Periodically Modulated Thermal Convection

Many natural and industrial turbulent flows are subjected to time-dependent boundary conditions. Despite being ubiquitous, the influence of temporal modulations (with frequency $f$) on global transport properties has hardly been studied. Here, we perform numerical simulations of Rayleigh-Bénard convection with time periodic modulation in the temperature boundary condition and report how this modulation can lead to a significant heat flux (Nusselt number Nu) enhancement. Using the concept of Stokes thermal boundary layer, we can explain the onset frequency of the Nu enhancement and the optimal frequency at which Nu is maximal, and how they depend on the Rayleigh number Ra and Prandtl number Pr. From this, we construct a phase diagram in the 3D parameter space ($f$, Ra, Pr) and identify the following: (i) a regime where the modulation is too fast to affect Nu; (ii) a moderate modulation regime, where Nu increases with decreasing $f$, and (iii) slow modulation regime, where Nu decreases with further decreasing $f$. Our findings provide a framework to study other types of turbulent flows with time-dependent forcing.

Figure 1

(a) Modulated frequency dependence of the Nusselt number $\mathrm{Nu}(f)$, normalized by ${\mathrm{Nu}}_0=\mathrm{Nu}(f=0)$, for different Rayleigh numbers and fixed $\mathrm{Pr}=4.3$. (b) $\mathrm{Nu}(f)/{\mathrm{Nu}}_0$ for different Prandtl numbers and fixed $\mathrm{Ra}={10}^8$. (c) Global $\mathrm{Re}(f)$ normalized by the ${\mathrm{Re}}_0=\mathrm{Re}(f=0)$ for different Rayleigh numbers and fixed $\mathrm{Pr}=4.3$.

Figure 2

(a) Instantaneous temperature fields at different phases in one modulation period for $\mathrm{Ra}={10}^8$, $\mathrm{Pr}=4.3$, $f={10}^{-3}$. (b)–(d) Phase-averaged temperature profiles during one period for $\mathrm{Ra}={10}^8$, $\mathrm{Pr}=4.3$ and different modulation frequencies, namely (b) without modulation; (c) $f={10}^{-1}$; (d) $f={10}^{-4}$. The horizontal axis is the temperature and the vertical axis is the height. The colorbar shows the bottom temperature (phase angle) from $0(-\pi /2)$ to $2(\pi /2)$. (e) Sketch of the relations between the three BLs [Stokes thermal BL (${\lambda }_S$), thermal BL (${\lambda }_{\theta }$), momentum BL (${\lambda }_u$)] for the three regimes ($\mathrm{Pr}=4.3$): (i) ${\lambda }_u>{\lambda }_{\theta }>{\lambda }_S$; (ii) ${\lambda }_u>{\lambda }_S>{\lambda }_{\theta }$; (iii) ${\lambda }_S>{\lambda }_u>{\lambda }_{\theta }$. Arrows represent the flow in the bulk. (b) Sketch of two different phases during one period for regime ii: (a) heating phase when ${\theta }_{\text{bot}}>1$ and (b) cooling phase when ${\theta }_{\text{bot}}<1$. (f) Phase-averaged center temperature for $\mathrm{Ra}={10}^8$, $\mathrm{Pr}=4.3$. The red (blue) curve represents the phase when the bottom temperature is maximal (minimal). The dashed lines (from right to left) correspond to $f_{\text{onset}}$ and $f_{\mathrm{opt}}$ for Nu.

Figure 3

(a) Normalized Nu as a function of $fR{a}^{-1/6}$, for different Ra and $\mathrm{Pr}=4.3$, (b) $f{\mathrm{Pr}}^{1/2}$. (c) $f{\mathrm{Pr}}^{3/2}$ for different Pr and $\mathrm{Ra}={10}^8$. Dashed lines show the onset frequency [where $\mathrm{Nu}(f)$ starts to be affected, $\mathrm{Nu}(f)/{\mathrm{Nu}}_0=1.01$] or optimal frequency [where $\mathrm{Nu}(f)$ reaches the maximum], averaged for different Ra or Pr. Phase diagram (a) in the $f$ vs Ra and (b) in the $f$ vs Pr parameter spaces. In (a), the lower dashed line shows the optimal frequency $f_{opt}=0.65{\mathrm{Ra}}^{-0.22}$ that corresponds to the maximal Nu. The upper dashed line shows the onset frequency $f_{\text{onset}}=0.015{\mathrm{Ra}}^{0.14}$ that corresponds to the onset of the heat flux enhancement. In (b), the lower dashed line shows the optimal frequency $f_{\mathrm{opt}}=0.06{\mathrm{Pr}}^{-1.35}$, while the upper one shows the onset frequency $f_{\text{onset}}=0.45{\mathrm{Pr}}^{-0.65}$. The prefactors originate from fits to the DNS data for $f_{opt}$ and $f_{\text{onset}}$ [set to occur when $\mathrm{Nu}(f)/{\mathrm{Nu}}_0=1.01$]; see Supplemental Material [42] for details on the fitting.