Viscoelastic swirling flow with free surface in cylindrical chambers

The flow of a viscoelastic liquid driven by the steadily rotating bottom cover of a cylindrical cup is investigated. The flow field and the shape of the free surface are determined at the lowest significant orders of the regular domain perturbation in terms of the angular velocity of the bottom cap. The meridional field superposed on a primary azimuthal field shows a structure of multiple cells. The velocity field and the shape of the free surface are strongly effected by the cylinder aspect ratio and the elasticity of the liquid. The use of this flow configuration as a free surface rheometer to determine the first two Rivlin-Ericksen constants is shown to be promising.Nomenclature R, ,Z Coordinates in the physical domain D - , , Coordinates in the rest stateD ₀ - r, ,z Dimensionless coordinates in the rest stateD ₀ - Angular velocity - Zero shear viscosity - Surface tension coefficient - Density - Dimensionless surface tension parameter - ₁, ₂ The first two Rivlin-Ericksen constants - Stream function - Dimensionless second order meridional stream function - ^* Dimensionless second normal stress function - 2 Dimensionless sum of the first and second normal stress functions - N ₁,N ₂ The first and second normal stress functions - n Unit normal vector - D Stretching tensor - A _n nth order Rivlin-Ericksen tensor - S Extra-stress - u Velocity field - U Dimensionless second order meridional velocity field - V Dimensionless first order azimuthal velocity field - p Pressure - Modified pressure field - P Dimensionless second order pressure field - J Mean curvature - a Cylinder radius - d Liquid depth at rest - D Dimensionless liquid depth at rest - h Free surface height - H Dimensionless free surface height at the second order     

Fluid flow in a rotating cylindrical container with a rotating disk at the fluid surface

Fluid flow in a rotating cylindrical container of radius R_w and height H with a co-axially rotating disk of radius R_d at the fluid surface is numerically investigated. The container and the disk rotate with angular velocities Ω_w and Ω_d, respectively. We solve the axisymmetric Navier-Stokes equations using a finite-volume method. The effects of the relative directions and magnitudes of the disk and container rotations are studied. The calculations are carried out with various ratios of Ω_w and Ω_d for H/R_w = 2 and R_d/R_w = 0.7. Streamlines and velocity vectors in the meridional plane and azimuthal velocities are obtained. The flow fields in the meridional plane are discussed with relation to azimuthal velocities in the interior of the container. The numerical results are also compared with experimental data.     

Viscous flow in a cylindrical vessel in the presence of a rotating disk

Viscous flows in the cylinder-disk system have been investigated theoretically and experimentally, over a broad range of Reynolds numbers Re, H/R_T, and R_k/R_T in order to explore the characteristics of the flow, which is a function of time, the depth of the liquid, the Reynolds number, the radii of the disk and the cylinder, and their geometry (flat, convex or concave disk). The results of comparing the data of numerical and laboratory simulations are presented. The appearance of secondary eddies in the axial region at large Reynolds numbers has been detected and diagrams of flows of different spatial configuration constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 33–40, September–October, 1985.     

Control of vortex breakdown in a closed cylinder with a rotating lid

The flow within a closed cylinder with a rotating lid is considered as a prototype for fundamental studies of vortex breakdown. Numerical simulations for various parameter values have been carried out to reproduce the known effect of a thin rotating rod positioned along the center axis as well as analyze the influence of local vorticity sources. As expected, the results show that the breakdown bubbles in the steady axisymmetric flow can be affected dramatically, i.e., fully suppressed or significantly enhanced, by rotating the rod. The main contribution of this article is to show that the observed behavior can be explained by the vorticity generated by the rod locally near the rotating lid and near the fixed lid, as analogous behavior is caused by the introduction of local vorticity sources in the flow without a rod. Moreover, we describe the influence on the breakdown bubbles of the vorticity sources by an analytical model. In addition to improving our understanding, this finding should also open the door to other types of flow control devices capable of generating localized vorticity.     

Observations of the flow produced in a cylindrical container by a rotating endwall

Observations made using the laser-induced fluorescence technique are presented of the steady swirling flow produced in a closed cylindrical container completely full of fluid (a glycerine/water mixture for the experiments reported here) by rotating one endwall. The flow behaviour is determined by two parameters: the height-to-radius ratioH/R and a rotation Reynolds number R ²/. In an earlier study, Vogel (1968) defined the stability limit in the (H/R, R²/) plane within which a vortex breakdown bubble occurred on the axis of symmetry. The results of Vogel's investigation are confirmed and extended by the present work. In particular, it is found that asH/R is increased two further stability limits can be determined within which two and ultimately three breakdown bubbles occur in succession. It is also found that there is a Reynolds number boundary above which the flow is oscillatory and at even higher Reynolds number the flow becomes turbulent. Until well into the unsteady-flow domain, the flow shows negligible departure from axisymmetry.     

