A class of exact solutions of a nonlinear boundary-value problem for fluid flow in a rotating cylinder

The flow of an ideal incompressible weightless fluid that fills a rotating cylinder is investigated. The rotation axis of the cylinder is outside it and parallel to the cylinder generator, and the form of the cylinder section is determined in the process of solution of the problem. In the paper, a class of exact solutions of the problem is obtained in terms of elementary functions for different angular velocities of the cylinder. In these solutions, the flow field is formed by two straight vortex filaments parallel to the cylinder generator. The intensities of the vortex filaments are determined by the angular velocity . Investigations of ideal fluid flow in rotating vessels were begun already in the last century by Stokes and Zhukovskii [1]. The subject has been reviewed in monographs [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 1, pp. 71–75, January–February, 1984.     

Forms of partial breakdown of an oil body in a compound vortex

Over a wide range of parameters, the pattern of a two-layer (sunflower oil-water) fluid flow in a compound vortex is visualized. The vortex is formed by a disk mounted on the bottom of a cylindrical container and rotating at a constant angular velocity. The shapes of the interface of two media (water-oil, oil-air) are determined for different flow regimes, and the conditions and types of their breakdown are studied. Under supercritical conditions, the individual oil droplets, detached from the oil-water interface, form a direct water-oil emulsion near the lower edge of the body. With increase in the angular velocity of the disk, an invert emulsion is formed, i.e. cells containing water with oil shells. The dimensions of typical structural components of the flow are determined.     

Experiment on Vortex Formation in a Fluid Flowing out of a Reservoir

Experimental data on vortex formation near the sinkhole in a fluid flowing out of a reservoir are analyzed. An experimental facility which makes it possible to study the influence of various factors on vortex formation near a sinkhole has been designed and built. In the experiments performed, a vortex was observed only if before the sinkhole was opened a rotating disk was immersed in the fluid. In the steady-state flow regime, a brief rotation of the disk resulted in the formation and subsequent damping of a vortex. For a preswirled fluid, the formation of a vortex crater depends on the sinkhole location.     

Reconnections of vortex filaments

The process of break-down and reconnection of vortex filaments is considered by the method of three-dimensional vortex singularities (vortons) in various situations, including oblique interaction of a vortex ring with a boundary in shear flow, shedding of a vortex ring from a horseshoe vortex, instability of elliptic vortex ring, Crow instability of two perturbed antiparallel vortex filaments, merging and subsequent splitting of vortex rings. Special attention is paid to the global integrals (vorticity, momentum, angular momentum) and to the inviscid dissipation of energy. The visualization of the effective vortex core, created by the interference of the vorticity fields of vortons, is presented. The comparisons with other methods of simulation of three-dimensional vortex interactions and with the observations have been made.     

Rotationally symmetric flow over a rotating disk

The rotationally symmetric flow over a rotating disk in an incompressible viscous fluid is analyzed by a new method when the fluid at infinity is in a state of rigid rotation (in the same or in the opposite sense) about the same axis as that of the disk. Asymptotic expansions for the velocity field over the entire flow field are obtained for the general class of one-parameter rotationally symmetric flows. This method is further extended to the case when a uniform suction or injection is assumed at the rotating disk. Fluid motion induced by oscillatory suction of small amplitude at the rotating disk is also discussed.An initial-value analysis reveals that resonance is possible only when the angular velocity of the rotating fluid is greater than that of the rotating disk.     

Marker transport in a composite vortex

Using a coloring method, the pattern of the transport of a compact spot of markers in a composite vortex created by a uniformly rotating disk, which is located on the bottom of a cylindrical tank, is investigated. The markers were formed by droplets of an aqueous solution of uranyl or ink, which fell down on the free surface of the fluid. In a quiescent fluid, the spot of the markers was transformed into a slowly sinking cascade of vortices. The markers introduced at the center of the cavity on the composite-vortex surface formed a spiral branch. For a non-central introduction of the markers, two spiral branches were formed, one growing in the direction of the fluid rotation and the other growing in the direction, opposite to the rotation. The markers penetrate into the interior of the composite vortex along isolated cylindrical surfaces. The geometrical parameters of the flow are presented.     

