Numerical investigation of the instability of Benard problem

In this work, the dynamics of instability of a liquid layer heated from below, which is know as Benard problem is investigated. It is a prototype of nonlinear problem where the instability is governed by the two parameters: the Grashof number Gr, and the Prandtl number Pr. To shed some light on the instability of the problem and to understand the route to chaos, a small perturbation was introduced to the flow field using a sinusoidal function with small amplitude. The effect of this perturbation was then studied by changing the amplitude regularly. Finite difference method was employed to solve numerically the associated system of partial differential equations. Results of these calculations were analyzed using the modern theory of dynamical systems. Numerical results indicate that for fixed values of the two parameters: Pr and Gr and for relatively large values of the amplitude the system will become chaotic. Numerical results indicate that the system will become chaotic through intermittency.