Stochastic natural convection in square enclosures with horizontal isothermal walls

The effectiveness of two well established stochastic approaches, i.e. the generalized polynomial chaos method and the multi-element generalized polynomial chaos method, is investigated to simulate the onset of convection in bidimensional square enclosures with horizontal isothermal walls. The Boussinesq's approximation for the variation of physical properties is assumed. The stability analysis is first carried out in a deterministic sense, to determine steady state solutions and the primary bifurcation which identifies the transition from conduction to convection regime. Stochastic simulations are then conducted around discontinuities and transitional regimes. Randomness is introduced into the system through the Rayleigh number which is assumed to be a random variable following a uniform distribution. By a comparison with an accurate Monte-Carlo simulation it is shown that the statistics for the velocity and the temperature fields can be efficiently captured by the multi-element generalized polynomial chaos method.