Competition and bistability of ordered undulations and undulation chaos in inclined layer convection

Abstract

Experimental and theoretical investigations of undulation patterns in high-pressure inclined layer gas convection at a Prandtl number near unity are reported. Particular focus is given to the competition between the spatiotemporal chaotic state of undulation chaos and stationary patterns of ordered undulations. In experiments, a competition and bistability between the two states is observed, with ordered undulations most prevalent at higher Rayleigh number. The spectral pattern entropy, spatial correlation lengths and defect statistics are used to characterize the competing states. The experiments are complemented by a theoretical analysis of the Oberbeck–Boussinesq equations. The stability region of the ordered undulations as a function of their wave vectors and the Rayleigh number is obtained with Galerkin techniques. In addition, direct numerical simulations are used to investigate the spatiotemporal dynamics. In the simulations, both ordered undulations and undulation chaos were observed dependent on initial conditions. Experiment and theory are found to agree well

Pattern formation in spatially forced thermal convection

“…Longitudinal forcing (ϕ = 0 • ): For inclination angles 28 • < θ < 72 • , forced LR became unstable to uniformly drifting undulations (UN) as shown in figure 7. This state, which was also observed for unforced ILC [26], is characterized by the modes q 1 = q f (1, s y ) and q 2 = q f (1, −s y ), where s y decreased with increasing inclination angle from s y = 0.47 for θ = 30 • to s y = 0.25 for θ = 65 • (taken at undulation onset). While unforced UN were often defect turbulent (undulation chaos) [18,26], forced UN were mostly ordered and only rarely perturbed by fast traveling defects.…”

“…This state, which was also observed for unforced ILC [26], is characterized by the modes q 1 = q f (1, s y ) and q 2 = q f (1, −s y ), where s y decreased with increasing inclination angle from s y = 0.47 for θ = 30 • to s y = 0.25 for θ = 65 • (taken at undulation onset). While unforced UN were often defect turbulent (undulation chaos) [18,26], forced UN were mostly ordered and only rarely perturbed by fast traveling defects. This fact, together with the previously mentioned shift of the instability line to higher values of ε, (section 5) is a clear indication of the stabilizing effect of forcing.…”

“…For comparison, the calculated instability of LR in unforced ILC is plotted as a dashed line in figure 3(a). It is significantly lower, implying that forcing stabilizes LR and suppresses the transition to UN in this θ range (see [26]).…”

“…For the longitudinal forcing case (ϕ = 0, figure 3(a)), the instability line that separates LR at lower ε and undulations (UN; see [26]) at higher ε decreases monotonically with increasing θ for θ < θ cd . For comparison, the calculated instability of LR in unforced ILC is plotted as a dashed line in figure 3(a).…”

Pattern Forming System in the Presence of Different Symmetry-Breaking Mechanisms

“…As a result, depending on the inclination angle , either longitudinal rolls (buoyancy driven) or transverse rolls (shear-flow driven) set in at onset [11]. The wealth of nonlinear states observed with increasing ÁT has been the focus of recent experimental and theoretical studies [12][13][14].In this Letter, we applied periodic forcing at an angle ' with respect to the in-plane gravity component [ Fig. 1(a)], with a wave vector q f .…”