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Volume 8, Issue 8, August 1996

Experimental investigations on the nature of the first wavy instability in liquid‐fluidized beds
View Description Hide DescriptionExperiments are described which suggest that the first wavy instability of fluidized beds is convective in nature. In particular, this instability is shown to be sensitive to a harmonic forcing localized at the bottom of the bed.

An O(N) algorithm for Stokes and Laplace interactions of particles
View Description Hide DescriptionA method for computing Laplace and Stokes interactions among N spherical particles arbitrarily placed in a unit cell of a periodic array is described. The method is based on an algorithm by Greengard and Rokhlin [J. Comput. Phys. 73, 325 (1987)] for rapidly summing the Laplace interactions among particles by organizing the particles into a number of different groups of varying sizes. The far‐field induced by each group of particles is expressed by a multipole expansion technique into an equivalent field with its singularities at the center of the group. The resulting computational effort increases only linearly with N. The method is applied to a number of problems in suspension mechanics with the goal of assessing the efficiency and the potential usefulness of the method in studying dynamics of large systems. It is shown that reasonably accurate results for the interaction forces are obtained in most cases even with relatively low‐order multipole expansions.

Experimental measurement of shear‐induced diffusion in suspensions using long time data
View Description Hide DescriptionA method based upon Taylor dispersion theory is used to determine the shear‐induced diffusion coefficient in concentrated suspensions. The experiments are performed in a cylindrical Couette device with a suspension consisting of polystyrene spheres in a density‐matched solution of glycerin and water. A sequence of several hundred transit times for a single tagged sphere to complete successive orbits within the device is measured. The data are analyzed to compute the azimuthal Taylor dispersion coefficient from which the coefficient of shear‐induced diffusivity is obtained. In our experiments the particle Reynolds numbers are O(10^{−1}). The experimental results are compared to the existing measurements of the shear‐induced diffusion coefficient obtained at lower particle Reynolds numbers and based upon short‐time data. We find a shear‐enhanced diffusion coefficient D _{⊥}/γ̇a ^{2}=O(0.1) for a volume fraction of φ≊0.25; this is comparable to existing results from previous low particle Reynolds number studies (R<10^{−3}).

Flow due to a periodic array of point forces, and the motion of small particles within a cylindrical tube of arbitrary cross section
View Description Hide DescriptionThe properties and computation of Stokes flow due to a periodic array of point forces exerted in the interior of a fluid‐filled cylindrical tube with an arbitrary cross‐sectional shape are discussed. It is shown that the relationship between the pressure drop and the axial flow rate occurring when the point forces have a component parallel to the generators can be deduced immediately from a knowledge of the velocity profile corresponding to unidirectional pressure‐driven flow. A boundary‐integral method for computing the associated Green’s function of Stokes flow is developed and implemented in a numerical procedure that exploits the cylindrical boundary geometry to improve the accuracy of the results and efficiency of the computations. Streamline patterns of the flow within tubes with circular, elliptical, and nearly square shapes are presented and discussed with reference to flow reversal. In the limit as the separation between the point forces becomes increasingly larger than the typical size of the cross section of the tube, we recover the flow due to a solitary point force, and the numerical result are in agreement with those derived by previous authors for the particular case of a tube with a circular cross‐sectional shape. The flow due to the point forces is then coupled with the boundary integral representation to develop asymptotic expansions for the surface stress, force, torque, and higher moments of the traction exerted on a small suspended particle that belongs to a periodic array. Each particle may translate and rotate while the ambient fluid undergoes pressure‐driven flow. The coefficients of the asymptotic expansion are related to the non‐singular part of the Green’s function and its spatial derivatives, evaluated at the location of the point force. These quantities are computed and plotted for several cross‐sectional shapes.

