriented perpendicular to the flow.Charge deposition and removal with the conducting surface are included as possible charge transfer mechanisms. The analysis is limited to ions of positive charge. Five coupled partial differential equations govern the numerical model including continuity, momentum, gas phase conservation of charge, Poisson’s equation of electrostatics, and conservation of charge at the solidinterface. The governing equations together with empirical breakdown and current–voltage relationships (Φ–I characteristic) were evaluated by finite differencing schemes. The calculated results predict “corona thinning” of the boundary layer for a downstream ion flow and a corresponding reduction in drag, in agreement with previous theoretical studies. Various parameters of flow, electricity, and geometry, relating to coronainduceddrag, are investigated.

On the unsteady lowRossby number flow of a rotating fluid past a circular cylinder
View Description Hide DescriptionThe unsteady flow of a homogeneous viscous fluid past a straight circular cylinder (radius confined between two infinite parallel plates (a distance apart) relative to a rapidly rotating frame is considered. The cylinder is impulsively started from rest to a uniform velocity. The unsteady form of the boundarylayer equations for a rotating fluid is used to examine the flow Rossby number where is the Ekman number. A range of values of the nondimensional parameter (where is considered. For the flow pattern resembles that of the nonrotating case Initially, the wall shear around the cylinder is positive everywhere. After a time, flow reversal begins at the rear stagnation point and then the position of zero wall shear moves upstream, towards the front stagnation point. The boundarylayer thickness in the region of reversed flow grows with time until a singularity/eruption at a point in the flow occurs. The boundarylayer equations are written in terms of Lagrangian coordinates in order to numerically investigate the finitetime singularity for The flow close to the rear stagnation point is also examined in detail for a range of values of and results are compared with the largetime asymptotic forms for the growth of the displacement thickness. The analysis suggests the displacement thickness in this region grows exponentially with time, for certain ranges of For the displacement thickness grows exponentially with time in a manner similar to the nonrotating case. For the wall shear remains positive for all time. However, for the displacement thickness of the boundary layer close to the rear stagnation point again grows exponentially with time. For the flow close to the rear stagnation point also grows exponentially with time, although the form of solution differs from that for For the solution tends to a truly steady limit, consistent with previous studies on the steady problem.

Sideband instabilities of mixed barotropic/baroclinic waves growing on a midlatitude zonal jet
View Description Hide DescriptionThe instability of an arbitrarily shaped zonal jet on a midlatitude plane is considered within a twolayer quasigeostrophic model with linear friction. Depending on the horizontal and vertical shear of the jet, it is susceptible to both barotropic and baroclinic instabilities. The linear stability boundaries are determined numerically for a parameter regime relevant to the Gulfstream. The weakly nonlinear (finite amplitude) evolution of the instabilities is shown to be governed by a GinzburgLandau equation and for arbitrary jet shapes the coefficients in this equation are computed numerically. The finite amplitude state is shown to become unstable to BenjaminFeir sideband instabilities. The mixed baroclinic/barotropic character of the primary instability is crucial to this sideband instability which is shown to lead to complicated spatiotemporal behavior of the jet.

Chaotic properties of internal wave triad interactions
View Description Hide DescriptionWe discuss the stochasticities of twotriad interactions that occur in twodegreeoffreedom autonomous Hamiltonian systems. The system we study is a twotriad testwave system consisting of a single internal wave mode (testwave) interacting with a spectrum of ambient internal wave modes; the ambient modes, however, do not interact among themselves except through a threewave interaction which includes the testwave. The present study concerns the effect of nonlinearities on the oceaninternal wave field. Our numerical results using the physical parameters appropriate for the deep ocean confirm that the testwave system is nonintegrable. Moreover, there exists a certain separatrix net that fills the phase space and is covered by a thin stochastic layer for a twotriad pure resonant interaction. The stochastic web implies the existence of diffusion of the Arnold type for the minimal dimension of a nonintegrable autonomous system. For the nonresonant case, the stochastic layer is formed where the separatrix from KAM theory is disrupted. However, the stochasticity does not increase monotonically with increasing energy.

Stabilization of longitudinal vortex instabilities by means of transverse flow oscillations
View Description Hide DescriptionLongitudinal vortices with axes in the direction of a mean shear flow can arise due to body force instabilities associated with heating or centrifugal effects. In this paper we are concerned with possible stabilization or destabilization of such instabilities by means of a controlledflow oscillation in the spanwise direction. For the heated case, stabilization occurs up to the values of Rayleigh and Reynolds numbers at which streamwise twodimensional disturbances become critical. For the centrifugal case, stabilization occurs only for sufficiently large values of the Reynolds number associated with the spanwise oscillation; for smaller values, destabilization occurs.

