G. P. Neitzel

Centrifugal instabilities during spin-down to rest in finite cylinders. Numerical experiments

A cylinder filled with a viscous, incompressible fluid is in an initial state of rigid-body rotation about its axis of symmetry. If the container is brought to rest impulsively, the resulting unsteady spin-down flow may be subject to sidewall instabilities due to an imbalance between centrifugal and pressure gradient forces. These instabilities are examined numerically using a finite-difference simulation to integrate the axisymmetric Navier–Stokes equations for a variety of aspect ratios and Reynolds numbers. The Taylor–Gortler vortex-wavelength spectrum, the torque and the angular momentum histories are calculated. Criteria for the onset time for instability and the spin-down time are given. The effects of the enhanced mixing due to instability on the spin-down characteristics and torque are discussed. The results are compared with experiment.

Streak‐line motion during steady and unsteady axisymmetric vortex breakdown

Numerical calculations of vortex breakdown generated within a closed circular cylinder by rotation of one of the end walls have been performed. The solutions are used to compute steady streak‐line patterns for dye introduced both symmetrically and asymmetrically about the symmetry axis upstream of the breakdown bubble. Beginning from such a steady state, the end‐wall angular speed is impulsively increased and the asymmetric streak‐line pattern is observed during a portion of the subsequent period of unsteady flow.

The influence of initial condition on the linear stability of time‐dependent circular Couette flow

The effect of the starting condition on the linear stability properties of circular Couette flow with a time‐dependent inner‐cylinder motion is investigated. In addition to the WKBJ approach employed previously by Eagles [Proc. R. Soc. London, Ser. A 355, 209 (1977)] for slowly varying flow, an initial‐value method is also used. Results are presented for different stability criteria.

Marginal stability of impulsively initiated Couette flow and spin‐decay

A viscous, incompressible fluid is contained within an infinitely long circular cylinder or between a pair of infinitely long concentric cylinders. In both cases, the entire system is in a state of rigid‐body rotation. At time t = 0 the outer flow boundary is impulsively brought to rest, giving rise to a potentially unstable, unsteady swirl flow. The stability of these flows is examined by employing energy theory with a marginal stability criterion to obtain lower bounds on the onset times for instability. In some cases, the asymptotic steady‐state flow will be stable, with any instability merely a transient, while in other cases, stability of the asymptotic state is not guaranteed.

Energy stability theory of decelerating swirl flows

An infinite circular cylinder, filled with viscous, incompressible fluid, is rotating as a solid body. At time t=0 the angular velocity ω (t) of the cylinder is decreased in a prescribed fashion. The resulting swirl flow is susceptible to centrifugal instabilities. The method of energy is used to determine sufficient conditions, R<RG, where R is a Reynolds number, such that rotationally symmetric disturbances of arbitrary amplitude decay to zero with time. Both impulsive and smooth angular velocity histories of the container are considered. The analysis further provides a lower bound t+ on the onset time before which all disturbances must decay to zero, for R≳RG. If the final state (as t→∞) also corresponds to solid body rotation (e.g., when the cylinder is brought to rest), then the analysis simultaneously provides an upper bound t‡ on the decay time after which all disturbances decay zero, for R≳RG. In this case centrifugal instabilities are confined to times t in the interval t+<t<t‡.

Experiments on impulsive spin‐down to rest

Experiments were performed to determine the stability characteristics of impulsive spin‐down to rest in a circular cylinder of moderate aspect ratio. The characteristics measured include the onset time, onset wavelength, and time history of the wavelength for the Taylor–Gortler instability, as well as the global stability limit of the apparatus. Results compare favorably with earlier theoretical, numerical, and experimental investigations.

Experiments on the onset of instability in unsteady circular Couette flow

Experiments have been performed to study the hydrodynamic stability of unsteady circular Couette flow generated by monotonic time-dependent inner-cylinder motions. The onset of instability was determined by measuring the axial component of velocity using laser-Doppler velocimetry; deviations from the pure-swirl value of zero are indicative of the initiation of Taylor vortices. The measurement technique was found to have an ‘intrusive’ effect on the flow stability, which was eliminated by the design of the experimental procedure. Significant enhancement of stability was found, in qualitative, but not quantitative, agreement with earlier theoretical predictions.

Stability of circular Couette flow with variable inner cylinder speed

Energy & ability theory is employed to study the finite-amplitude stability of a viscous incompressible fluid occupying the space between a pair of concentric cylinders when the inner-cylinder angular velocity varies linearly with time. For the case with a fixed outer cylinder and increasing inner-cylinder speed, we find an enhancement of stability, consistent with a linear-theory result due to Eagles. When the inner-cylinder speed decreases, we find an initially decreased stability bound, indicating the possibility of hysteresis, while, if the inner cylinder is allowed to reverse direction and linearly increase in speed, we find significant stability enhancement.

Onset of convection in impulsively heated or cooled fluid layers

Energy stability theory is employed to determine lower bounds on onset times and global stability bounds for initially isothermal fluid layers subjected to impulsive changes in surface temperature. Various combinations of rigid and free boundary conditions and heating or cooling are considered.

Numerical experiments on the stability of unsteady circular Couette flow with random forcing

The stability of unsteady circular Couette flow generated by monotonically increasing the angular speed of the inner cylinder is examined through a series of numerical experiments. Random disturbances are imposed continually on the basic state by the inclusion of random forcing terms in the axisymmetric Navier–Stokes equations. Results of the calculations are compared with previous theoretical and experimental results.