Temperature response to periodic modulation in internal heating convection

Abstract

Thermal convection in nature and industrial applications usually encounters time-varying internal heating (IH); however, the effect of temporal modulation on temperature responses and heat fluxes of the system has been rarely explored. Here, we numerically studied the IH convection with a temporally periodic heating source. We conducted direct numerical simulations over Rayleigh number ( Ra) range [Formula: see text] at fixed Prandtl number Pr =  1 with dimensionless modulation frequency [Formula: see text] and amplitude fixed at Ω = 1. We first 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 [Formula: see text] 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 [Formula: see text] and A keeping at constant value; (ii) transition regime, where both [Formula: see text] and A decrease with increasing f for moderate frequencies; and (iii) half anti-phase regime, where [Formula: see text] attains the minimal value [Formula: see text]. 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 [Formula: see text]. To explain the regime transition, we further theoretically deduce an analytical solution for A and [Formula: see text], which agrees well with the numerical results. This solution can explain why [Formula: see text] gives a good description of the transition behavior. Our present findings provide a fundamental understanding of the underlying mechanism of temporal modulation on IH systems and have substantial implications for the investigation of convective system with periodic heating source.