Numerical investigation on Benard instability in a finite liquid layer

A numerical procedure for directly simulating the Benard-Marangoni instabilities (B-M-I) in a bounded liquid layer is presented in this paper. The procedure consists of applying a finite amplitude disturbance to the basic static state, and then integrating the Navier-Stokes equations to determine whether the disturbance will die down or will reach a state of finite-strength steady convection. The critical Marangoni number (Mac) for the onset of B-M-I can thus be determined and can be correlated as a function of the aspect ratio (Ar), Prandtl number (Pr), and Rayleigh number (Ra). The Biot number (B) between the liquid and the air is analyzed to approximate the heat transfer condition along the free surface. A 2D calculation is performed to investigate the effect of various initial disturbances, and the Mac is determined for Ar = 2, Ra = 0, and Pr = 0.7. Current results show that disturbances of different nature and amplitude have little effect on Mac. The Mac determined in this study also clearly demonstrates the dominant effect of the sidewalls.