Vortex dynamics in jets from inclined nozzles
View Description Hide DescriptionExperimental tests were performed on round jets exiting inclined nozzles at a Reynolds number of 9000. Both natural jets and jets forced with single frequencies corresponding to 0.5, 0.75, and 1.0 were examined. In the natural case, the nozzle incline caused a mild increase in the radial spreading in the plane of azimuthal symmetry. The forcing amplified the asymmetric radial spreading by altering the vortex structure. In general, the inclined vortex rings rolled up at an angle slightly smaller than the nozzle incline angle. As the rings moved downstream, they migrated away from the jet centerline and their incline angle increased. Vortex rings generated at did not pair because that Strouhal number was near the “preferred” mode. For nozzles with slight inclines, forcing at larger Strouhal numbers led to pairing near in order to achieve the “preferred” mode. For nozzles with larger inclines, the vortex cores broke down before pairing could occur. Forcing at a lower Strouhal number yielded ring formation at and subsequent pairing. Increasing the incline angle moved the pairing location closer to the nozzle lip. Also, the pairing process was found to depend on the nozzle incline angle.

Mixing enhancement in a lobed injector
View Description Hide DescriptionAn experimental investigation of the nonreactive mixing processes associated with a lobed fuel injector in a coflowing air stream is presented. The lobed fuel injector is a device which generates streamwise vorticity, producing high strain rates which can enhance the mixing of reactants while delaying ignition in a controlled manner. The lobed injectors examined in the present study consist of two corrugated plates between which a fuel surrogate, , is injected into coflowing air. Acetone is seeded in the supply as a fuel marker. Comparison of two alternative lobed injector geometries is made with a straight fuel injector to determine net differences in mixing and strain fields due to streamwise vorticity generation. Planar laserinduced fluorescence (PLIF) of the seeded acetone yields twodimensional images of the scalar concentration field at various downstream locations, from which local mixing and scalar dissipation rates are computed. It is found that the lobed injector geometry can enhance molecular mixing and create a highly strained flowfield, and that the strain rates generated by scalar energy dissipation can potentially delay ignition in a reacting flowfield.

Absolute/convective transition of wakedominated supersonic shear layers
View Description Hide DescriptionThe absolute and convective instabilitycharacteristics of supersonic wake dominated shear layers, described by a hyperbolictangent profile plus a wake component represented by a Gaussian distribution, are investigated. The effects of the Mach number, the freestream temperature ratio and the wake deficit on the boundary of the absolute/convective transition and the branch point parameters (such as frequency, wave number, and the spatial growth rate) are studied using linear stability theory. For supersonic mixing layers, it is found that at a given wake deficit the amount of backflow necessary to cause absolute instability decreases as the convective Mach number increases and that all the branch point parameters vary slowly as functions of the convective Mach number as opposed to subsonic mixing layers. As the wake deficit decreases, the curve separating the absolute and the convective regions moves down to the values of more negative freestream velocity ratio for both subsonic and supersonic mixing layers and the branch point spatial growth rate decreases for a given convective Mach number. The effect of decreasing the wake deficit on the absolute/convective transition boundary is similar to that of increasing the freestream temperature ratio.

Asymptotic analysis of a vertical Bridgman furnace at large Rayleigh number
View Description Hide DescriptionA vertical Bridgman furnace, through which an ampoule containing the melt of a certain dilute binary alloy is pulled at a fixed, and predetermined speed, provides a means of improving certain alloys. Radial segregation in the finished crystalline material can make it unusable. In this paper, we construct an asymptotic theory for the flow and solidification in the ampoule, for large Rayleigh numbers but at a small Biot number. We find largeRayleighnumber solutions to be, to leading order, completely insensitive to the character of the sidewall boundary condition on vertical velocity. Twodimensional equivalents of optimization conditions found by Tanveer are recovered for his two limiting cases—large thermal Rayleigh number, and large negative solutal Rayleigh number. Moderate surface tension at the crystal–melt interface is found to have no effect on the optimization conditions for the two limit cases, but it does somewhat reduce the magnitude of the segregation in both limits. In addition, we present new results for the case for which the two Rayleigh numbers are of comparable magnitude and show that there is an optimization possible for this case too. Conversion of the results of this paper to an axisymmetric geometry is shown to be trivial. Keeping careful track of the ordering, we indicate how to proceed to first effects of nonlinearity at small Biot and/or Prandtl number.

Nonlinear Lagrangian equations for turbulent motion and buoyancy in inhomogeneous flows
View Description Hide DescriptionLinear and nonlinear Lagrangianequations are derived for stochastic processes that appear as solutions of the averaged hydrodynamicequations, since their moments satisfy the budgets given by these equations. These equations include the potential temperature, so that nonneutral flows can be described. They will be compared with nonlinear and nonMarkovian equations that are obtained using concepts of nonequilibrium statistical mechanics. This approach permits the description of turbulent motion and buoyancy, where memory effects and driving forces with arbitrary colored noise may occur. The equations depend on assumptions that concern the dissipation and pressure redistribution. In the approximations of Kolmogorov and Rotta for these terms, the dissipation time scale remains open, which can be determined by the calculation of the production–dissipation ratio of turbulent kinetic energy. The features of these equations are illustrated by the calculation of turbulent states in the space of invariants.