Experimental investigations of the unsteady rotating flow field in a cylindrical vessel

The rotating flow field in a cylindrical vessel — the so-called whirlpool — is widely used in food engineering as a method for separating particles out of a suspension (Cup-of-tea-method). However many of these whirlpools do not operate adequately or fail entirely. In order to solve this problem, the first step was to investigate the flow field and its time dependency which has not been sufficiently understood until now.The rotating flow in a cylindrical vessel — induced by a fluid jet during the filling period of this vessel — is slowed down by fluid friction after the closing of the inlet valve. The velocity fields to be found mainly near, and pressure distributions at the bottom of the vessel, are measured during this unsteady flow. The results, especially those which describe vortex systems, are used to improve the separation system. This paper is restricted to the hydrodynamic aspect. Therefore success in industrial applications can only be indicated.     

Period-doubling route to chaos for a swirling flow in an open cylindrical container with a rotating disk

Conclusion The period-doubling route to chaos for a swirling vortex flow in an open cylindrical container, at an aspect ratio of 2, driven by a rotating bottom disk was recognized by using laser-Doppler velocimetry. The onset of periodic motion for the flow was found when Re was in the range between 1850 and 1900. The flow was subharmonically bifurcated into a double-period motion when Re was about 2150. When the Reynolds number was in the range from 2300 to 2400, the flow bifurcated again through the period-doubling mode. When the Reynolds number was further increased, the flow eventually showed chaotic motion. The existence of a free surface promotes the onset of periodicity, and the difference of the critical Reynolds number with and without a free surface was estimated to be about 600.This work was supported by the National Science Council of the Republic of China under grant No. NSC-82-0410-E-002-191     

Numerical solution of the navier-stokes equations for a rotating fluid flow in a cylindrical pipe

Summary A numerical investigation of a viscous incompressible rotating fluid flow in a circular cylinder with a net axial flow rate is presented. The cylinder is opened at the bottom and closed at the top by an ideal porous rotating disk, through which the incoming fluid flows receiving an action of solid body rotation. This physical model has been chosen for a realistic representation of fluid direct prerotation which takes place in the suction pipe of a centrifugal pump. The flow field was solved by the direct solution of the complete Navier-Stokes equations for values of Reynolds tangential numberRe _t up to 5 000. The numerical results give a detailed and complete sight of the principal features of such kind of flow, and, compared with experimental results in the turbolence range ofRe _t. show a significant qualitative agreement. A comparison with previous numerical investigations of other authors about rotating fluids is reported.

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Sommario In questa nota si presenta uno studio numerico del moto di un fluido viscoso ed incomprimibile in un tubo cilindrico, chiuso in uscita da un disco ideale poroso rotante, che trascina e mantiene in rotazione il fluido che lo attraversa. Si è adottato il suddetto modello per rappresentare la prerotazione diretta che ha luogo nel tubo di aspirazione di una pompa centrifuga. Si è determinato il campo di velocità per via numerica attraverso la soluzione diretta delle equazioni di Navier-Stokes in forma completa fino a valori del numero di Reynolds tangenzialeRe _t pari a 5 000. I risultati numerici danno un quadro dettagliato e completo delle principali caratteristiche di tale tipo di flusso, e, confrontati con risultati sperimentali disponibili nel campo diRe _t turbolento, mostrano un significativo accordo qualitativo. Si riporta infine un confronto con i risultati di precedenti indagini numeriche sul moto di fluidi rotanti presenti in letteratura.

     

Viscoelastic flow between eccentric rotating cylinders: unstructured control volume method

This paper reports a convergent numerical algorithm for the Upper-Convected Maxwell (UCM) fluid between two eccentric cylinders at various eccentricity ratios (?); the outer cylinder is stationary, and the inner one rotating. The problem is solved by an unstructured control volume method (UCV), which is designed for a general viscoelastic flow problem with an arbitrary computational domain. A self-consistent false diffusion technique and an iteration scheme are used in combination to solve the problem. The computations of the UCM fluid using the numerical algorithm are carried out to a higher value of the Deborah number (De) at each eccentricity tested than hitherto possible with previous numerical simulations. The solutions are compared with previous numerical results, confirming the effectiveness of the UCV method as a general technique for solving viscoelastic flow problems.     

