Abstract:

Numerous studies have addressed the stability of fluid between two thermally conducting plates (abbreviate frequently as Rayleigh-Bénard convection), the Lorenz system serves as the classical model of Rayleigh-Bénard convection problems and provides a paradigm from the laminar to turbulent transition. In this paper we study the dynamical mechanism and energy conversion of the Lorenz equation, the Lorenz chaotic system is transformed into a Kolmogorov-type system, which is decomposed into four types of torques: inertial torque, internal torque, dissipation and external torque. By combining different torques, the key factors for the generation of chaos in the Lorenz system-the mathematical model corresponding to the Rayleigh-Bénard convection problem have been studied. We further investigate the conversion among Hamiltonian, kinetic and potential energies, as well as the correlation between the energies and the Reynolds number. It is concluded that the combination of the four torques is necessary to produce chaos, and the system can produce chaos only when the dissipative torques match the driving (external) torques. While any combination of three types of torques cannot produce chaos. The external torque, driven by heat from the bottom plate, supplies energy, and that leads to produce roll vortex and chaos. Moreover, we introduce the Casimir function to analyze the system dynamics, and choose its derivation formulate the energy conversion. The bound of chaotic attractor is obtained by the Casimir function and Lagrange multiplier. It is found that the Casimir function reflects the energy conversion and the distance between the orbit and the equilibria.

Key words: Rayleigh-Bénard convection, dynamical mechanism, Kolmogorov system, chaos