Frequency spectra of turbulent thermal convection with uniform rotation

The frequency spectra of the entropy and kinetic energy along with the power spectrum of the thermal flux are computed from direct numerical simulations for turbulent Rayleigh-Bénard convection with uniform rotation about a vertical axis in low-Prandtl-number fluids $\left(\mathrm{Pr}<0.6\right)$. Simulations are done for convective Rossby numbers $\mathrm{Ro}\ge 0.2$. The temporal fluctuations of these global quantities show two scaling regimes: (i) ${\omega }^{-2}$ at higher frequencies for all values of $\text{Ro}$ and (ii) ${\omega }^{-{\gamma }_1}$ at intermediate frequencies with ${\gamma }_1\approx 4$ for $\mathrm{Ro}>1$, while $4<{\gamma }_1<6.6$ for $0.2\le \mathrm{Ro}<1$.

Figure 1

Frequency spectra $E_{\Phi }\left(\omega \right)$ of the spatially averaged convective entropy computed from DNS for different values of $\mathrm{Ro}$ (curves of different colors). The spectra show two scaling regimes: $E_{\Phi }\left(\omega \right)$ scales as $\sim {\omega }^{-{\gamma }_1^{\Phi }}$ at the moderate frequencies, with the exponent ${\gamma }_1^{\Phi }$ showing a weak dependence on $\mathrm{Ro}$ for $\mathrm{Ro}>1 \left(3.37\le {\gamma }_1\le 4.30\right)$ and a strong dependence on $\mathrm{Ro}$ for $\mathrm{Ro}<1 \left(3.93\le {\gamma }_1\le 6.59\right)$, and $E_{\Phi }\left(\omega \right)$ scales with $\omega$ as $\sim {\omega }^{-{\gamma }_2^{\Phi }}$, with ${\gamma }_2^{\Phi }\sim 2$ at higher frequencies for all values of $\mathrm{Ta},\mathrm{Pr}$, and $\mathrm{Ro}$ considered here. The upper and lower black straight lines show the scaling behavior as ${\omega }^{-4}$ and ${\omega }^{-2}$, respectively. The inset shows the variation of the exponent ${\gamma }_1^{\Phi }$ with $\mathrm{Ro}$.

Figure 2

Frequency spectra $E\left(\omega \right)$ of the spatially averaged kinetic energy, as computed from DNS, for the several values of the Rossby number $0.2\le \mathrm{Ro}<\infty$: $E\left(\omega \right)$ scales with $\omega$ as ${\omega }^{-{\gamma }_1}$ at moderate frequencies with $3.06\le {\gamma }_1\le 6.54$ and $E\left(\omega \right)$ scales with $\omega$ as ${\omega }^{-{\gamma }_2}$ with ${\gamma }_2\sim 2$. The exponent ${\gamma }_2$ is found to be independent of $\mathrm{Ta},\mathrm{Pr}$, and $\mathrm{Ro}$. The upper and lower black lines stand for the scaling behaviors ${\omega }^{-4}$ and ${\omega }^{-2}$, respectively. The inset displays the variation of ${\gamma }_1$ with $\mathrm{Ro}$.

Figure 3

Power spectra of temporal fluctuations of the heat flux (Nusselt number $\mathrm{Nu}$), as computed from DNS, for different values of $\mathrm{Ro}$. Two scaling regimes are also shown, which scale as ${\omega }^{-{\gamma }_1^{\mathrm{Nu}}}$ with $3.57\le {\gamma }_1^{\mathrm{Nu}}\le 6.35$ at intermediate frequencies and ${\omega }^{-{\gamma }_2^{\mathrm{Nu}}}$ with ${\gamma }_2^{\mathrm{Nu}}\sim 2$ at higher frequencies. The inset shows the variation of ${\gamma }_1^{\mathrm{Nu}}$ with $\mathrm{Ro}$.

Figure 4

(a) Mean energy $\tilde{K}$ and (b) Nusselt number $\text{Nu}$ scale with Taylor number $\mathrm{Ta}$ as ${\mathrm{Ta}}^{-0.19}$ and ${\mathrm{Ta}}^{0.13}$, respectively.