This paper reports on a theoretical analysis of the rich variety of spatio-temporal patterns observed recently in inclined layer convection at medium Prandtl number when varying the inclination angle γ and the Rayleigh number R. The present numerical investigation of the inclined layer convection system is based on the standard Oberbeck-Boussinesq equations. The patterns are shown to originate from a complicated competition of buoyancy driven and shear-flow driven pattern forming mechanisms. The former are expressed as longitudinal convection rolls with their axes oriented parallel to the incline, the latter as perpendicular transverse rolls. Along with conventional methods to study roll patterns and their stability, we employ direct numerical simulations in large spatial domains, comparable with the experimental ones. As a result, we determine the phase diagram of the characteristic complex 3-D convection patterns above onset of convection in the γ -R plane, and find that it compares very well with the experiments. In particular we demonstrate that interactions of specific Fourier modes, characterized by a resonant interaction of their wavevectors in the layer plane, are key to understanding the pattern morphologies.

We report the first observations of superlattices in thermal convection. The superlattices are selected by a four-mode resonance mechanism that is qualitatively different from the three-mode resonance responsible for complex-ordered patterns observed previously in other nonequilibrium systems. Numerical simulations quantitatively describe both the pattern structure and the stability boundaries of superlattices observed in laboratory experiments. In the presence of the inversion symmetry, superlattices are found numerically to bifurcate supercritically directly from conduction or from a striped base state.

We report and analyze complex patterns observed in a combination of two standard pattern forming experiments. These exotic states are composed of two distinct spatial scales, each displaying a different temporal dependence. The system is a fluid layer experiencing forcing from both a vertical temperature difference and vertical time-periodic oscillations. Depending on the parameters these forcing mechanisms produce fluid motion with either a harmonic or a subharmonic temporal response. Over a parameter range where these mechanisms have comparable influence the spatial scales associated with both responses are found to coexist, resulting in complex, yet highly ordered patterns. Phase diagrams of this region are reported and criteria to define the patterns as quasiperiodic crystals or superlattices are presented. These complex patterns are found to satisfy four-mode ͑resonant tetrad͒ conditions. The qualitative difference between the present formation mechanism and the resonant triads ubiquitously used to explain complex-ordered patterns in other nonequilibrium systems is discussed. The only exception to quantitative agreement between our analysis based on Boussinesq equations and laboratory investigations is found to be the result of breaking spatial symmetry in a small parameter region near onset.

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

The evolution of three-dimensional, cellular convective flows in a plane horizontal layer of a Boussinesq fluid heated from below is studied numerically. Slow motion in the form of a spatially periodic pattern of hexagonal cells is introduced initially. In a further development, the flow can undergo a sequence of transitions between various cell types. The features of the flow evolution agree with the idea of the flow seeking an optimal scale. In particular, two-vortex polygonal cells may form at some evolution stages, with an annular planform of the upflow region and downflows localized in both central and peripheral regions of the cells. If short-wave hexagons are stable, they exhibit a specific, stellate fine structure.