Prandtl number dependence of the small-scale properties in turbulent Rayleigh-Bénard convection

We analyze the Prandtl number (Pr) dependence of spectra and fluxes of kinetic energy, as well as the energy injection rates and dissipation rates, of turbulent thermal convection using numerical data. As expected, for a flow with $\mathrm{Pr}⪅1$, the inertial-range kinetic-energy flux is constant, and the kinetic-energy spectrum is Kolmogorov-like ($k^{-5/3}$). More importantly, we show that the amplitudes of the kinetic-energy fluxes and spectra and those of structure functions increase with the decrease of Pr, thus exhibiting stronger nonlinearity for flows with small Prandtl numbers. Consistent with these observations, both the kinetic-energy injection rates and the dissipation rates increase with the decrease of Pr. Our results are in agreement with earlier studies that report the Reynolds number to be a decreasing function of Prandtl number in turbulent convection. On the other hand, the tail of the probability distributions of the local heat flux grows with the increase of Pr, indicating increased fluctuations in the local heat flux with Pr.

Figure 1

(a) For $\mathrm{Ra}={10}^7$ and $\mathrm{Pr}=0.02$ (green), 0.1 (red), 1 (blue), 6.8 (brown), and 100 (purple): plots of the kinetic-energy spectra vs the wave number $k$. The spectra exhibits fluctuations at small and intermediate wave numbers.

Figure 2

For $\mathrm{Ra}={10}^7$ and $\mathrm{Pr}=0.02$, 0.1, 1, 6.8, and 100: (a) Integral kinetic-energy spectrum, ${\sum }_k^{\infty }{E}_u\left(k^{\prime }\right)$ vs wave number $k$, (b) kinetic-energy flux ${\mathrm{\Pi }}_u\left(k\right)$ vs $k$, (c) energy injection rate due to buoyancy, ${\mathcal{F}}_B\left(k\right)$, vs $k$, and (d) $d_k{\mathrm{\Pi }}_u\left(k\right)/{\mathrm{\Pi }}_u\left(k\right)$ vs $k$. The amplitudes of the energy spectrum and flux decrease with Pr. For $\mathrm{Pr}\le 1$, the energy spectrum exhibits Kolmogorov's scaling.

Figure 3

For $\mathrm{Ra}={10}^7$: Plots of ${\epsilon }_u$ and maximum kinetic-energy flux ${\mathrm{\Pi }}_{u,\max }$ vs $k$. The dissipation rates decrease with the increase of Pr, similar to the energy spectrum and flux. The difference between the kinetic-energy flux and the dissipation rate increases as Pr is increased.

Figure 4

For $\mathrm{Pr}=0.02$, 0.1, 1, 6.8, and 100, (a) plots of ${\epsilon }_u$ vs Ra, and (b) plots of ${\epsilon }_u/(U^3/d)$ vs Ra (data taken from Bhattacharya et al. [82]). For small Pr, ${\epsilon }_u\sim U^3/d$ as in hydrodynamic turbulence. However, ${\epsilon }_u$ has an additional Ra dependence for larger Prandtl numbers.

Figure 5

For $\mathrm{Pr}=0.02$, 0.1, 1, 6.8, and 100: longitudinal velocity structure functions of orders (a) 2, (b) 3, (c) 5, and (d) 7 vs $l$. The amplitudes of the structure functions decrease with the increase of Pr.

Figure 6

For $\mathrm{Pr}=0.02$, 0.1, and 1: The scaling exponents ${\zeta }_q$ for the velocity structure functions vs order $q$. The exponents closely match the predictions of [40] (SL94). The figure also exhibits the extrapolated ${\zeta }_q$'s for K41 ($q/3$) and Bolgiano-Obukhov (BO59) ($3q/5$).

Figure 7

For $\mathrm{Ra}={10}^7$: The probability distribution functions (PDFs) of normalized local convective heat flux in the (a) $x$ direction, (b) $y$ direction, and (c) $z$ direction for different Pr. The fluctuations of the local heat flux increase with Pr.

Figure 8

For $\mathrm{Ra}={10}^7$: The probability distribution functions (PDFs) of the local convective heat flux normalized with their respective standard deviations (${\sigma }_x$, ${\sigma }_y$, ${\sigma }_z$) in the (a) $x$ direction, (b) $y$ direction, and (c) $z$ direction. The normalized PDFs for different Prandtl numbers collapse into one curve.

Figure 9

For $\mathrm{Ra}={10}^7$ and $\mathrm{Pr}=1$, 6.8, and 100: (a) Entropy spectrum $E_{\theta }$ (with dual branches) and (b) entropy flux ${\mathrm{\Pi }}_{\theta }$ vs $k$. The amplitudes of the entropy spectrum do not vary with Pr, but the amplitudes of the entropy flux decrease with increase of Pr.

Figure 10

For $\mathrm{Ra}={10}^7$ and $\mathrm{Pr}=0.02$ and 0.1: Semilogarithmic plots of (a) normalized entropy spectrum $k{E}_{\theta }$ and (b) entropy flux ${\mathrm{\Pi }}_{\theta }$ vs $k$. The lower branch of the entropy spectrum and the entropy flux fit well with the exponential function.