Nonlinear invariant solutions underlying spatio-temporal patterns in thermally driven shear flows*

Driven wall-bounded fluid flows transitioning to turbulence are spatially extended chaotic dissipative non-equilibrium systems that support a large variety of self-organized patterns with regular spatial and temporal structure. In linearly stable parallel shear flows, patterns such as long-studied spontaneously emerging turbulent-laminar oblique stripes remain only partly understood. On the contrary, thermal convection in a fluid layer between two horizontal plates kept at different temperature, exhibits patterns that can often be described via a sequence of bifurcations off a base state undergoing a linear instability. If a Rayleigh-Bénard convection cell is inclined against gravity, buoyancy forces drive hot and cold fluid up and down the incline leading to a shear flow. In this so-called inclined layer convection (ILC) system, the competition of buoyancy and shear gives rise to a large variety of complex spatio-temporal flow patterns.

We study the dynamics of ILC using a fully nonlinear dynamical systems approach based on a state space analysis of the governing equations. Exploiting the computational power of highly parallelized numerical continuation tools based on matrix-free Newton methods (www.channelflow.ch), we compute a large set of invariant solutions of ILC and discuss their bifurcation structure. Specifically, equilibria, travelling waves, periodic orbits and heteroclinic orbits will be shown to form dynamical networks that support moderately complex dynamics at intermediate angles of inclination. At high inclination angles, where shear forces dominate, localized patches of weakly turbulent convection within a background of straight longitudinal convection rolls are observed. We present exact invariant solutions capturing both the dynamics and the spatial localization of these so-called transverse bursts.