Some scientists have performed direct numerical simulations (DNS) of homogeneous Rayleigh-Bénard (RB) convection to seek an asymptotic high-Rayleigh number heat transport scaling law Nu ~ Raγ. By applying periodic boundary conditions, the goal has been to focus on the ‘ultimate’ regime where boundary layers are negligible and convective heat transport is limited only by the turbulence in the bulk. The value γ = 1/ 2 obtained in the DNS is consistent with the analytical conjecture established in the past several decades. However, it should be pointed out that the tri-periodic model possesses exact exponentially growing solutions which transport unlimited heat. Such runaway solutions and their possible secondary instabilities are manifest above a critical Ra in the DNS. Thus, the relevance of computations on tri-periodic domains as models for the RB ultimate state is arguable. In this thesis, to understand the secondary instability mechanism, four systems have been constructed by (1) multiple scalings based on the aspect-ratio of the growing modes; (2) a modal truncation of the dynamical equations for the exact exploding solutions; (3) random noisy horizontal velocity fields; (4) a combination of the modal truncation and multiple scalings (two time-scales). Numerical studies of (1), (2) and (3) have revealed that, respectively, the growing modes are unbounded; boundedness or unboundedness is uncertain; unbounded if the noise strength is small. (4) suggests that when the nonlinear terms are dominant in the governing equations, there exist several constants of the motion for the ODEs obtained by modal truncations. This is possibly one explanation of the bursting cycles of the exponentially growing solutions observed in the DNS. The latest DNS results of homogeneous RB convection (obtained from our co-workers) are also reported, which conclude that the secondary instabilities observed in the earlier homogeneous RB DNS might be caused by numerical errors.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:484735 |
| Date | January 2007 |
| Creators | Tanabe, Aya |
| Contributors | Gibbon, J. D. |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/1270 |
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