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 ΩR
2/ν. In an earlier study, Vogel (1968) defined the stability limit in the (H/R, Ω R2/ν) 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.

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

The flow between two finite rotating disks enclosed by a cylinder is investigated both numerically and experimentally. For this finite geometry the full stationary Navier–Stokes equations are solved numerically without similarity assumptions. Experimental results are obtained by means of stereophotography of small tracer particles. The results are in good agreement with the numerical solution. Owing to the presence of the cylinder sidewall, the solution is found to be unique for all values of the parameters considered. When the disks rotate in opposite senses with counter-rotation above 15%, a stagnation point appears at the slower-rotating disk. This stagnation point is associated with a two-cell structure in the meridional plane and is experimentally observed as a ring of particles at the slower-rotating disk. Near the axis of rotation the solution is found to satisfy similarity demands; for weak counter-rotation the solution is of Batchelor type near the axis of rotation, but for strong counter-rotation a Stewartson profile is found to be more adequate for the description of the tangential velocity near the axis.

The flow between two finite rotating disks enclosed by a cylinder

A flow circulation in a closed circular-cylindrical container is produced by a rotating lid. After a transient phase from an initial state at rest a steady-flow situation is reached for a certain parameter range. In a subspace of this parameter range an undulating meridional flow occurs that may exhibit at the axis of rotation one or several separation bubbles which are interpreted as vortex breakdown. Numerical calculations on the basis of the Navier-Stokes equations for incompressible homogeneous and Boussinesq fluids enable the study of the influence of various flow parameters on the properties of these separation bubbles. The parameters varied are the Reynolds, Prandtl, Rayleigh, and Eckert numbers together with the ratio of height to radius of the container. The numerical results are in good agreement with experiments performed by Vogel, Ronnenberg, and Escudier. The stability of the fluid motions in these experiments with respect to non-axisymmetric disturbances strongly suggests that the corresponding axisymmetric solutions of the Navier-Stokes equations are stable configurations.

Axisymmetric vortex breakdown with and without temperature effects in a container with a rotating lid

An Efficient Spectral-Projection Method for the Navier–Stokes Equations in Cylindrical Geometries: I. Axisymmetric Cases

The relationship of the axisymmetric flow between large but finite coaxial rotating disks to the von Karman similarity solution is studied. By means of a combined asymptotic – numerical analysis, the flow between finite disks of arbitrarily large aspect ratio, where the aspect ratio is defined as the ratio of the disk radii to the gap width separating the disks, is examined for two different end conditions: a ‘closed’ end (shrouded disks) and an ‘open’ end (unshrouded or free disks). Complete velocity and pressure fields in the flow domain between the finite rotating disks, subject to both end conditions, are determined for Reynolds number (based on gap width) up to 500 and disk rotation ratios between 0 and – 1. It is shown that the finite-disk and similarity solutions generally coincide over increasingly smaller portions of the flow domain with increasing Reynolds number for both end conditions. In some parameter ranges, the finite-disk solution may not be of similarity form even near the axis of rotation. It is also seen that the type of end condition may determine which of the multiple similarity solutions the finite-disk flow resembles, and that temporally unstable similarity solutions may qualitatively describe steady finite-disk flows over a portion of the flow domain. The asymptotic – numerical method employed has potential application to related rotating-disk problems as well as to a broad class of problems involving flow in regions of large aspect ratio.

On rotating disk flow

A numerical computation of a viscous incompressible fluid confined in a circular cylindrical chamber has been carried out, where the top disk is rotating with a constant angular velocity, and the bottom disk and side wall are held fixed. Using the full Navier‐Stokes equations, steady secondary flows were calculated for various Reynolds numbers (Re = Ωb2/ν) up to a maximum value of 400. It is found that, for the Reynolds numbers below 10, the governing equations are essentially linear and the numerical solution agrees very well with the analytic solution. For higher Reynolds numbers, the flow near the boundary is intensified. The detailed flow patterns are compared with previous work of a single disk and two infinite disks. The volume flow rate of the secondary flow due to the centrifugal action and the frictional moment of the disk were also calculated.

Hsien‐Ping Pao

Papers

Numerical Solution of the Navier-Stokes Equations for Flows in the Disk-Cylinder System