Flow structure in a flat vortex chamber

We investigate a flow in a flat vortex chamber in which the distance between the end walls is smaller than the radius of the chamber. The study was mainly performed by optical methods: a Töpler device was employed, with the Foucault knife replaced by a diaphragm. It is shown that the flow in the chamber has a complicated spatial structure. In addition to the basic helical flow, an intense “transverse” rotation of the type of Taylor-Görtler vortices occurs. In contrast to previously studied flows, where these vortices were observed near a concave surface, in the motion considered transverse vortices occur in the entire working volume of the chamber. In this case, four parallel vortex filaments are formed. The high intensity of the vortices has allowed one to visualize them by the Töpler method and by “tinting” the flow by highly disperse particles. Quantitative dependences of the dimensions of the vortex cells on the flow regime, i.e., on the pressure of gas deceleration, were obtained.     

On the development of lift and drag in a rotating and translating cylinder

The two-dimensional flow around a rotating cylinder is investigated numerically using a vorticity forces formulation with the aim of analyzing quantitatively the flow structures, and their evolutions, that contribute to the lift and drag forces on the cylinder. The Reynolds number considered, based on the cylinder diameter and steady free stream speed, is Re=200, while the non-dimensional rotation rate (ratio of the surface speed and free stream speed) selected was α=1 and 3. For α=1 the wake behind the cylinder for the fully developed flow is oscillatory due to vortex shedding, and so are the lift and drag forces. For α=3 the fully developed flow is steady with constant (high) lift and (low) drag. Each of these cases is considered in two different transient problems, one with angular acceleration of the cylinder and constant speed, and the other one with translating acceleration of the cylinder and constant rotation. We characterize quantitatively the contributions of individual fluid elements (vortices) to aerodynamic forces, explaining and quantifying the mechanisms by which the lift is generated in each case. In particular, for high rotation (when α=3), we explain the relation between the mechanisms of vortex shedding suppression and those by which the lift is enhanced and the drag is almost suppressed when the fully developed flow is reached.     

Convective flow with uniform vorticity in a vertical layer

The free convective circulation of liquid in plane vertical slits of circular and square cross section with a longitudinal horizontal temperature gradient at the boundaries was investigated experimentally. It was found that under such heating conditions there is a uniform-vorticity flow with a region of quasirigid rotation, which has the shape of a disk in a circular slit and the shape of a cross in a square slit; in each longitudinal section of this zone the liquid moves along concentric trajectories with constant angular velocity. Dimensionless numbers for the problem were established by tests with various liquids and cavities of different dimensions. In dimensionless form, the angular velocity of the vortex and the temperature gradient in it depend linearly on the temperature difference at the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 160–165, May–June, 1984.     

Three-dimensional evolution of vortices and early features of coherent structure in the turbulent wake behind a 2-D circular cylinder

3-D evolution of Kármán vortex filaments and vortex filaments in braid regions in the turbulent wake of a 2-D circular cylinder is investigated numerically based on inviscid vortex dynamics by analyzing the response of the initially 2-D spanwise vortex filaments to periodic spanwise disturbance of varying magnitude, wavelength and initial phase angles. Our results reveal a kind of 3-D vortex system in the wake which consists of large scale horseshoe-shaped vortices and small scale γ-shaped vortex filaments as well as vortex loops. The mechanism and the dynamic process about the generation of streamwise vortical structure and the 3-D coherent structure are reported. currently published in the Chinese Edition of Acta Mechanica Sinica, Vol.25, No.3, 1993 The project supported by National Natural Science Foundation of China and the National Basic Research Project “Nonlinear Science”     

Flow of a third grade fluid due to an accelerated disk

The magnetohydrodynamic (MHD) flow induced by non‐coaxial rotation of porous disk and a third grade fluid at infinity is investigated. The disk is moving with uniform acceleration and rotating with a uniform angular velocity. Numerical solution of the governing nonlinear initial and boundary value problem is obtained. The effects of physical parameters on the velocity profiles are examined in detail. The present study shows that the constant acceleration part has a greater influence than the time part of the assumed variable velocity of the disk. Copyright © 2009 John Wiley & Sons, Ltd.     

A crack in a perfect fluid

An analytic particular solution is obtained for a plane flow problem. The plane flow is forced in an incompressible perfect fluid by a rigid moving wall surrounding the fluid. The wall has the shape of an elliptic cylinder and rotates about its axis. It is found that, during the rotation of such a cylinder, there appears a “hanging” vortex sheet such that its intensity is directly proportional to the angular velocity of rotation.     

