Georgios H. Vatistas

A note on liquid vortex sloshing and Kelvin's equilibria

Observations of liquid vortex sloshing and Kelvins equilibrium states were made inside a cylindrical container using a spinning disk near its base. Both steady and periodic free-surface sloshing phenomena were found to take place. During periodic sloshing, the air core sustained shape transformations, assuming an elliptical cross-section at the end, and then collapsed forming a pair of vortices. Kelvins equilibrium states emerged at lower liquid levels. These were stable within an interval of rotational speeds. The bandwidth of stationary states decreased as the wavenumber ( N ) increased. For N greater than six, the states appeared critically stable. Between equilibria, unstable transitional regions were found to exist. As the liquid level was decreased, the core shape spectrum shifted towards smaller frequencies.

Reynolds Stress Model in the Prediction of Confined Turbulent Swirling Flows

Strongly swirling vortex chamber flows are examined experimentally and numerically using the Reynolds stress model (RSM). The predictions are compared against the experimental data in terms of the pressure drop across the chamber, the axial and tangential velocity components, and the radial pressure profiles. The overall agreement between the measurements and the predictions is reasonable. The predictions provided by the numerical model show clearly the forced and free vortex modes of the tangential velocity profile. The reverse flow (or back flow) inside the core and near the outlet, known from experiments, is captured by the numerical simulations. The swirl number has been found to have a measurable impact on the flow features. The vortex core size is shown to contract with the swirl number which leads to higher pressure drop, higher peak tangential velocity, and deeper radial pressure profiles near the axis of rotation. The adequate agreement between the experimental data and the simulations using RSM turbulence model provides a valid tool to study further these industrially important swirling flows.

Dynamics of Liquid Sloshing in Horizontal Cylindrical Road Containers

A study of the liquid behaviour in horizontal cylindrical road containers undergoing a steady turning manoeuvre is presented and discussed. The steady state solutions are derived analytically from the hydrostatic equations. The transient solutions are obtained by numerical integration of the Navier-Stokes, continuity and free-surface equations. The non-dimensional governing equations are solved in primitive variables by using a modified marker-and-cell technique which involves the interpolation-reflection type boundary conditions developed for this investigation. The mathematical model of the liquid motion includes all essential non-linear effects and allows the damped natural frequencies of liquid vibrations to be obtained as well as the magnitudes of the liquid slosh loads. This study also enables the coupled directional dynamics of the ‘vehicle-liquid tank’ system undergoing different road manoeuvres to be investigated by integrating the non-linear fluid slosh model and an appropriate vehicle model simultaneously.

Liquid sloshing in rectangular road containers

Abstract A study of liquid behaviour in rectangular road containers undergoing a turning or braking manoeuvre is presented and discussed. The steady-state solution in terms of liquid heights, forces and overturning moments is derived analytically from the hydrostatic equations. The transient response of the liquid is obtained via numerical solution of the continuity, Navier-Stokes and free-surface equations. The governing equations are discretized in a Eulerian mesh and solved with respect to the nondimensional primitive variables together with the boundary conditions at rigid walls and the free surface using a modified marker-and-cell technique. Such an approach allows one to take into account all basic nonlinearities proper to the sloshing problem and to obtain the damped frequencies and magnitudes of the sloshing parameters. The present study is a contribution to the overall dynamics of coupled “vehicle-liquid” systems performing some road manoeuvres.

Simple Model for Turbulent Tip Vortices

V ORTICES are interwoven into the fabric of fluid mechanics transporting mass momentum and energy in the majority of natural and industrial flows. In technology these are either produced deliberately to accomplish the task, improve the function of devices, or emerge as a parasitic by-product of fluid motion. Aeronautical applications are very susceptible to these flow manifestations. Consequences of lift, tip vortices cause considerable drag, noise, and/or hazard in fixedand/or rotating-wing aircraft. Although complicated in nature simple mathematical models are routinely used to elaborate on some of their fundamental properties [1,2]. It is a well-known fact that, irrespective of the host flowfield, intense vortices are analogous. Theoretically, intense (or strong) vortices are those where the tangential velocity component is orders of magnitude larger than the radial and axial. In practice, however, the property is also maintained by vortices where the swirl velocity component is dominant, but not necessarily of a magnitude enormously greater than the other two. Under these conditions the vortex appears to develop in amanner that the swirling action ignores the secondarylike flow in the azimuthal plane. Consequently, simple Rankine-like formulations have been used widely to model a variety of geophysical, wingtip, cyclone chamber, ship propeller, intake, and other types of vortices. The physics of laminar vortices is relatively well known. Simple exact solutions of the Navier–Stokes equations are due to Rankine [3], Oseen [4], Lamb [5], Burgers [6], and others. Every one of them represents a possible solution, applicable to low vortex Reynolds numbers (Re ) defined as the total circulation divided by the kinematic viscosity. In antithesis, our grasp reduces exponentially for turbulent vortices. Certainly this is not accidental. Turbulent flows are considerably more complex both analytically and experimentally. Even today the turbulent vortex is among the not very well-explored territories in aerodynamics. Not long ago, Ramasamy and Leishman [7] examined turbulent helicopter tip vortices using state of the art instruments and experimental techniques. Their high-resolution visualizations revealed that these types of whirls display the already familiar path to turbulence (probably slow, through spectral development). Inside the core, they have confirmed that helicopter tip vortices do enjoy laminarlike conditions. Approaching the core from the origin, past the laminar core and when a critical local Reynolds number is reached, the flow enters a transition region. This condition persists until a second critical Reynolds number where the flowfield changes into the “turbulent state” at larger radii. Most important of all, the accompanied high-fidelity velocity data (at Re 48; 000) exposed the following particular behavior for the azimuthal velocity component. In the region where the vortex is turbulent, the velocity decreases at a rate noticeably smaller than that of a laminar vortex. As a consequence of the new experimental evidence it became amply evident that high Re helicopter tip vortices cannot be modeled by the laminar formulations of the past. Instead one has to seek analytical representations of the phenomenon where at least the most fundamental effects of turbulence are included. Previous, theoretically more involved developments on the subject are those of Newman [8], Inversen [9], Tang [10], and Ramasamy and Leishman [7]. Here we present a new simple mathematically convenient formulation that accounts for the flattening effect in the tangential velocity profile.

