Marginal stability curve and linear growth rate for rotating Couette–Taylor flow and Rayleigh–Bénard convection

Abstract

Results of high numerical accuracy are reported for the growth rate σ of axisymmetric disturbances to circular Couette flow between concentric cylinders with the inner one rotating and of two-dimensional disturbances of the pure conduction state of a Boussinesq fluid between horizontal parallel plates heated from below. When σ is positive, the former case leads to Couette–Taylor (CT) flow and the latter to Rayleigh–Bénard (RB) convection. The curve of vanishing growth rate (marginal stability curve) Tc(q) or Rc(q), where q is the wave vector of the disturbance and T and R are the Taylor and Rayleigh number, respectively, is obtained for both cases. The correlation length amplitude ξ0 is calculated by obtaining the curvature of the marginal stability curve. The amplitude σ0 of σ was determined from data with T>Tc(q) or R>Rc(q). In that range, the wave vector qm of maximum σ is also found. For the CT system, the calculations were performed systematically as a function of radius ratio η for 0.975≥η≥0.1. For the RB system, those parameters which depend on the Prandtl number P were obtained for 10−3≤P≤103. Some of the RB results could be compared with analytic calculations and served to test this numerical procedure. It is expected that these results will be useful in calculations using an amplitude equation for the weakly nonlinear state and in studies of wavenumber-selection processes.

Spiral defect chaos in large aspect ratio Rayleigh-Bénard convection

“…The mean wavevector < k > of the pattern decreased with increasing ǫ in such a way that it stayed well within the theoretically stable region 3,16 (the Busse balloon) as shown in of the disordering effects of dynamics intrinsic to the pattern, rather than as a response to boundary conditions. At higher ǫ, the wavevector distribution extends over both boundaries of the Busse balloon, and we do occasionally observe, for example, the nucleation of a pair of dislocations in places where rolls are pushed close together.…”

Phase slip solutions in magnetically modulated Taylor–Couette flow

“…Two further comments are also in order regarding this initial bifurcation. First, the slight reduction from 68.2 in the non-magnetic case [36] to 67.4 here is caused by the presence of the background cells, and does not indicate a subcritical bifurcation. Second, as a transition from a steady to a periodic solution, it is a true bifurcation.…”

Boundary limitation of wave numbers in Taylor-vortex flow

“…We depended upon the measured time evolution of the phase of our system to decide whether the data were quasi-static. Amplitude equilibration after a change of f is generally much faster and typically occurs on a time scale of τ 0 t ν /ǫ with [35] τ 0 = 0.076, which for ǫ > ∼ 10 −2 is only about a minute. A CCD camera was used to digitally record an image of the Taylor-vortex pattern.…”