An integral equation approach to internal (2‐layer) solitary waves
View Description Hide DescriptionThe exact within potential flowintegral equation approach of Evans and Ford [Proc. R. Soc. London Ser. A 452, 373 (1996)] for the normal solitary wave, is here generalized to 2‐layer, ‘‘internal’’ solitary waves. This differs in its mathematical form from other exact integral equation methods based on the complex velocity potential. For both ‘‘rigid lid’’ (i.e., flat toplayer surface) and ‘‘free‐surface’’ boundary conditions, a set of coupled non‐linear integral equations are derived by an application of Green’s theorem. For each point on the layer interface(s), these describe functional constraints on the profiles and interface fluid velocity moduli; the exact profiles and velocities being those forms that satisfy these constraints at all such interface points. Using suitable parametric representations of the profiles and interface velocity moduli as functions of horizontal distance, x, and utilizing tailored quadrature methods [Int. J. Comput. Math. B 6, 219 (1977)], numerical solutions were obtained by the Newton–Raphson method that are highly accurate even at large amplitudes. For ‘‘rigid lid’’ boundary conditions,internal wave solutions are presented for layer density and depth ratios typical of oceanicinternal wave phenomena as found in the Earth’s marginal seas. Their various properties, i.e., mass, momentum, energy, circulation, phase and fluid velocities, streamline profiles, internal pressures, etc., are evaluated and compared, where possible, with observed properties of such phenomena as reported, for example, from the Andaman Sea. The nature of the limiting (or ‘‘maximum’’) internal wave is investigated asymptotically and argued to be consistent with two ‘‘surge’’ regions separating the outskirts flow from a wide mid‐section region of uniform ‘‘conjugate flow’’ as advocated by Turner and Vanden‐Broeck [Phys. Fluids 31, 286 (1988)].

Spin‐up in a circular tank with a radial barrier
View Description Hide DescriptionThe time‐dependent motion of fluid in a circular tank with a radial barrier as a result of an increase in angular velocity of the tank is investigated. The length of the barrier is considered as the main experimental parameter. The flow field immediately after the increase in angular velocity is calculated analytically. Experiments have been performed with a tank placed on a rotating table. Quantitative results for the time‐dependent flow were obtained by the tracking of small particles floating at the free surface of the fluid. The flow appears to be characterized by separation from the end of the barrier and the subsequent formation of a stable vortex pattern. The trajectory of the vortex that is shed from the end of the barrier is determined with dye visualization, and compared with analytical results from a point‐vortex model.

The generation of edge waves by a wave‐maker
View Description Hide DescriptionA theory is developed to describe the generation of edge waves on a uniform beach by a wave‐maker. The theory is based on the linear shallow‐water equations. The wave‐maker is a vertical plate spanning between the shoreline and the paddle axis offshore, and oscillates periodically in the alongshore direction. It is found theoretically that only propagating modes exist; the evanescent modes are always accompanied by incoming waves from offshore and are not permissible in this case. For each propagation mode the cross‐shore variation is described by the Laguerre polynomial with an exponentially decaying amplitude. Laboratory experiments are performed and experimental data are compared with theoretical solutions. Since the viscous damping is ignored in the theory, the wave amplitudes for the experimental data are usually lower than the theoretical predictions. However, the cross‐shore variations of the wave form are predicted well by the theory. Furthermore, from both theoretical and experimental data, it is shown that wave fields are dominated by the Stokes edge‐wave mode in the low frequency range, f<0.5 Hz.

Different approximations of shallow fluid flow over an obstacle
View Description Hide DescriptionThree different sets of shallow water equations, representing different levels of approximation are considered. The numerical solutions of these different equations for flow past bottom topography in several different flow regimes are compared. For several cases the full Euler solutions are computed as a reference, allowing the assessment of the relative accuracies of the different approximations. Further, the differences between the dispersive shallow water (DSW) solutions and those of the highly simplified, hyperbolic shallow water (SW) equations is studied as a guide to determining what level of approximation is required for a particular flow. First, the Green‐Naghdi (GN) equations are derived as a vertically‐integrated rational approximation of the Euler equations, and then the generalized Boussinesq (gB) equations are obtained under the further assumption of weak nonlinearity. A series of calculations, each assuming different values of a set of parameters—undisturbed upstream Froude number, and the height and width of the obstacle, are then presented and discussed. In almost all regions of the parameter space, the SW and DSW theories yield different results; it is only when the flows are entirely subcritical or entirely supercritical and when the obstacles are very wide compared to the depth of the fluid that the SW and DSW theories are in qualitative and quantitative agreement. It is also found that while the gB solutions are accurate only for small bottom topographies (less than 20% of the undisturbed fluid depth), the GN solutions are accurate for much larger topographies (up to 65% of the undisturbed fluid depth). The limitation of the gB approximation to small topographies is primarily due to the generation of large amplitude upstream propagating solitary waves at transcritical Froude numbers, and is consistent with previous analysis. The GN approximation, which makes no assumptions about the size of the nonlinearity, is thus verified to be a better system to use in cases where the bottom topographies are large or when the bottom topographies are moderate but the flow transcritical.

