Large-eddy simulations of turbulent thermal convection using renormalized viscosity and thermal diffusivity

In this paper we employ renormalized viscosity and thermal diffusivity to construct a subgrid-scale model for large eddy simulation (LES) of turbulent thermal convection. For LES, we add ${\nu }_{\mathrm{ren}}\propto {\mathrm{\Pi }}_u^{1/3}{(\pi /\mathrm{\Delta })}^{-4/3}$ to the kinematic viscosity; here ${\mathrm{\Pi }}_u$ is the turbulent kinetic energy flux, and $\mathrm{\Delta }$ is the grid spacing. We take subgrid thermal diffusivity to be same as the subgrid kinematic viscosity. We performed LES of turbulent thermal convection on a ${128}^3$ grid and compare the results with those obtained from direct numerical simulation (DNS) on a ${512}^3$ grid. We started the DNS with random initial condition and forked a LES simulation using the large wave number modes of DNS initial condition. Though the Nusselt number is overestimated in LES as compared to that in DNS, there is a good agreement between the LES and DNS results on the evolution of kinetic energy and entropy, spectra and fluxes of velocity and temperature fields, and the isosurfaces of temperature.

Figure 1

Plots for LES on a ${128}^3$ grid (thin red lines) and DNS on a ${512}^3$ grid (thick black lines) with $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=1$: temporal evolution of (a) total kinetic energy $E_u\left(t\right)$, (b) total entropy $E_{\theta }\left(t\right)$.

Figure 2

Plots for LES on a ${128}^3$ grid (thin red lines) and DNS on a ${512}^3$ grid (thick black lines) with $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=1$: temporal evolution of (a) total viscosity ${\nu }_{\mathrm{tot}}$, (b) Nusselt number $\mathrm{Nu}$. Note that ${\nu }_{\mathrm{tot}}=\nu$ for DNS.

Figure 3

Comparison of Nu-Ra scaling of our DNS and LES with the DNS by Pandey and Verma [49] for free-slip boundary condition, the DNS by Bhattacharya et al. [56] for no-slip boundary condition, the transient Reynolds-averaged Navier Stokes (TRANS) runs by Kenjereš and Hanjalic [27], and the experiments by Niemela et al. [57]. We plot $\mathrm{Nu}{\mathrm{Ra}}^{-0.3}$ vs $\mathrm{Ra}$. The dashed line denotes $\mathrm{Nu}{\mathrm{Ra}}^{-0.3}=0.25$. The exponents and coefficients of these runs are listed in Table 2.

Figure 4

Plots for LES (thin red lines) and DNS (thick black lines) of RBC with $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=1$ at $t=45$ free-fall time: (a) normalized kinetic energy spectrum $E_u^{\prime }\left(k\right)=E_u\left(k\right)k^{5/3}{\mathrm{\Pi }}^{-2/3}$, (b) kinetic energy flux ${\mathrm{\Pi }}_u\left(k\right)$. We observe these quantities to be approximate constants (see flat line) in the inertial range $k=[10,70]$.

Figure 5

Plots for LES and DNS of RBC with $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=1$ at $t=45$ free-fall time: (a) entropy spectra $E_{\theta }\left(k\right)$ exhibit a bispectrum where the upper branch scales as $k^{-2}$ (thick blue line), while the lower branch is fluctuating with neither $k^{-5/3}$ (green line with medium thickness) nor $k^{-3/2}$ spectrum (thin pink line). Red triangles represent LES, and black circles represent DNS. (b) Entropy flux ${\mathrm{\Pi }}_{\theta }\left(k\right)$ is constant in the inertial range with the thick black line representing the DNS, and the thin red line representing the LES.

Figure 6

Plots for LES (thin red lines) and DNS (thick black lines) of RBC with $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=1$: (a) horizontally averaged temperature $T_m\left(z\right)$; (b) planar averaged rms values of the temperature, $T_{\sigma }\left(z\right)$; (c) planar averaged vertical velocity, $w_m\left(z\right)$, which is 0 due to continuity and wall constraint; (d) planar averaged rms values of the vertical velocity, $w_{\sigma }\left(z\right)$.

Figure 7

Plots for LES and DNS of RBC with $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=1$: The temperature isosurfaces at $t=45$ (top row) and at $t=80$ (bottom row). The left configurations represent the LES profiles, while those on the right are from DNS.

Figure 8

Plots for LES (thin red lines) and DNS (thick black lines) of RBC with $\mathrm{Ra}={10}^7$ and $\mathrm{Pr}=0.5$ at $t=45$ free-fall time: (a) normalized kinetic energy spectrum; (b) kinetic energy flux ${\mathrm{\Pi }}_u\left(k\right)$.

Figure 9

Plots for LES and DNS of RBC with $\mathrm{Ra}={10}^7$ and $\mathrm{Pr}=0.5$ at $t=45$ free-fall time: (a) entropy spectrum $E_{\theta }\left(k\right)$ exhibiting bispectra with red triangles for LES and black circles for DNS. The blue, green, and pink lines with decreasing thickness represent $k^{-2}, k^{-5/3}$, and $k^{-7/5}$ curves; (b) entropy flux ${\mathrm{\Pi }}_{\theta }\left(k\right)$ with the thick black line representing the DNS, and the thin red line representing the LES.

Figure 10

Plots for LES (thin red lines) and DNS (thick black lines) of RBC with $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=5.0$ at $t=45$ free-fall time: (a) normalized kinetic energy spectrum $E_u^{\prime }\left(k\right)=E_u\left(k\right)k^{5/3}{\mathrm{\Pi }}^{-2/3}$; (b) kinetic energy flux ${\mathrm{\Pi }}_u\left(k\right)$.

Figure 11

Plots for LES and DNS of RBC with $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=5.0$ at $t=45$ free-fall time: (a) entropy spectrum $E_{\theta }\left(k\right)$ exhibiting bispectra with red triangles for LES and black circles for DNS. The blue, green, and pink lines with decreasing thickness represent $k^{-2}, k^{-5/3}$, and $k^{-7/5}$ curves; (b) entropy flux ${\mathrm{\Pi }}_{\theta }\left(k\right)$ with the thick black line representing the DNS, and the thin red line representing the LES.