Numerical Study of Turbulent Rayleigh-Benard Convection with Cubic confinement
Turbulent Rayleigh-Bénard convection (RBC) occurs when a shallow layer of fluid is heated from below. It is a challenging subject in non-linear physics, with many important applications in natural and engineering systems. Because of the complexity of the governing equations, analytical progress in understanding convection has been slow, and laboratory experiments and numerical simulations have assumed increased importance. In regard to numerical work, Large-Eddy Simulation (LES) techniques have proved to be reliable and powerful tool to understand the physics since it provides better coverage for measurements, that are not as easily obtained in physical experiments or the other numerical approaches. This thesis addresses different aspects of Rayleigh-Bénard convection in fully developed turbulent regime through Large Eddy Simulation (LES) to shed light on some important aspect of the geometrical shape of the convection cell. The layout of the thesis is as follows: In Chapter 1, we first introduce Rayleigh-Bénard convection and the equations and parameters that govern it. This is followed by a discussion on different types of boundary conditions used in numerical and theoretical studies of RBC. Subsequently we present various convection states that are observed analytically and experimentally in RBC as a function of Ra and Ʈ. To this end we present a brief survey of the analytical, experimental and numerical works on confined thermal convection. We introduce different regimes and related scaling according to Grossman and Lohse theory. We also present the experimental and numerical results related to the Large Scale Circulation (LSC) within different geometries. In Chapter 2, we present the details of the numerical methods used to solve the governing non-linear equations . In the second part, we provide the details of the solver and the algorithm used to solve the RBC dynamical equations in a Cartesian geometry together with boundary conditions. In Chapter 3, we demonstrate that our numerical method and solver give results consistent with earlier numerical results. Results from the Direct Numerical Simulations (DNS) and Large Eddy Simulation (LES) with constant and dynamic subgrid scale Prandtl number (P_sgs) are presented and compared. We observe close agreement with Lagrangian dynamic approaches. In the first part of Chapter 4 we analyse the local fluctuations of turbulent Rayleigh-Bénard convection in a cubic confinement with aspect ratio one for Prandtl number Pr = 0.7 and Rayleigh numbers (Ra) up to 10^9 by means of LES methodology on coarse grids. Our results reveal that the scaling of the root-mean-square density and velocity fluctuations measured in the cell center are in excellent agreement with the unexpected scaling measured in the laboratory experiments of Daya and Ecke (2001) in their square cross-section cell. Moreover we find that the time-averaged spatial distributions of density fluctuations show a fixed inhomogeneity that maintains its own structure when the flow switches from one diagonal to the other. The largest level of rms density fluctuations corresponds to the diagonal opposite that of the Large Scale Circulation (LSC) where we observed strong counter-rotating vortex structures. In the second part we extended our simulations and Ra up to 1011, in order to identify the time periods in which the orientation of LSC is constant. Surprisingly we find that the LSC switches stochastically from one diagonal to the other. In Chapter 5, we study the effect of 3D-roughness on scaling of Nu(Ra) and consequently on the fluctuations of density. Moreover we present the effect of roughness shape when the tip has a wide angle and the other one is smooth. We study two types of elements, one of which is a pyramid and the other is a sinusoidal function spread over the bottom (heated) and top (cooled) plates, in a cubic confinement. However preliminary results suggest that the effect of roughness appears evident at high Ra numbers when the thermal boundary layer is thin enough to shape around the obstacles.