Vertical water entry of disks at low Froude numbers
View Description Hide DescriptionAs basilisk lizards (Basiliscus basiliscus) and shore birds run along the water surface they support their body weight by slapping and stroking into the water with their feet. The foot motions exploit the hydrodynamic forces of low‐speed water entry. To determine the forces that are produced during water entry at low speeds, we measured directly the impact and drag forces for disks dropped into water at low Froude numbers (u ^{2}/gr=1–80). Also, we measured the period during which the air cavity behind the disk remains open to atmospheric air. We found that the force impulse produced during the impact phase is due to the acceleration of the virtual mass of fluid associated with a disk at the water surface. A dimensionless virtual mass M, defined as M=m _{virtual}/(4/3)πρr ^{3}, has a value near 1/π for disks. After impact, as penetration depth of the disk increases, the drag force can rise by as much as 76% even though the downward velocity is steady. However, a dimensionless force which includes the contribution from hydrostaticpressure [C _{ D } ^{*}=Drag(t)/(ρSgh(t)+0.5ρSu ^{2})] takes a constant value near 0.7 regardless of disk size, speed, or cavity depth. Over the entire range of disk sizes and velocities, the period between impact and cavity closure, T _{seal}, can be described by a single value of dimensionless time, τ=T _{seal}(g/r)^{0.5}, near 2.3. We con‐ clude that the fundamental phenomena associated with the low‐speed water entry of a disk can be characterized by three dimensionless numbers (M, C _{ D } ^{*}, and τ).

On the three‐dimensional instability of elliptical vortex subjected to stretching
View Description Hide DescriptionIt is known that two‐dimensional vortices are subject to generic three‐dimensional instabilities. This phenomenon is located near the core of vortices and depends on the eccentricity of their streamlines. In this paper we are concerned with the modification of this instability when stretching is applied to such vortices. We describe this instability by linearizing the Navier–Stokes equations around a basic state, which is an exact time‐dependent solution. The complete system for the perturbations is reduced to a single equation for the perturbed velocity along the vortex span. In the limit of weak stretching, a perturbation theory can be performed and leads to a WKBJ approximation for the solution. This procedure demonstrates that a small amount of stretching is able to prevent the appearance of three‐dimensional instabilities for vortices with a low enough eccentricity. Since most vortices are slightly elliptical in turbulent flows, the above computations are expected to cover a wide range of experimental cases. In particular, it is tentatively argued that this mechanism may explain recent experimental observations [Phys. Fluids 7, 630 (1995)].

Thermal and electrohydrodynamic plumes. A comparative study
View Description Hide DescriptionIn this paper we deal with self‐similar thermal and electrohydrodynamic (EHD) plumes. The former arises from hot lines or points, whereas the latter arises when sharp metallic contours submerged in nonconducting liquids support high electrostatic potential, resulting in charge injection. Although the motive force is buoyancy in one case and Coulomb force in the other, it is shown that the solution for EHD plumes is the same as for thermal plumes in the limit of large Prandtl numbers. We present the analysis of axisymmetric plumes for large values of Prandtl number, and this analysis is subsequently applied to EHD plumes. The validity of the approximations for EHD plumes is discussed in the light of experimental data.

Collective behavior of wakes downstream a row of cylinders
View Description Hide DescriptionThis experimental study is devoted to visualisation and ultrasonic velocitymeasurement of the wakes formed behind a row of parallel cylinders placed side by side, perpendicular to an incoming flow at low Reynolds numbers. When the distance separating the cylinders is small compared to their diameter, two instability mechanisms, associated with different patterns and dynamics compete. A first spatial symmetry breaking appears when the stationary wakes behind each cylinder are deviated towards one side or the other and form large clusters containing from two to sometimes more than ten wakes. These clusters are separated by intense recirculating zones. When the Reynolds number is increased, the wakes belonging to the widest clusters experience a secondary temporal oscillatory bifurcation. Classical Bénard‐Von Kármán vortex streets are thus shed in phase by these cylinders (acoustic mode), by contrast with the wakes outside these cells which stay stationary. Finally, the flow around far apart cylinders is also investigated. The primary instability does not occur in this case and a perfect optical mode of vortex shedding, with neighbours in phase opposition, takes place in the flow.