Purely accelerating and decelerating flows within two flat disks

SummaryThis paper deals with the power series solutions of the steady, laminar, radial flows either purely accelerating, or purely decelerating, that develop in the gap formed by two flat disks. The results include velocity profiles and static pressure distributions. These are compared with previously reported approximate solutions or the experimental data for the pressure obtained by others. The development of the two types of flows is shown to be entirely different except for λ (parameter that combines the non-dimensional radial distance and Reynolds number) close to zero where both behave as Poiseuilles flows between two infinite plates. For the inflow, the radial velocity flattens near the mid-plane diffusing towards the walls as the parameter λ increases. In contrast to the inflow, the magnitude of the maximum velocity of the outflow is shown to increase with λ, indicating that most of the fluid motion is taking place near the central channel region. For the outflow, two critical values of λ are used to indicate notable flow field transformations. The first marks the point where the pressure difference changes sign, while the second denotes when the derivative of the velocity (in the axial direction) on the wall becomes zero. Beyond the second value, purely decelerating flow cannot exist. The sign change of the pressure is attributed to the interaction between the inertia, viscous, and pressure forces.

Recent findings on Kelvin's equilibria

SummaryThe linearized stability analysis is applied to investigate the wave behavior in a water vortex produced in a cylindrical tank with a flat disk rotating at the bottom. Two flow cases are considered herein. The first case deals with waves developed on the free surface of a hollow vortex, while the second with waves generated in the core of a Rankine vortex. It is evident from the analysis that the experimental dispersion velocity approaches the calculated one when the wave amplitude becomes smaller. The latter is consistent with the small perturbation assumption that is inherent in the theory. For the case where the core is flooded, the presence of a cylindrical wall is shown to enhance the wave speed. A hypothesis as to how the core develops in the mixed state regions is proposed. The graphical simulations appear to predict reasonably well the main features of the observations.

On Flow Development in Jet-Driven Vortex Chambers

This work presents the study of the flow in a jet-driven vortex chamber over a wide range of Reynolds numbers, contraction ratios, inlet angles, area and aspect ratios. Dimensional analysis furnishes the general functional relationships between the fundamental dimensionless quantities. Application of the integral equations of continuity and energy over the control volume, along with the minimum-pressure-drop or maximum flow rate postulate, provide the required analytical means to relate the predominant non-dimensional parameters such as the chamber geometry, the core size, pressure drop, Reynolds number, and viscous losses. Both the n = 2 vortex model, with reverse and non-reverse flow, and the free vortex model have been used at the vortex chamber exit plane. The theoretical results are found to successfully capture most of the salient properties of the flow. The influence of vortex chamber geometry, such as contraction ratio, inlet angle, area ratio, aspect ratio, and Reynolds number, on the flow field has been analyzed and compared with the present experimental data. A parametric study explores how the pressure coefficient and the core size vary with the different dimensionless properties. The observations show the pressure drop to decrease with the length. At first this appears to be counterintuitive since one habitually expects the pressure drop to be larger for longer pipes. A closer examination however, reveals that in addition to the radial-axial plane flow there is also a substantial centrifugal force, which decays with the length, thus shaping the development of the overall flow-field. The pressure drop across the vortex chamber differs from that in pipe flow, due to the mechanism of swirl flow. It depends mainly on intensity of tangential velocity. If the chamber length is increased, the vortex decay factor decreases, which leads to less pressure drop. The current theory confirms that the previous published models are only applicable for high Reynolds numbers where the inertia dominates the viscous forces. Based on the present theory, a new approach to determine the tangential velocity and radial pressure profiles inside the vortex chamber is developed and compared with the available experimental data. The n = 2 vortex model with reverse flow gives better results for strongly swirling flow.

New Model for Compressible Vortices

A new analytical solution for self-similar compressible vortices is derived. Based on the previous incompressible formulation of intense vortices, we derived a theoretical model that includes density and temperature variations. The governing equations are simplified assuming strong vortex conditions. Part of the hydrodynamic problem (mass and momentum) is shown to be analogous to the incompressible kind and as such the velocity is obtained through a straightforward variable transformation. Since all the velocity components are bounded in the radial direction, the density and pressure are then determined by standard numerical integration without the usual stringent simplification for the radial velocity

Radial inflow between two flat disks

SummaryA numerical solution to the radial inflow between two flat disks is presented. The pressure and the radial velocity distribution are shown to be functions of solely one non-dimensional parameter that combines the Reynolds number and the radial distance. The numerically predicted pressure correlates well with the experiment. By comparison to the results of the linearized theory, the radial velocity profile is seen to be lower at the mid-plane and larger near the disk surface.