Gravitational thermal flows of liquid metals in 3D variable cross-section containers : transition from low-dimensional to high-dimensional chaos

This study extends the numerical results presented in past author's work (Lappa and Ferialdi, Phys. Fluids, 29(6), 064106, 2017) about the typical instabilities of thermogravitational convection (the so-called Hadley flow) in containers with inclined (converging or diverging) walls. The flow is now allowed to develop along the third dimension (z). In a region of the space of parameters where the two-dimensional solutions were found to be relatively regular in time and with a simple structure in space (supporting transverse waves propagating either in the downstream or in the upstream direction), the 3D flow exhibits either waves travelling along the spanwise direction or spatially disordered and chaotic patterns. In order to identify the related mechanisms, we analyse the competition between hydrodynamic and hydrothermal (Oscillatory Longitudinal Roll) modes of convection for different conditions. A peculiar strategy of analysis is implemented, which, on the one hand, exploits the typical properties of systems developing coexisting branches of solution ("multiple" states) and their sensitivity to a variation of the basin of attraction and, on the other hand, can force such systems to select a specific category of disturbances (by enabling or disabling the related "physical" mechanisms). It is shown that hydrodynamic modes can produce early transition to chaos. The dimensionality of such states is investigated through evaluation of the "fractal" (correlation) dimension on the basis of the algorithm by Grassberger and Procaccia. When low-dimensional chaos is taken over by high-dimensional chaos, the flow develops a recognisable interval of scales where turbulence obeys the typical laws of the so-called "inertial range" and produces small-scale features in agreement with available Kolmogorov estimates