Near‐wake of a perturbed, horizontal cylinder at a free‐surface
View Description Hide DescriptionA horizontal cylinder intersecting a free surface is subjected to controlled vertical perturbations, and the consequent vortex formation is characterized by high‐image‐density particle image velocimetry, which leads to instantaneous patterns of velocity, vorticity, and streamlines. For the limiting case of the stationary cylinder, the near wake does not exhibit rapid formation of organized vortical structures in a manner similar to Kármán vortices. Application of perturbations, however, generates phase‐locked vortex formation over a wide range of excitation frequencies, even at relatively low amplitudes, indicating that the near wake in presence of a free surface is convectively, rather than absolutely, unstable. At a sufficiently high value of excitation frequency, the formation of the initial vortex undergoes an abrupt change in timing, which is analogous to that occurring for Kármán vortex formation from a completely submerged cylinder. All of these features of the near wake are interpreted in terms of foci, saddle points, and reattachment points of the streamwise topology.

On secondary vortices in the cylinder wake
View Description Hide DescriptionThe wake of a circular cylinder is investigated for Reynolds numbers between 160 and 500 by means of particle image velocimetry(PIV). For the first time cross‐stream velocity fields are determined for two classes of secondary vortices (A‐mode and B‐mode). The circulation of the A‐mode secondary vortices in this plane is approximately twice the circulation of the B‐mode secondary vortices. The spanwise wavelength of the secondary vortices is four to five cylinder diameters for the A‐mode and one diameter for the B‐mode. The spatio‐temporal development of the wake is analyzed by acquiring a time sequence of PIV images covering several Kármán periods. On the basis of the vorticity field, the A‐ and B‐modes can be identified as topologically different vortex structures. Two vortex models are developed to explain the differences between these modes.

Large eddy simulation of the proximal region of a spatially developing circular jet
View Description Hide DescriptionLarge eddy simulations(LES) of spatially developing circular jets were carried out. The subgrid scale (SGS) model was of a dynamic type and was based on an assumed asymptotic behaviour of the SGS‐stress. This assumption is valid only for adequate spatial and temporal resolutions. The effects of the SGS‐model were studied by comparing simulations with and without SGS‐model. LES with different spatial resolutions were performed to study the effects of the spatial resolution on the numerical solution. The numerical results were compared with experimental data. Simulations were performed for the Reynolds numbers 1⋅10^{4}, 5⋅10^{4} and 50⋅10^{4} to study the Reynolds numbers effects in the proximal region of the jet. The turbulent intensity increases from a low initial level, given by a low amplitude white noise disturbance in the inlet, to a high level in the studied proximal region of the jet. For the lower Reynolds numbers certain amplified frequencies were found, at Strouhal numbers about 0.3 and the corresponding first two harmonics, which agree well with experimental observations. The spatial resolution was found to be adequate to support the longitudinal and transversal Taylor length scales. A new bound for the dynamic model parameter is proposed and it is studied a priori using the computed flow fields. This bound is based on the non‐negativity of the total dissipation, i.e. an entropy condition for the sum of viscous‐, SGS‐ and numerical‐ dissipation, in the discrete transport equation of the resolved scale‐energy.

