Liquid parametric sloshing, known also as Faraday waves, has been a long standing subject of interest. The development of the theory of Faraday waves has witnessed a number of controversies regarding the analytical treatment of sloshing modal equations and modes competition. One of the significant contributions is that the energy is transferred from lower to higher harmonics and the nonlinear coupling generated static components in the temporal Fourier spectrum, leading to a contribution of a nonoscillating permanent sinusoidal deformed surface state. This article presents an overview of different problems of Faraday waves. These include the boundary value problem of liquid parametric sloshing, the influence of damping and surfactants on the stability and response of the free surface, the weakly nonlinear parametric and autoparametric sloshing dynamics, and breaking waves under high parametric excitation level. An overview of the physics of Faraday wave competition together with pattern formation under single-, two-, three-, and multifrequency parametric excitation will be presented. Significant effort was made in order to understand and predict the pattern selection using analytical and numerical tools. Mechanisms for selecting the main frequency responses that are different from the first subharmonic one were identified in the literature. Nontraditional sources of parametric excitation and Faraday waves of ferromagnetic films and ferrofluids will be briefly discussed. Under random parametric excitation and g-jitter, the behavior of Faraday waves is described in terms of stochastic stability modes and spectral density function.

The results of the experimental study of the Reynolds number effect on the process of the Rayleigh–Taylor (R-T) instability transition into the turbulent stage are presented. The experimental liquid layer was accelerated by compressed gas. Solid particles were scattered on the layer free surface to specify the initial perturbations in some experiments. The process was recorded with the use of a high-speed motion picture camera. The following results were obtained in experiments: (1) Long-wave perturbation is developed at the interface at the Reynolds numbers Re < 104. If such perturbation growth is limited by a hard wall, the jet directed in gas is developed. If there is no such limitation, this perturbation is resolved into the short-wave ones with time, and their growth results in gas-liquid mixing. (2) Short-wave perturbations specified at the interface significantly reduce the Reynolds number Re for instability to pass into the turbulent mixing stage.

The flow field in a cylindrical container driven by a flat bladed impeller was investigated using particle image velocimetry (PIV). Three Reynolds numbers (0.02, 8, 108) were investigated for different impeller locations within the cylinder. The results showed that vortices were formed at the tips of the blades and rotated with the blades. As the blades were placed closer to the wall the vortices interacted with the induced boundary layer on the wall to enhance both regions of vorticity. Finite time lyapunov exponents (FTLE) were used to determine the lagrangian coherent structure (LCS) fields for the flow. These structures highlighted the regions where mixing occurred as well as barriers to fluid transport. Mixing was estimated using zero mass particles convected by numeric integration of the experimentally derived velocity fields. The mixing data confirmed the location of high mixing regions and barriers shown by the LCS analysis. The results indicated that mixing was enhanced within the region described by the blade motion as the blade was positioned closed to the cylinder wall. The mixing average within the entire tank was found to be largely independent of the blade location and flow Reynolds number.

The development of a columnar vortex and its attenuation using radial rods at the bottom boundary of a stationary container are experimentally studied. The fluid motion is achieved combining two independent flows: a global circulation around the cylinder axis and a meridian flow generated by recirculating fluid through a central nozzle located at the vessel bottom. The resulting velocity field is analyzed under two conditions: with and without the meridian or suction flow. It is shown that in the second condition a columnar vortex merges and that its intensity increases when the suction flow rate is increased. The key role played by the bottom boundary layer in the vortex formation is demonstrated. In the last part of the work, the attenuation of the vortex intensity produced by a set of rods located at the vessel bottom is investigated. It is found that obstacles with heights of the order of the boundary layer thickness are enough to produce the total annihilation of the vortex column.

This paper presents a numerical analysis of the free surface liquid metal flow driven by an alternating current magnetic field in a spinning cylindrical container. The axisymmetric flow structure is analyzed for various values of the magnetohydrodynamic interaction parameter and Ekman numbers. The governing hydrodynamic equations are solved by a spectral collocation method, and the alternating magnetic field distribution is found by a boundary-integral method. The electromagnetic and hydrodynamic fields are fully coupled via the shape of the liquid free surface. It is found that the container rotation may reduce the meridional flow significantly.