Strong cylindrical shocks in a rotating gas

Similarity solutions describing the flow behind a diverging strong cylindrical shock wave, advancing into a nonuniform gas having solid body rotation, are studied. The effects of the angular velocity variation on the shock velocity are shown graphically. It is found that an increase in the initial angular velocity leads to a decrease in the shock velocity.Nomenclature c ₀ sound velocity in unperturbed state - c sound velocity in unperturbed state at the axis of symmetry - D nondimensional density in unperturbed state - E energy release per unit length - f nondimensional radial velocity in perturbed state - g nondimensional pressure in perturbed state - h nondimensional density in perturbed state - k nondimensional azimuthal velocity in perturbed state - M an integral - N another integral - P nondimensional pressure in unperturbed state - p pressure in perturbed state - p ₀ pressure in unperturbed state - p pressure at the axis in unperturbed state - p ₁ pressure immediately behind the shock front - R shock front radius - r radial coordinate - R ₀ a characteristic length parameter - t time coordinate - U shock front velocity - u particle velocity (radial) in perturbed state - u ₀ particle velocity (radial) in unperturbed state - u ₁ particle velocity (radial) immediately behind the shock front - v particle velocity (azimuthal) in perturbed state - v ₀ particle velocity (azimuthal) in unperturbed state - v ₁ particle velocity (azimuthal) immediately behind the shock - w nondimensional azimuthal velocity in unperturbed state - x a nondimensional independent variable - z axial coordinate of cylindrical coordinates - Z a nondimensional independent variable - ₀ angular velocity in unperturbed state - ₁ angular velocity immediately behind the shock - density in perturbed state - ₀ density in unperturbed state - ₁ density immediately behind the shock - density at r=0 in unperturbed state - adiabatic index of the gas - ₀ R ₀ ² ₀ ² /(c)²     

On the dynamics of liquids in a cylindrical tank with a flexible bottom

Summary The problem of hydroelastic sloshing of a liquid, partially filling a cylindrical tank with rigid side walls and a flat flexible bottom, has been investigated. The liquid is assumed to be inviscid, homogeneous and incompressible; the flexible bottom is assumed to be a stretched membrane. An approximate solution is given and numerical examples are discussed. The data obtained show good agreement with experimental results reported by other authors.

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Übersicht In der vorliegenden Arbeit wird das hydroelastische Problem des Schwappens von flüssigem Treibstoff in einem kreiszylindrischen Tank mit starrer Wand und ebenem, flexiblem Boden untersucht. Die Flüssigkeit wird als reibungsfrei, homogen und inkompressibel, der Boden als gespannte Membran angenommen. Eine Näherungslösung wird angegeben und numerische Rechnungen werden diskutiert. Die erhaltenen Resultate zeigen gute Übereinstimmung mit experimentellen Befunden anderer Autoren.

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Paper presented in part at the Canadian Congress of Applied Mechanics, Centennial Year 1967, 22–26 May 1967, Lavai University at Quebec City. This work was performed under National Aeronautics and Space Administration Grant No. NsG-542 to the University of Florida. Numerical calculations were carried out at the University of Florida computing center.     

Flow of a viscous fluid in a cylindrical vessel with a rotating cover

The problem considered arises in solving various technical problems associated with flows of a viscous fluid in a closed space near rotating plane surfaces, turbomachine disks, thrust bearings, rotational viscosimeters, etc. The approximate solution of the problem on the basis of a simplified flow scheme was first obtained by Schultz-Grunow [1], The most complete investigation has been made recently by Grohne [2], who outlined a program for solving the problem by joining several partial solutions on the basis of definite hypotheses concerning the flow core.With the development of electronic digital computers and the necessary numerical methods, the most effective means of solving the considered problem is the use of the grid methods for solving partial differential equations. The present paper is devoted to presenting the results of the solution of the problem using the grid method on a digital computer.     

Heat transfer in rotating cylindrical cells with partitions

A numerical analysis and an experimental study of heat transfer rate in rotating cylindrical cells with partitions are performed. The work is done mainly in the Ekman suction regime, where the Coriolis force dominates over centrifugal buoyancy. It is shown that the heat transfer rate increases substantially by placing partitions in the cell. The partitions suppress the Coriolis force so that convection induced by the centrifugal buoyancy becomes important. It is found that the Nusselt number correlates with the parameter PrβΔTEk^?1/2 with the partitions. The partitions have no effect on the heat transfer in the centrifugal buoyancy convection regime.     

Time-series PIV measurements in a very large rotating tank

 We describe a non-intrusive PIV system developed for performing high-resolution measurements in a field of view of 2 m×3 m, as required on the LEGI-Coriolis 13 m diameter rotating platform. Particle preparation, laser illumination, photographic digitization, and cross-correlation analysis techniques are explained. Some results on the wake behind a cylinder illustrate the possibilities of this PIV system. Received: 29 October 1996/Accepted: 5 April 1997     

Flow of a shear-thinning liquid in a cylindrical container with a rotating end wall

The flow patterns produced by rotating one end wall of a circular cylinder completely filled with a strongly shear-thinning viscoelastic liquid have been investigated using the laser-induced fluorescence flow visualization technique. An intense toroidal vortex is produced in the vicinity of the rotating end wall with outward spiraling flow over the end wall itself. This vortex drives a second countercirculating vortex of low intensity in the region of the stationary end wall. Under some circumstances an axial jet of fluid is observed moving away from the rotating end wall. This jet showed evidence of instability, whereas all flows were otherwise completely steady. The double-vortex structure is different from those recently observed in either a Newtonian or slightly shear-thinning liquid or in the low Reynolds number flow of an elastic liquid. There are, however, similarities with older work for a viscoelastic liquid at relatively high Reynolds numbers. The observations highlight the suitability of the cylinder/rotating end wall configuration as a sensitive test case for computational work.