Analytical and phenomenological studies of rotating turbulence
View Description Hide DescriptionA framework, which combines mathematical analysis, closure theory, and phenomenological treatment, is developed to study the spectral transfer process in turbulent flows that are subject to rotation. First, we outline a mathematical procedure that is particularly appropriate for problems with two disparate time scales. The approach that is based on the Green’s method leads to the Poincaré velocity variables and the Poincaré transformation when applied to rotating turbulence. The effects of the rotation are now conveniently included in the momentum equation as the modifications to the convolution of nonlinear term. The Poincaré transformed equations are used to obtain a time‐dependent Taylor–Proudman theorem valid in the asymptotic limit when the nondimensional parameter μ≡Ωt→∞ (Ω is the rotation rate and t is the time). The ‘‘split’’ of the energy transfer in both direct and inverse directions is established. Second, we apply the Eddy‐Damped‐Quasinormal‐Markovian (EDQNM) closure to the Poincaré transformed Euler/ Navier–Stokes equations. This closure leads to expressions for the spectral energy transfer. In particular, a unique triple velocity decorrelation time is derived with an explicit dependence on the rotation rate. This provides an important input for applying the phenomenological treatment of Zhou [Phys. Fluids 7, 2092 (1995)]. In order to characterize the relative strength of rotation, another nondimensional number, a spectral Rossby number, which is defined as the ratio of rotation, and turbulence time scales, is introduced. Finally, the energy spectrum and the spectral eddyviscosity are deduced.

Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite‐difference analysis of the nonlinear Boltzmann equation for hard‐sphere molecules
View Description Hide DescriptionHeat flow and temperature and density distributions in a rarefied gas between two parallel plates at rest with different uniform temperatures are analyzed numerically on the basis of the full nonlinear Boltzmann equation for hard‐sphere molecules and the Maxwell‐typeboundary condition by a finite difference method where the collision term is computed direct numerically. The accurate results are presented for the case in the density measurement by Teagan and Springer [Phys. Fluids 11, 497 (1968)], where the temperature ratio is 1.326, the value of the accommodation coefficient is 0.826, and the ratio of mean free path to plate spacing (Knudsen number Kn) is 0.0658≤Kn≤0.7582. It is found that there is a considerable difference between the present density distribution and the experimental data. The reason for this discrepancy is also discussed. The accurate numerical results of the linearized problem are also presented for comparison.

A DNS study of turbulent mixing of two passive scalars
View Description Hide DescriptionWe employ direct numerical simulations to study the mixing of two passive scalars in stationary, homogeneous, isotropic turbulence. The present work is a direct extension of that of Eswaran and Pope from one scalar to two scalars and the focus is on examining the evolution states of the scalar joint probability density function (jpdf) and the conditional expectation of the scalar diffusion to motivate better models for multi‐scalar mixing. The initial scalar fields are chosen to conform closely to a ‘‘triple‐delta function’’ jpdf corresponding to blobs of fluid in three distinct states. The effect of the initial length scales and diffusivity of the scalars on the evolution of the jpdf and the conditional diffusion is investigated in detail as the scalars decay from their prescribed initial state. Also examined is the issue of self‐similarity of the scalar jpdf at large times and the rate of decay of the scalar variance and dissipation.

Effects of acceleration on turbulent jets
View Description Hide DescriptionEffects of acceleration on turbulent jets were investigated in a series of flow visualization experiments. Prior to the initiation of acceleration, a steady jet with a Reynolds number of 3000 was established. Three distinct acceleration schemes of linear, quadratic, and exponential were utilized to increase the nozzle exit velocity by an order of magnitude. As the flow accelerated, a discernible ‘‘front’’ was established. The parcels constituting the front were less diluted than the steady jet parcels at the same location. For each acceleration scheme, the temporal evolution of the front position had the same functional form as the nozzle velocity. The front velocity increased linearly with the acceleration rate for the linear and quadratic cases. In comparison with a steady jet, the front’s lateral growth rate was reduced by 16% in the linear case and by 25% in the two nonlinear cases, even though the linear cases had generally larger acceleration rates. A model, based on the scaling of centerline velocity in steady jets, appears to correctly predict the time dependence of the front position.

A note on the vorticity spectrum
View Description Hide DescriptionThe vorticity spectrum has been inferred, using local isotropy, from measurements of the lateral vorticity components in a turbulent wake over a small Reynolds number range. The high wavenumber part of the spectrum supports Kolmogorov’s [C. R. Akad. Sci. USSR 30, 301 (1941)] similarity theory. Among the different published analytical expressions for the three‐dimensional energy spectrum, the model of Kida and Murakami [Phys. Fluids 30, 2030 (1987)] is closest to the present data.