Particle simulation of three-dimensional convection patterns in a Rayleigh-Bénard system

The transition of convection patterns in the three-dimensional Rayleigh-B\'enard system is simulated using the direct simulation Monte Carlo method. The simulation region is a rectangular box with an aspect ratio of $8:8:1$, and the number of simulation particles is 2.048×10⁷. A hexagonal flow pattern is observed in the convection state with the Rayleigh number slightly higher than the critical value. It is found that the hexagonal flow pattern evolves into a roll pattern with an increase in the Rayleigh number. A hysteresis is seen in the transition between the hexagonal and roll patterns.