Lorenz Model For The Rotating Rayleigh-Bénard Problem And Its Numerical Simulation

A four-dimensional Lorenz model for rotating Rayleigh-Benard problem is derived in this paper and the solution of the model is numerically investigated. The impact of the system parameters on the Lorenz attractor is studied by numerical simulation.The Rayleigh-Benard problem is an infinite dimensional dynamical system modelling a fluid layer heated from below, which includes the Navier-Stokes equations and a heat conduction equation. The direct study of the infinite-dimensional nonlinear system is very difficult and many researchers often select truncated forms of the Fourier expansion to transform the system into a system of ordinary differential equations(i.e. Lorenz model). Lorenz took a low-dimensional truncated form to convert the partial differential equations of the Rayleigh- Benard problem to a model of three-dimensional ordinary differential equations. The dynamical property of the ODE model is studied numerically and the Lorenz chaotic-attractor was first advanced. Rotating Rayleigh-Benard problem refers to the case when the fluid layer rotates about a vertical axis. In this case the PDEs contain three contral parameters:Prandtl number(P) reflects the physical properties of the fluid, Rayleigh number (R) is the ratio of viscosity force and buoyancy, Taylor number (T) denotes the rotation. A four-dimensional Lorenz system for the rotating Rayleigh-Benard problem is derived by taking an effective truncated form in the Fourier expansion. On this basis, the range of parameters of the Lorenz equation in the case of Pr>1 is estimated. The occurrence of Lorenz attractor and its dependence on three contral parameters of the system are investigated by numerical simulation. The main conclusions are as follows:(1) The four-dimensional Lorenz model:(2) The control parameters in the four-dimensional Lorenz model and the Rayleigh-Benard system have the following relation: whereτ∈[0,(?)], r∈(0,∞), b∈[0,8/3] and ac is a root of an algebraic equation. In the case of Pr>1 the classical three-order model is just the particular case of our four-order Lorenz model without rotation.(3) Lorenz attractor and its dependence on the control parameters are investigated by means of numerical simulation, the results are compared with those of theoretical analysis of the partial differential equations.The case of large Prandtl number:The rotation has stability effect and in the case without rotation, the instability always sets in as stationary convection.The case of small Prandtl number:It's found that there exists a critical Prandtl number Pr*≈0.6, when Pr>Pr* the instability will always manifest itself first as stationary convection. For a given Pr<Pr*, there exists a critical Taylor number TPr, depending on Pr, such that for T≤TPr, the onset of instability will be as stationary convection, while for T>Tpr, it will be as oscillation of increasing amplitude i.e. overstability.