Rayleigh-Bénard turbulence modified by two-way coupled inertial, nonisothermal particles

Direct numerical simulation (DNS) combined with the Lagrangian point particle model is used to study Rayleigh-Bénard convection in order to understand modifications due to the interaction of inertial, nonisothermal particles with buoyancy-driven turbulence. In this system, turbulence can be altered through direct momentum coupling, as well as through buoyancy modification via thermal coupling between phases. We quantify the effect of the dispersed phase by changes to the total integrated turbulent kinetic energy (TKE) and Nusselt number (Nu). The dispersed particles experience gravitational settling and are introduced at the lower wall so that turbulence must overcome the settling velocity for the particles to vertically distribute throughout the domain. We focus primarily on particle inertia, settling velocity, mass fraction, and the ratio of the particle to fluid specific heat. Furthermore, individual contributions by the momentum coupling and thermal coupling are studied to see which most significantly changes Nu and TKE. Our results show that particles with Stokes number of order unity maximize Nu, corresponding to a peak of clustering and attenuation of TKE. Increased mass fractions lead to a linear increase of Nu and decrease of TKE. With varying specific heat ratio, Nu and TKE exhibit monotonic behaviors, where in the high limit particles become isothermal and depend upon the initialized particle temperature. It is also shown that particles two-way coupled only through momentum attenuate Nu and weaken TKE, while thermal-only coupling also weakens TKE but enhances Nu. When both couplings are present, however, thermal coupling overwhelms the momentum coupling attenuation, and the net result is an enhancement of Nu.

Figure 1

Instantaneous snapshots of normalized temperature contours in a top-down view at a height of $z/H=0.8$ for (a) $\text{St}\approx 0.1$, (b) $\text{St}\approx 1$, and (c) $\text{St}\approx 10$. Black dots represent particles, and the slice is taken at a height of $z/H=0.85$. $V_g/U_{\mathrm{buoy}}={10}^{-3}$ for these cases.

Figure 2

Profiles of normalized number concentration for (a) varying St at a constant $V_g/U_{\mathrm{buoy}}={10}^{-3}$ and (b) varying $V_g/U_{\mathrm{buoy}}$ at a constant $\text{St}\approx 1$.

Figure 3

Normalized, mean vertical profiles of (a) fluid temperature $〈T〉$, (b) particle temperature $〈T_p〉$, (c) difference between average particle temperature and average fluid temperature seen by the particles $〈T_p〉-〈T_f〉$, and (d) difference between fluid temperature and fluid temperature seen by the particles $〈T〉-〈T_f〉$. The dashed horizontal reference line represents the maximum reinjection height of $z/H=0.1$. Shown are three different Stokes numbers at a constant $V_g/U_{\mathrm{buoy}}={10}^{-3}$.

Figure 4

(a) Instantaneous Voronoï diagram for $\text{St}\approx 1$ taken at $z/H=0.5$ for a slice of thickness $2\eta$ (the Kolmogorov length scale) with $V_g/U_{\mathrm{buoy}}={10}^{-3}$. (b) Standard deviation of the normalized Voronoï area ${\sigma }_V$, normalized by that of a random Poisson process, ${\sigma }_{RPP}$, as a function of St.

Figure 5

Normalized (a) Nu and (b) TKE as a function of $V_g/U_{\mathrm{buoy}}$ for $\text{St}\approx 1$ with two-way thermal and mechanical coupling. The dashed horizontal lines are the unladen values and reflect the unladen values.

Figure 6

Normalized (a) Nu and (b) TKE as a function of varying St for $V_g/U_{\mathrm{buoy}}={10}^{-3}$ with two-way coupling. The horizontal lines are the unladen values.

Figure 7

Overall heat flux contributions due to (a) turbulent flux and (b) particle source flux.

Figure 8

Dissipation rate $\epsilon$ and thermal dissipation rate ${\epsilon }_T$ normalized by the vertical average (va) of its corresponding components. Plots have been zoomed in to highlight behavior in the domain center.

Figure 9

Normalized (a) Nu and (b) TKE as a function of both St and ${log}_{10}\left(V_g/U_{\mathrm{buoy}}\right)$. The solid lines represent evenly spaced contour levels for better clarity of the features.

Figure 10

Normalized (a) Nu and (b) TKE for all three types of couplings.

Figure 11

Normalized (a) Nu and (b) TKE for varying $C_{p,p}/C_{p,f}$, holding $\text{St}\approx 1$ and $V_g/U_{\mathrm{buoy}}={10}^{-3}$ with both couplings.

Figure 12

Normalized (a) Nu and (b) TKE with all three types of couplings while varying ${\varphi }_m$ at constant $\text{St}\approx 1$ and $V_g/U_{\mathrm{buoy}}={10}^{-3}$.