Very-Low-Frequency State in a Short Annulus Taylor-Couette Flow

Taylor-Couette flow between two concentric rotating cylinders continues to provide a canonical physical system that has been instrumental in developments in nonlinear dynamics, routes to chaos, and equivariant dynamical systems (e.g. [1, 2, 3]). The study of the influence of endwalls and reflection symmetry at the annulus half-height [4, 5] opened up a new perspective into the importance of Z2 symmetry (reflection) in Taylor-Couette flow, as well as in many other equivariant problems. The impact of Z2 symmetry is enhanced as the aspect ratio of the annulus is reduced [6]. Most theoretical and numerical studies of short annulus Taylor-Couette flow have been restricted to an axisymmetric subspace and have primarily considered steady Z2-symmetry breaking. Experimentally however, very rich and complex spatiotemporal dynamics, including global bifurcations, have been observed that are intrinsically associated with the Z2 equivariance of the system (e.g. [7, 8, 9, 10]). Equivariant bifurcation theory [11, 12, 13] provides a classification of possible bifurcation scenarios that may occur in the presence of symmetries. Two recent studies on short annulus Taylor-Couette flow with Z2 reflection symmetry have identified two of the ways the basic state can become unstable, both breaking the Z2 symmetry; in [14] the symmetry breaking is via steady-state pitchfork bifurcations and in [15] it is via a Hopf bifurcation. In both investigations, the analyses of the results were restricted to an axisymmetric SO(2) subspace. One of the main conclusions from [14] is the conjecture that at small aspect ratios (less than 0.5), the steady axisymmetric flow is unique, and yet they report observing time dependence and disorder at higher Reynolds number (Re) and claim that an understanding of the evolution remains an outstanding challenge. In this paper we investigate numerically the dynamics at aspect ratio Γ = 0.5 using a threedimensional Navier-Stokes spectral solver. At small Re we found the steady-state pitchfork bifurcation that breaks the Z2 symmetry, as well as the inverse pitchfork where the bifurcated states revert to a Z2-symmetric state, as reported in [14]. At slightly higher Re, the flow becomes unstable via two Hopf bifurcations that break Z2, one leading to an axisymmetric limit cycle, LC0, and the other to a nonaxisymmetric rotating wave, RW1. The short aspect ratio of the annulus investigated here results in a basic state with a single outwards jet at the mid-height plane; much larger aspect ratios lead to states with Taylor vortex-like flows with multiple jets. From group theoretic considerations, we describe the four ways in which the basic state can become unstable via primary symmetry breaking bifurcations. Three of these are identified as solutions for this particular geometry. One is a steady axisymmetric pitchfork bifurcation, which has been previously investigated [14]. The other two are via Hopf bifurcations producing either an axisymmetric limit cycle or a rotating wave with azimuthal wave number m = 1, which are systematically investigated here for the first time. The two Hopf bifurcation curves intersect at a double Hopf bifurcation, at which point an unstable mixed mode also bifurcates. The role of this mixed mode is to either stabilize or destabilize the periodic solutions from the two Hopf bifurcations at Neimark-Sacker bifurcations. A second mixed mode, in the form of a modulated rotating wave (MRW), not directly associated with the double Hopf bifurcation, also bifurcates from the two periodic solutions at higher Reynolds numbers. It is stable (unstable) when it bifurcates from a stable (unstable) periodic state. These stable and unstable (a) η < ηdH LC MRW VLF 0