Temperature response to periodic modulation in internal heating convection

Thermal convection usually encounters time-varying internal heating(IH), however, the effect of temporal modulation on the IH system has been rarely explored. Here, we numerically studied the IH convection with a temporally-periodic heating source. We conducted DNSs over Rayleigh number range 105{less than or equal to} Ra{less than or equal to}109 at fixed Prandtl number Pr=1 with modulation frequencies 4x10-5{less than or equal to} f{less than or equal to}10-1 and fixed amplitude Ω=1. We find that the introduction of periodic modulation has a slight effect on the heat transport over the individual plates and flow strength except for the lowest Ra. We then focus on the characteristics of the amplitude A and phase lag ΔΦ of the globally-averaged temperature response to the periodic modulation. Three regimes of the phase response are identified: (i) in-phase regime, where synchronous response is found at small frequencies with the vanished phase lag ΔΦ and A keeping at constant value; (ii) transition regime, where both ΔΦ and A decrease with increasing f for moderate frequencies; (iii) anti-phase regime, where ΔΦ attains the minimal value -π/2. We also find that the transition behavior between three regimes can be well described using the normalization of the Ra-dependent critical frequency with the scaling Ra-0.30. To explain the regime transition, we further theoretically deduce an analytical solution for A and ΔΦ, which agrees well with the numerical results. This solution can explain why fRa0.30 gives a good description of the transition behavior. Our findings reveal the underlying mechanism of temporal modulation on IH systems and have substantial implications for the investigation of convective systems with periodic heating.