We report analysis and measurements of the torque and flow of a ferrofluid in a cylindrical annulus subjected to a rotating magnetic field perpendicular to the cylinder axis. The presence of the inner cylinder results in a nonuniform magnetic field in the annulus. An asymptotic analysis of the ferrohydrodynamic torque and flow assuming linear magnetization and neglecting the effect of couple stresses indicated that the torque should have a linear dependence on field frequency and quadratic dependence on field amplitude. To the order of approximation of the analysis, no bulk flow is expected in the annular gap between stationary cylinders. Experiments measured the torque required to restrain a polycarbonate spindle surrounded by ferrofluid in a cylindrical container and subjected to the rotating magnetic field generated by a two-pole magnetic induction motor stator, as a function of the applied field amplitude and frequency, and for various values of the geometric aspect ratios of the problem. The ultrasound velocity profile method was used to measure the azimuthal and axial velocity profiles in the ferrofluid contained in the annular gap of the apparatus. Flow measurements show the existence of a bulk azimuthal ferrofluid flow between stationary coaxial cylinders with a negligible axial velocity component. The fluid was found to corotate with the applied magnetic field. Both the torque and flow measurements showed power-of-one dependence on frequency and amplitude of the applied magnetic field. This analysis and these experiments indicate that the action of antisymmetric stresses is responsible for the torque measured on the inner cylinder, whereas the effect of body couples is likely responsible for bulk motion of the ferrofluid.

Granular materials exhibit unusual kinds of behavior, including pattern formations during the shaking of the granular materials; the characteristics of these various patterns are not well understood. Vertically shaken granular materials undergo a transition to convective motion that can result in the formation of bubbles. A detailed overview is presented of collective processes in gas-particle flows that are useful for developing a simplified model for molecular dynamic type simulations of dense gas-particle flows. The governing equations of the gas phase are solved using large eddy simulation technique. The particle motion is predicted by a Lagrangian method. Particles are assumed to behave as viscoelastic solids during interactions with their neighboring particles. Interparticle normal and tangential contact forces are calculated using a generalized Hertzian model. The other forces that are taken into account are gravitational and drag force resulting from velocity difference with the surrounding gas. A simulation of gas-particle flow is performed for predicting the flow dynamics of dense mixtures of gas and particles in a vertical, pentagonal, prism shaped, cylindrical container. The base wall of the container is subjected to sinusoidal oscillation in the vertical direction that spans to the bottom of the container. The model predicts the formation of oscillon type structures on the free surface. In addition, the incomplete structures are observed. Interpretations are proposed for the formation of the structures, which highlights the role played by the surrounding gas in dynamics of the shaken particles.

The linear instability of a power law liquid emerging as a jet from an orifice on the surface of a rotating container is investigated, with applications to industrial prilling. Asymptotic methods are used to examine the growth rate and wavenumber of the most unstable traveling wave mode for different flow index numbers. Comparison with Newtonian liquids show that for small rotation rates shear thinning liquids are most stable to disturbances. In contrast for higher rotation rates we find shear thickening liquids are more stable than shear thinning liquids. The influence of viscosity, surface tension, and rotation rate on the growth rates and most unstable wavenumbers associated with both types of liquids are also examined.

The swirling flows of water and CTAC (cetyltrimethyl ammonium chloride) surfactant solutions ( 50 - 1000 ppm ) in an open cylindrical container with a rotating disc at the bottom were experimentally investigated by use of a double-pulsed PIV (particle image velocimetry) system. The flow pattern in the meridional plane for water at the present high Reynolds number of 4.3 × 10 4 differed greatly from that at low Reynolds numbers, and an inertia-driven vortex was pushed to the corner between the free surface and the cylindrical wall by a counter-rotating vortex caused by vortex breakdown. For the 1000 ppm surfactant solution flow, the inertia-driven vortex located at the corner between the bottom and the cylindrical wall whereas an elasticity-driven reverse vortex governed the majority of the flow field. The rotation of the fluid caused a deformation of the free surface with a dip at the center. The dip was largest for the water case and decreased with increasing surfactant concentration. The value of the dip was related to determining the solution viscoelasticity for the onset of drag reduction.

Numerical studies are reported on the vortex breakdown in a differentially-rotating cylindrical container in which the top endwall rotates at a high angular velocity Ω t and the cylinder and bottom endwall rotate at a low angular velocity Ω sb . Critical boundaries and the location and size of the vortex breakdown bubble are quite different from the case when the top endwall rotates and the cylinder and the bottom endwall are stationary. As | Ω sb / Ω t | is increased, the breakdown bubble moves toward downstream for Ω sb / Ω t < 0 , whereas the bubble moves toward upstream for Ω sb / Ω t > 0 . The Brown and Lopez criterion is extended to a differentially-rotating container.

