Rayleigh-B\'enard convection is numerically simulated in two- andthree-dimensions using a recently developed two-component lattice Boltzmannequation (LBE) method. The density field of the second component, which evolvesaccording to the advection-diffusion equation of a passive-scalar, is used tosimulate the temperature field. A body force proportional to the temperature isapplied, and the system satisfies the Boussinesq equation except for a slightcompressibility. A no-slip, isothermal boundary condition is imposed in thevertical direction, and periodic boundary conditions are used in horizontaldirections. The critical Rayleigh number for the onset of the Rayleigh-B\'enardconvection agrees with the theoretical prediction. As the Rayleigh number isincreased higher, the steady two-dimensional convection rolls become unstable.The wavy instability and aperiodic motion observed, as well as the Nusseltnumber as a function of the Rayleigh number, are in good agreement withexperimental observations and theoretical predictions. The LBE model is foundto be efficient, accurate, and numerically stable for the simulation of fluidflows with heat and mass transfer.
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