Rayleigh–Bénard convection instability in the presence of spatial temperature modulation on both plates

• Direct numerical simulation of secondary instability of Rayleigh-Bénard convection.
• random temperature perturbation of order 10-unsteady vorticity.
• stream function and velocity fields for four instant of time.

This paper is devoted to the analysis of two-dimensional viscous fluid flow between two parallel, horizontal plates, where the lower plate is heated and the upper one is cooled. The temperature difference between the plates increases gradually during the initial time period, and after that period it is temporarily constant. The temperature distribution on the lower and upper plates is not constant in the x-direction, there is longitudinal temperature modulation imposed on the mean temperature. Temperature modulation on the plates leads to periodic convection patterns for very small Rayleigh numbers. We investigate the instability of Rayleigh–Bénard convective cells by direct numerical simulation on the coarse grid, for the two-dimensional Navier–Stokes equation in the stream function–vorticity form and the thermal energy equation.