Longitudinal vortices in unsteady Taylor-Couette flow: solution to a 60-year-old mystery

Applying a sufficiently rapid start-stop to the outer cylinder of the Couette-Taylor system, structures approximately aligned with the axis were recorded in the classic work of Coles (1965). These short-lived rolls are oriented perpendicular to the classic Taylor-vortex rolls. In this work we report numerical observation of this instability, guided by a more recent experimental observation. The instability is shown to be related to an inflection in the azimuthal velocity profile, a finding consistent with the experimental observations of its emergence during the deceleration phase. Despite the transient nature of start-stop experiments, we show that the instability can be linked to that of the oscillating boundary layer problem of Stokes. There are several reasons why the instability may have remained elusive, both for experimental observation and intrinsic to the idealized system. We look in more detail at dependence on the radius ratio for the Taylor-Couette system and find that, in the case where the size of the rolls scales with the gap width, for radius ratios any lower than that used by Coles, R_i/R_o=0.874, the instability is quickly overrun by axisymmetric rolls of Gortler type.

💡 Research Summary

This paper revisits a long‑standing puzzle first noted by Coles (1965): the appearance of short‑lived, axially aligned vortical structures when the outer cylinder of a Taylor‑Couette apparatus is abruptly stopped. Those structures, which run parallel to the axis rather than the classic Taylor‑vortex rolls that encircle the annulus, have remained largely undocumented because the original experiments provided no quantitative parameters and the phenomenon is highly transient. Recent observations by Burin & Czarnocki (2012) reported a similar instability during the deceleration phase of a start‑stop protocol, prompting the authors to investigate the underlying physics through high‑resolution numerical simulations.

The authors model the flow between concentric cylinders of radii (R_i) and (R_o) (radius ratio (\eta = R_i/R_o)) using the incompressible Navier–Stokes equations non‑dimensionalised by the gap width (d = R_o - R_i) and the viscous diffusion time (d^2/\nu). The base azimuthal flow (V_0(r,t)) follows the instantaneous Reynolds numbers of the inner and outer cylinders, while the perturbation field (\mathbf{u}’) satisfies a rotational form of the governing equations (Eq. 2.4). A double‑Fourier (azimuthal and axial) – Chebyshev (radial) spectral discretisation is employed, with axial wavenumber set to zero (i.e., two‑dimensional, axially invariant calculations) because the observed rolls are aligned with the axis. The time integration uses a Crank–Nicolson scheme for diffusion and an Euler predictor for the nonlinear terms; a fixed timestep of (10^{-6}) is used throughout. Linear stability is probed by freezing the mean profile at selected times and applying an Arnoldi iteration to obtain growth rates of infinitesimal disturbances.

The start‑stop protocol is idealised as a linear ramp up of the outer‑cylinder Reynolds number (Re_o(t)) from zero to a peak value (U_1) over a short interval (T_1), followed by a linear ramp down to zero over (T_2). The inner cylinder remains stationary ((Re_i=0)). Parameter values chosen to mimic the Burin & Czarnocki experiments are (U_1=8000), (T_1=0.01), (T_2=0.05) (in viscous time units), and a narrow gap (\eta=0.97). The azimuthal wavelength observed experimentally is about (3.5) gap widths, which corresponds to an azimuthal mode number (m\approx 60).

During the ramp‑up phase the viscous diffusion penetrates only a fraction of the gap ((\delta/d\sim\sqrt{t})), leaving a non‑zero velocity profile in the interior. When the outer cylinder decelerates, the mean azimuthal profile develops a pronounced inflection point in its second derivative. This inflection point satisfies the Rayleigh criterion for centrifugal instability and simultaneously creates a shear layer reminiscent of the Stokes oscillating boundary layer. Linear analysis of the frozen profiles shows that the most unstable azimuthal mode is (m’=1) (i.e., a single pair of counter‑rotating rolls) with growth rates (\sigma) in the range (0.02)–(0.04) (non‑dimensional). The peak growth occurs around (t\approx0.04), precisely during the deceleration stage, and the preferred wavelength is (\lambda\approx2.5)–(3) gap widths, matching the experimental observations.

A systematic study of the radius ratio reveals that for (\eta) lower than the value used by Coles ((\eta=0.874)), Görtler‑type centrifugal rolls, which arise from curvature‑induced instability on the inner wall, grow faster and overrun the longitudinal vortices. Thus the longitudinal instability is confined to relatively narrow gaps (large (\eta)). Varying the peak Reynolds number (U_1) changes the amplitude of the base flow but does not significantly shift the preferred wavelength; higher amplitudes simply increase the growth rate. However, excessively large accelerations generate strong end‑cap turbulence that can mask the rolls, explaining why the phenomenon is difficult to capture experimentally.

The authors conclude that the longitudinal vortices are a transient manifestation of a combined centrifugal‑inflection‑point instability that is mathematically analogous to the Tollmien‑Schlichting instability of the Stokes oscillating boundary layer. The rapid start‑stop creates a thin, highly sheared boundary layer on the outer cylinder; as the cylinder decelerates, the shear layer’s inflection point triggers the growth of axially aligned rolls. This mechanism resolves the 60‑year‑old mystery of Coles’s “instability following start‑stop motion of the outer cylinder.” The work also highlights why the instability has been elusive: it requires a narrow gap, a sufficiently rapid deceleration, and a low‑noise environment; otherwise it is quickly suppressed by Görtler rolls or end‑cap turbulence.

Future directions suggested include exploring nonlinear saturation, interaction with end‑cap driven secondary flows, and extending the analysis to higher Reynolds numbers where turbulent background may coexist with the transient rolls. Such studies could have practical implications for flow control in rotating machinery and for understanding transient phenomena in geophysical and astrophysical rotating flows.