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.