The effect of boundary conditions on the onset of chaos in RayleighBard convection using energy-conserving Lorenz models
- Title
- The effect of boundary conditions on the onset of chaos in RayleighBard convection using energy-conserving Lorenz models
- Creator
- Kanchana C.; Siddheshwar P.G.; Yi Z.
- Description
- The influence of 16 boundary conditions on linear and nonlinear stability analyses of RayleighBard system is reported. A StuartLandau amplitude equation for the RayleighBard system between stress-free, isothermal boundary conditions is derived and the procedure used in this derivation serves as guidance for constructing an appropriate FourierGalerkin expansion for the other 15 boundary conditions to derive a generalized Lorenz model. The influence of the boundary conditions comes within the coefficients of the generalized Lorenz model. It is shown that the obtained generalized Lorenz model is energy conserving and for certain boundary conditions it retains features of the classical Lorenz model. Further, the principle of exchange of stabilities is shown to be valid for the present problem and hence it is the steady-state, linearized version of the generalized Lorenz model which yields an analytical expression for the Rayleigh number. On minimizing this expression with respect to wave number the critical Rayleigh number at which the onset of regular convective motion occurs in the form of rolls is determined for all 16 boundary conditions. It is found that these values are in good agreement with those of previous investigations leading to the conclusion that the chosen minimal FourierGalerkin expansion is a valid one. Exhibition of chaotic motion in the generalized Lorenz system at the Hopf Rayleigh number is studied. The phase-space plots which indicate a clear-cut visualization of the transition from regular convective motion to chaotic motion in the generalized Lorenz system are presented. Further, existence of a developing region for chaos (mildly chaotic motion) and windows of periodicity are captured using the bifurcation diagrams. It is concluded from the phase-space plots and the bifurcation diagrams that the generalized Lorenz model for certain sets of boundary conditions retains all the features of the classical Lorenz model. Such a conclusion cannot be made by a linear stability analysis and the need thus for a nonlinear analysis is highlighted in the paper. 2020 Elsevier Inc.
- Source
- Applied Mathematical Modelling, Vol-88, pp. 349-366.
- Date
- 2020-01-01
- Publisher
- Elsevier Inc.
- Subject
- Chaos; FourierGalerkin; Hopf bifurcation; Lorenz model; RayleighBard convection; StuartLandau equation
- Coverage
- Kanchana C., Harbin Institute of Technology, Shenzhen, Nanshan District, Shenzhen, 518055, China; Siddheshwar P.G., CHRIST(Deemed to be University), Bengaluru, 560056, India; Yi Z., Harbin Institute of Technology, Shenzhen, Nanshan District, Shenzhen, 518055, China
- Rights
- Restricted Access
- Relation
- ISSN: 0307904X; CODEN: AMMOD
- Format
- Online
- Language
- English
- Type
- Article
Collection
Citation
Kanchana C.; Siddheshwar P.G.; Yi Z., “The effect of boundary conditions on the onset of chaos in RayleighBard convection using energy-conserving Lorenz models,” CHRIST (Deemed To Be University) Institutional Repository, accessed February 24, 2025, https://archives.christuniversity.in/items/show/16166.