Point vortex in a viscous incompressible fluid

A plane steady problem of a point vortex in a domain filled by a viscous incompressible fluid and bounded by a solid wall is considered. The existence of the solution of Navier-Stokes equations, which describe such a flow, is proved in the case where the vortex circulation Θ and viscosity ν satisfy the condition |Θ| < 2πν. The velocity field of the resultant solution has an infinite Dirichlet integral. It is shown that this solution can be approximated by the solution of the problem of rotation of a disk of radius Γ with an angular velocity ω under the condition 2πγ ² ω → Γ as γ → 0 and ω→∞.     

Numerical simulation of plate autorotation in a viscous fluid flow

A general formulation of the plane coupled dynamical and aerodynamical problem of the motion of a rigid body with a rotational degree of freedom in a viscous incompressible fluid flow is given. A computation technique for solving the Navier-Stokes equations based on the meshless viscous vortex domain method is used. The autorotation of a single plate and a pair of plates is investigated. The effect of the reduced moment of inertia and the Reynolds number on the angular rotation velocity is determined. The time dependences of the hydrodynamic loads are compared with the corresponding instantaneous flow patterns. The increased the autorotation velocity of two plates in tandem is detected.     

Dynamics of a two-fluid system in a rotating horizontal cylinder under longitudinal vibration

The average dynamics of immiscible fluids with different densities in a horizontal axially vibrating cylinder are studied. The angular velocity is sufficient to sustain the centrifuged state of the fluids. The vibration and rotation frequencies are of the same order. The generation of toroidal vortices periodic along the rotation axis and the formation of quasi-steady interface relief with axially periodic axisymmetric crimps are detected. It is shown that the vortex flow is associated with the generation of an inertial axisymmetric standing wave. The formation of the quasi-steady relief is induced by the development of the Kelvin-Helmholtz instability on the fluid interface under tangential vibration.     

Characteristics of ventilating flow generated by a rotating swirler in a vortex vent

A vortex ventilation system with a rotating annular disk installed coaxially with the exhaust inlet is a very effective local ventilator. A swirling flow generated by a rotating swirler makes the ventilation flow concentrated around the axis of rotation, which can increase the ventilation depth by a factor of five compared to a conventional exhaust hood. Despite the well-documented excellent ventilation performance of such a system, detailed flow characteristics are not well understood. In this study, the swirling flow field in the vortex vent was tested, and a number of peculiar flow characteristics were observed. When the rotational speed was varied, a series of different flow patterns appeared, and the changes in the flow pattern showed rapid transition, hysteresis, and flow instability similar to the vortex. The transition of the flow pattern could be explained based on the ratio of the centrifugal force to exhaust pressure. Hysteresis of the flow transition occurred in an unstable equilibrium mode between the two forces, and an unstable flow pattern occurred when the secondary recirculating flow was located beneath the swirler. A formula for the critical rotational speed was derived, which showed satisfactory agreement with experimental observation.     

MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

Electrochemical and optical studies of a stable to unstable transition in a polymer solution flowing close to a rotating disc

Summary A transition has been displayed in a concentrated solution of a linear chain polymer flowing close to a rotating disk. For a critical angular speed of rotation, a discontinuity of the mass transfer rate by diffusion and convection, as well as the existence of flow birefringence are observed. This transition is interpreted as an effect of conformational change (coiled to stretched shape) of the polymer under the local strain in the fluid.With 4 figures     

Some features of the flow around rapidly rotating bodies made of cellular-porous materials

Two types of gas flows arising near a rapidly rotating cellular-porous disk are studied numerically and experimentally. Steady-state limits for the flow around a disk rotating in free space and the type and scenario of the loss of stability are determined. Transitional flows are characterized by formation of a vortex sheet at the boundary of the exhausting jet. Numerical simulations of the flow around a cellular-porous disk rotating near a flat screen show that it is possible to form a closed swirl flow responsible for redistribution of swirl in the gap between the disk and the flat screen. The computed results offer an explanation for the experimentally observed excess of tangential velocity of the flow in the gap over the velocity of disk rotation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 86–96, January–February, 2007.