Pinning of rotating waves to defects in finite Taylor–Couette flow

Abstract

Experiments in small aspect-ratio Taylor-Couette flows have reported the presence of a band in parameter space where rotating waves become steady nonaxisymmetric solutions (a pinning effect) via infinite-period bifurcations. Previous numerical simulations were unable to reproduce these observations. Recent additional experiments suggest that the pinning effect is not intrinsic to the dynamics of the problem, but rather is an extrinsic response induced by the presence of imperfections. Here we present numerical simulations that include a small tilt of one of the endwalls, simulating the effects of imperfections that break the SO(2) axisymmetry of the problem, and indeed are able to reproduce the experimentally observed pinning of the rotating waves. Dynamical systems considerations suggest that any imperfection breaking the SO(2) axisymmetry of the problem must result in the formation of a pinning region of finite width. We have also found that the particulars of the pinning process, in particular the width of the pinning region, are extremely sensitive to the type of imperfection in the system. Almost identical flows respond in completely different ways to the same imperfection, depending on subtle differences in the weak secondary characteristics of the flow. The numerical simulations of the Navier-Stokes equations for the problem with an imposed tilt of an endwall together with normalform analysis of a Hopf bifurcation subjected to imposed symmetry-breaking help shed some light on previous experiments that reported a variety of different dynamical behaviour for which a clear explanation was lacking.

Three-dimensional instabilities in a discretely heated annular flow: Onset of spatio-temporal complexity via defect dynamics

“…One example in Taylor-Couette flow has been extensively studied, [36][37][38] where the small parameter gap between a symmetric limit cycle and asymmetric quasi-periodic states is populated by a complicated bifurcational process involving homoclinic cascades. Another example is in rotating convection or in small aspect ratio Taylor-Couette flow, 39 where the breaking of the SO(2) rotational symmetry results in a small parameter gap (a pinning region) that separates rotating wave solutions precessing in opposite directions. On approaching this pinning region close to the Hopf bifurcation results in very complex homoclinic and heteroclinic dynamics.…”

Mixing in thermally stratified nonlinear spin-up with uniform boundary fluxes

“…The governing equations (1)-(2) are discretized on a staggered grid with the velocities at the faces and all the scalars in the center of the computational cell; the resulting system of equations is solved by a fractional-step method. The finite-difference solver is based on that described by [34] and has been tested in a wide variety of enclosed cylindrical flows [18,20,24,25,32,33], establishing resolution requirements over a wide range of parameters.…”

Bifurcations with imperfect SO(2) symmetry and pinning of rotating waves

“…Figure 9. Bifurcation diagrams for the one-cell state from (a) the experimental results of Abshagen et al [16] with natural imperfections, and (b) the numerical results of Pacheco et al [23] with a tilt of 0.1…”

“…The correspondence between these results and the normal form theory described in this paper is excellent, strongly suggesting that the general remarks on pinning extracted from the analysis of the five particular cases are indeed realized both experimentally and numerically. These two studies [16,23] are the only cases we know of where quantitative data about the pinning region are available. Yet, even in these cases, the dynamics close to the intersection of the Hopf curve with the pinning region, which according to our analysis should include complicated bifurcational processes, has not been explored either numerically or experimentally.…”