Thermal Convection: Differential heating, travelling thermal waves and turbulent superstructures

Abstract

Turbulent thermal convection is ubiquitous in geophysical, astrophysical, and industrial applications. In the present work, various phenomena in the convective paradigm systems of horizontal convection and Rayleigh-Bénard convection are studied, with emphasis on the influence of thermal boundary conditions, scaling laws of heat transport and characterization of large-scale flow structures. The problems are studied theoretically and by means of direct numerical methods. In chapter 3, the main computational methods are presented with focus on the implementation details of computational codes and numerical solvers for linear stability analysis. A finite-volume code for direct numerical simulations has been improved towards massively parallel simulations and a parallel pseudospectral code has been developed and tested. In chapter 3, horizontal convection is investigated for various Rayleigh numbers ${\mathop{\rm Ra}}$ and Prandtl numbers $\Pr$. Several Nusselt number ${\mathop{\rm Nu}}$ vs. ${\mathop{\rm Ra}}$ scaling transitions are identified, validating the theoretical scaling model (Shishkina et al., 2016). Examining the global flow structures, we show that the onset of time dependence in horizontal convection is triggered either by oscillations (smaller $\Pr$) or by detaching plumes (larger $\Pr$). We analyze their dependence on ${\mathop{\rm Ra}}$ and $\Pr$, which is consistent with dimensional-analysis based estimates of their onset. In chapter 4, the formation of mean zonal flows in convective systems with traveling thermal waves is explored. Excellent agreement of the dependence of zonal flow strength on thermal wave propagation speed between the theoretical model and the fully nonlinear simulations is found for small ${\mathop{\rm Ra}}$, while for larger ${\mathop{\rm Ra}}$ it is overestimated by the model due to nonlinear effects. An important result of this chapter is the reversal of zonal flows from purely retrograde for small ${\mathop{\rm Ra}}$ to predominantly prograde for large ${\mathop{\rm Ra}}$. Stability analysis of convection rolls indicates that the tilted cell instability may play a key role in the formation of zonal flows in convection-dominated flows, even in the presence of traveling thermal waves. In chapter 5 we study different thermal sidewall boundary conditions in Rayleigh-Bénard convection from the onset of convection to the turbulent regime, with the main goal of mimicking imperfectly adiabatic sidewalls in experiments. Linear sidewall temperatures lead to a premature collapse of the single roll state, while constant sidewall temperatures lead to enhanced single roll stability. Enlargement of corner rolls is identified as the main collapse mechanism, and two distinct corner roll growth rate regimes are obtained. In the intermediate ${\mathop{\rm Ra}}$ range, vertically stacked double rolls (or double toroidal structures in cylindrical systems) are shown to predominate for linear and adiabatic sidewalls. The different flow structures leave their imprint on the global heat transport, however, at larger ${\mathop{\rm Ra}}$ heat transport and flow dynamics become increasingly alike for different sidewall boundary conditions, indicating a low sensitivity of very large ${\mathop{\rm Ra}}$ experiments with respect to spurious sidewall heat fluxes. In chapter 6 we address the strong spatial inhomogeneity of Rayleigh-Bénard convection arising from the presence of turbulent superstructures by decomposing the flow into large-scale plume ejecting and impacting zones. Using a conditional averaging algorithm based on pattern matching, we show the existence of a crossover in the wall heat transport from impacting-dominated to ejecting-dominated at ${\mathop{\rm Ra}}\approx 3\times10^{11}$ in a two-dimensional laterally periodic configuration with $\Pr=1$ and $\Gamma=2$. The heat transport increase in the ejecting region arises from the development of a turbulent mixing zone due to the emission of thermal plumes. This mixing zone reaches its peak heat transport efficiency at about five thermal boundary layer thicknesses and expands vertically and laterally with increasing ${\mathop{\rm Ra}}$, becoming more dominant for the total heat transfer.