Part 2 of this two-part paper provides an example case study following the recently developed comprehensive verification and validation approach presented in Part 1. The case study is for a RANS simulation of an established benchmark for ship hydrodynamics using a ship hydrodynamics CFD code. Verification of the resistance (integral variable) and wave profile (point variable) indicates iterative uncertainties much less than grid uncertainties and simulation numerical uncertainties of about 2 % S 1 ( S 1 is the simulation value for the finest grid). Validation of the resistance and wave profile shows modeling errors of about 8 % D ( D is the measured resistance) and 6 % ζ max ( ζ max is the maximum wave elevation), which should be addressed for possible validation at the 3 % D and 4 % ζ max levels. Reducing the level of validation primarily requires reduction in experimental uncertainties. The reduction of both modeling errors and experimental uncertainties will produce verified and validated solutions at low levels for this application using the present CFD code. Although there are many issues for practical applications, the methodology and procedures are shown to be successful for assessing levels of verification and validation and identifying modeling errors in some cases. For practical applications, solutions are far from the asymptotic range; therefore, analysis and interpretation of the results are shown to be important in assessing variability for order of accuracy, levels of verification, and strategies for reducing numerical and modeling errors and uncertainties.

We consider the boundary layer that forms on the wall of a rotating container of stratified fluid when altered from an initial state of rigid body rotation. The container is taken to have a simple axisymmetric form with sloping walls. The introduction of a non-normal component of buoyancy into the velocity boundary-layer is shown to have a considerable effect for certain geometries. We introduce a similarity-type solution and solve the resulting unsteady boundary-layer equations numerically for three distinct classes of container geometry. Computational and asymptotic results are presented for a number of parameter values. By mapping the parameter space we show that the system may evolve to either a steady state, a double-structured growing boundary-layer, or a finite-time breakdown depending on the container type, rotation change and stratification. In addition to extending the results of Duck et al. (1997) to a more general container shape, we present evidence of a new finite-time breakdown associated with higher Schmidt numbers.

A bathtub vortex is usually formed at the axis of a drain. In the presence of such a vortex, gravity separation of solid impurities lighter than the embedding fluid is modified by centrifugal separation and viscous resuspension. Both mechanisms are responsible for the agglomeration of impurities at the axis of the vortex. From there the impurities are easily sucked into the outlet. In the investigated case, a viscous fluid with a given initial rotation is spinning down in a container with endplates both at the bottom and the top. The amount of fluid withdrawn through a circular hole in the center of the vortex is constantly replaced by a radial influx. The resulting time-dependent flow was solved by means of a finite difference method taking into account the influence of Ekman layers at the bottom and the top. Subsequently, the process of centrifugal separation of particles lighter than the embedding fluid was studied in the aforementioned flow field. The results were compared with the particle motion in a classical Oseen vortex. For a simplified case an analytical solution was derived and compared with the corresponding numerical solution. Both results were found to be in good agreement.

We consider the nonlinear spin-up of a rotating stratified fluid in a conical container. An analysis of similarity-type solutions to the relevant boundary-layer problem (Duck et al, 1997) has revealed three types of behavior for this geometry. In general, the boundary-layer evolves to either a steady state, a growing boundary-layer, or a finite-time singularity depending on the initial to final rotation rate ratio, and a “modified Burger number.” We emphasize the experimental aspects of our continuing spin-up investigations and make some preliminary comparisons with the boundary-layer theory, showing good agreement. The experimental data presented is obtained through particle tracking velocimetry. We briefly discuss the qualitative features of the spin-down experiments which, in general, are dominated by nonaxisymmetric effects. The experiments are performed using a conical container filled with a linearly stratified fluid, the generation of which is nontrivial. We present a general method for creating a linear density profile in containers with sloping boundaries.

The flow field induced inside a cylindrical container by the rotation of the two end walls is described. It is shown that stagnation points leading to separation bubbles occur on the axis of rotation and/or the bottom end wall for certain ranges of the characteristic parameters; the Reynolds number, the aspect ratio of the container, and the ratio of the rotation rates of the end walls. Flow fields in a container of aspect ratio 2.0 are examined for Reynolds numbers from 100 to 3000 and ratios of the rotation rates of the top and bottom end walls from −0.10 to 1.0. For a range of ratios of the rotation rates of the top and bottom end walls and Reynolds numbers it is shown that ring vortices surrounding a columnar vortex core exist.

