In §3 we study the stability and bifurcation sequences of nonlinear solutions in supercrit-ical rotating plane Couette flow (RPCF). We present bifurcations of steady secondary solutions in rotation number and cross-flow energy space, (Ro, Ecf). In addition to well-known Taylor vortex solutions, we discuss steady oblique vortex flows (OVF), finding upper and lower solutions which bifurcate from unstable modes of the primary flow. The steady OVF are related to the travelling wave spiral vortex flows discussed for Taylor-Couette flow (TCF) by, for example, Hoffmann et al. (2009), Altmeyer et al. (2010) and Deguchi & Altmeyer (2013). Our analysis contributes to the understanding of these oblique flows, and elucidates the differences in bifurcation behaviour between the oblique and streamwise independent flow states.
6.2 Rotating plane Couette flow 187
An extensive stability analysis of Taylor vortex flow (TVF) in §3.6 yields an insight into the bifurcation sequences which develop as Taylor vortices lose stability. Stability calculations across a wide range of TVF allows the characterization of the instabilities affecting TVF and the tertiary flows into which they develop. Alongside the tertiary flows which have already been discovered in RPCF, such as wavy vortex flow (WVF) and twisted vortex flows (TWI and wTWI), in §3.7 we find an oscillating vortex struc-ture (oWVF) and a streamwise indepedent short spanwise wavelength strucstruc-ture which we have named skewed vortex flow (SVF). Bifurcation analyses in Ro and streamwise wavenumber αs reveal that for small αs wTWI emerges in a saddle-node bifurcation from TVF. A solution branch of wTWI is found to exist which connects TVF states with βs = 1.5 and βs = 3, suggesting a more intricate relationship between the two flows than has previously been appreciated. A bifurcation scenario is presented for oWVF where no stable solution exists and flow moves between unstable states in a quasi-chaotic orbit.
There is perhaps some novelty that such an orbit can exist in a transitional flow regime, and a related process could form the basis for the turbulent dynamics found at larger Re. In a bifurcation analysis of SVF, we find that it connects the first and second TVF which bifurcate from the primary flow. The stability properties of representative cases for all of these tertiary flows have been analyzed, suggesting the existence of quaternary structures to be investigated in future research. The existence of such a range of tertiary and higher order nonlinear structures could, at larger Reynolds numbers, be important in the understanding of fully turbulent flow.
The relevence of tertiary states to transitional RPCF is assessed in §3.8. Through nu-merical simulations on a domain which supports a range of wavenumbers and through a model which simplifies the transition process based on bifurcation sequences, we analyze the states observed in the experiments of Tsukahara et al. (2010) and Suryadi et al.
(2013). We find good agreement between model, simulation and experiment in most cases. However, a notable divergence between model and simulation occurs for Ro = 0.9, where the model predicts a final flowfield consisting of wTWI and the simulation finishes in a final flowfield with a localized twist vortex amongst streamwise independent TVF.
Despite this deviation, the model and simulation flowfields are good approximations for two experimentally observed flowfields at Ro = 0.9.
Further studies could include both a Floquet stability analysis of secondary oblique vor-tices and a study of the bifurcation scenarios which follow when it loses stability. We have some preliminary results in this direction, having found a tertiary oblique wave structure reminiscent of wavy spiral vortex flow of TCF discussed in Hoffmann et al. (2009) and Deguchi & Altmeyer (2013), shown in Figure 6.1. We name this structure wavy oblique
188 6. Conclusions & further work
vortex flow wOVF.
Figure 6.1: Two wOVF flowfields for (αs, βs, Re, Ro) = (0.2, 2, 100, 0.1) which bifurcate from steady OVF at the same parameters.
It would be interesting to explore the connection between wOVF and the turbulent stripe phenomenon in non-rotating plane Couette flow (PCF). Turbulent stripes are a curious feature known to the fluid dynamics community since they were reported in experiments by Coles (1965), whose images caught the attention of Richard Feynman, and prompted him to coin the phrase “barber-pole turbulence” in description of the stripes (Feynman (1964)). Turbulent stripes are found in experiments and simulations of PCF (see, for example, the recent studies of Prigent et al. (2002), Duguet et al. (2010), Duguet &
Schlatter (2013)), however, the origin of the oblique structure remains an open question.
If wOVF or a similar state can be continued from RPCF to PCF, it could serve as a co-herent structure from which obliqueness of the stripes could originate, and about which the turbulent dynamics could be organized.
A non-modal growth analysis across subcritical RPCF has been conducted in §4. By optimization over streamwise and spanwise wavenumbers α and β, we compute the maximum linear energy amplification, G, available to perturbations across subcritical (Ro, Re) space. We find that the effect of strong cyclonic (Ro < 0) or anti-cyclonic (Ro > 0) rotation is to hinder the lift-up and anti lift-up mechanisms, such that optimal perturbations rely on the Orr mechanism for energy growth. The restriction of the anti lift-up mechanism in anti-cyclonic RPCF has been noted before by Yecko (2004) and Rincon et al. (2007), whose results were interpreted in the context of transition in astro-physical accretion disks. Our contribution is therefore to show that strong cyclonic RPCF behaves comparably to strong anti-cyclonic RPCF, with respect to optimal disturbances.
We posit that the absence of the lift-up mechanism in strong cyclonic RPCF will alter the transition scenario as it is currently understood in PCF. The lift-up mechanism is