Torque measurements have been made on rotating three-dimensional bluff bodies in a cylindrical container from start-up to mean flow steady state. The flow is observed to pass through three distinct temporal regimes in the transient process. These regimes include a build-up period where the torque remains approximately constant, a decay period where the torque decreases, and a mean steady state where the mean torque remains at a constant level. Effects of body geometry, rotation rate, acceleration rate, and fluid height in the tank are quantified. The torque coefficient during the build-up and mean steady-state regimes is shown to be a function of Reynolds number and body geometry. A time scale marking the beginning of the decay period is presented in terms of the problem parameters. Over the range of parameters studied, the build-up, decay, and mean steady-state regimes are completely specified by the Reynolds number, geometric parameters, and decay time scale.

We describe the important structural features of swirling recirculating flows induced by a rotating boundary. A knowledge of this structure has allowed us to match the core flow to the boundary layer using a momentum-integral technique. In particular, we derive a single integral-differential equation, valid for any shape of container, which predicts the distribution of swirl, secondary recirculation, and wall shear stress. This momentum-integral approach has been applied to three cases: flow between parallel disks; flow in a cone; and flow in a hemisphere. The results compare favorably with published experimental data, and with computed numerical results. Our momentum-integral approach complements numerical solution methods. For simple geometries all the important information can, in principle, be derived using the momentum-integral approach, and this is particularly useful for establishing the scaling laws. In more complex geometries a numerical approach may be more appropriate. However, even in such cases, the scaling laws derived using the momentum-integral analysis are still useful as they allow extrapolation of a single computation to a wide range of high Reynolds number flows.

Numerical studies are made of three-dimensional flow of a viscous fluid in a cubical container. The flow is driven by the top sliding wall, which executes sinusoidal oscillations. Numerical solutions are acquired by solving the time-dependent, three-dimensional incompressible Navier-Stokes equations by employing very fine meshes. Results are presented for wide ranges of two principal physical parameters, i.e., the Reynolds number, Re ≤ 2000 and the frequency parameter of the lid oscillation, ω′ ≤ 10.0. Comprehensive details of the flow structure are analyzed. Attention is focused on the three-dimensionality of the flow field. Extensive numerical flow visualizations have been performed. These yield sequential plots of the main flows as well as the secondary flow patterns. It is found that the previous two-dimensional computational results are adequate in describing the main flow characteristics in the bulk of interior when ω′ is reasonably high. For the cases of high-Re flows, however, the three-dimensional motions exhibit additional complexities especially when ω′ is low. It is asserted that, thanks to the recent development of the supercomputers, calculation of three-dimensional, time-dependent flow problems appears to be feasible at least over limited ranges of Re.

A description is made of the transient shape of interface of a two-layer liquid in an abruptly rotating circular cylinder. The density of the lower layer is higher than that of the upper layer, but the viscosities may assume arbitrary values. The overall Ekman number is much smaller than unity, and the cylinder aspect ratio is 0(1). The classical Wedemeyer model, which deals with the spin-up from rest of a homogeneous fluid, is extended to tackle the two-layer liquid system. If the upper-layer fluid is of higher viscosity, the interface, at small and intermediate times, rises (sinks) in the center (periphery). After reaching a maximum height at the center, the interface tends to the parabolic shape characteristic of the final-state rigid-body rotation. If the lower-layer fluid is of higher viscosity, the interface, at small and intermediate times, sinks (rises) in the center (periphery). The deformation at the center reaches a minimum height, after which the interface approaches the final-state parabola. The gross adjustment process is accomplished over the spin-up time scale, E n −1/2 Ω −1 , where E n and Ω denote the lower value of the Ekman numbers of the two layers and the angular velocity of the cylindrical container, respectively. These depictions are consistent with the physical explanations offered earlier. A turntable experiment is performed to portray the transient interface shape. The model predictions of the interface form are in satisfactory agreement with the laboratory measurements.

The motion of small, monodisperse particles in fluid was studied in a horizontal, cylindrical container rotating about its axis. One instigation for the study was the common requirement for mixed-phase, chemical or biological reactors to maintain particles in suspension for extended periods. A cylindrical, rotating reactor can allow long-term particle suspension without particle collisions and resulting agglomeration. The purpose of this study was to verify parametric effects and optimize the time of particle suspension. The theoretical and experimental results were obtained for inert, constant-diameter particles of nearly neutral buoyancy. The centrifugal buoyancy and gravitation terms were both included in the equations of motion. Laser illumination, photography and computer imaging were used to measure experimental particle concentration.