Effects of particle settling on Rayleigh-Bénard convection

The effect of particles falling under gravity in a weakly turbulent Rayleigh-Bénard gas flow is studied numerically. The particle Stokes number is varied between 0.01 and 1 and their temperature is held fixed at the temperature of the cold plate, of the hot plate, or the mean between these values. Mechanical, thermal, and combined mechanical and thermal couplings between the particles and the fluid are studied separately. It is shown that the mechanical coupling plays a greater and greater role in the increase of the Nusselt number with increasing particle size. A rather unexpected result is an unusual kind of reverse one-way coupling, in the sense that the fluid is found to be strongly influenced by the particles, while the particles themselves appear to be little affected by the fluid, despite the relative smallness of the Stokes numbers. It is shown that this result derives from the very strong constraint on the fluid behavior imposed by the vanishing of the mean fluid vertical velocity over the cross sections of the cell demanded by continuity.

Figure 1

Area- and time-averaged normalized number density of thermally and mechanically coupled “cold” particles as a function of height in the cell; $d_p=25$ $\mu$m [(red) triangles], $d_p=100$ $\mu$m [blue (circles)], $d_p=175$ $\mu$m [(green) squares], and $d_p=200$ $\mu$m [(purple) crosses]. These are the only simulations described in this paper in which particles are injected at the top plate with the local fluid velocity.

Figure 2

Hot-plate Nusselt number as a function of the particle diameter $d_p$ for “cold” [circles and (blue) lines] and “warm” [diamonds and (red) lines] particles ($T_p=T_c$ and $T_p=T_m$, respectively). Dashed lines (open symbols; labeled $Q$) show results with thermal coupling only; solid lines (filled symbols; labeled $f+Q$), with combined thermal and mechanical coupling. The bottommost (green) line (open squares; labeled $f$) shows results for mechanical coupling only, which are independent of the particle temperature.

Figure 3

Fluid Reynolds number (based on the volume- and time-averaged r.m.s. velocity) as a function of particle diameter $d_p$ for “cold” [circles and (blue) lines] and “warm” [diamonds and (red) lines] particles ($T_p=T_c$ and $T_p=T_m$, respectively). See the caption to Fig. 2 for further details.

Figure 4

Vertical fluid velocity on the midplane cross section at $z=\frac{1}{2}H$ for purely mechanical coupling: (a) single phase, (b) $d_p=25$ $\mu$m, and (c) $d_p=100$ $\mu$m, $\left(d\right)$ $d_p=200$ $\mu$m; note the enlarged scale for the latter case.

Figure 5

Volume- and time-averaged fluid temperature as a function of the particle diameter $d_p$ for “cold” [circles and (blue) lines] and “warm” [diamonds and (red) lines] particles ($T_p=T_c$ and $T_p=T_m$, respectively). See the caption to Fig. 2 for further details.

Figure 6

Volume-averaged convective contribution to the hot-plate Nusselt number, Eq. (20), as a function of the particle diameter $d_p$ for “cold” [circles and (blue) lines] and “warm” [diamonds and (red) lines] particles ($T_p=T_c$ and $T_p=T_m$, respectively). See the caption to Fig. 2 for further details.

Figure 7

Ratio $E_1/E_0$ of the first two angular Fourier modes of the kinetic energy defined in Eq. (22) as a function of the particle diameter $d_p$ for “cold” [circles and (blue) lines] and “warm” [diamonds and (red) lines] particles ($T_p=T_c$ and $T_p=T_m$, respectively). See the caption to Fig. 2 for further details.

Figure 8

Trajectories of randomly selected fluid particles, with the color keyed to the local vertical velocity for purely mechanical coupling. Instantaneous vertical velocities on the cross sections at $0.05H$, $0.5H$, and $0.95H$ are shown similarly to Fig. 4: (a) single phase, (b) $d_p=25\mu$m, (c) $d_p=100$ $\mu$m, and (d) $d_p=200$ $\mu$m; note the enlarged scale for the last case.

Figure 9

Average vertical velocity of thermally and mechanically coupled “cold” particles as a function of height in the cell for $d_p=25$ $\mu$m [(red) triangles], $d_p=100$ $\mu$m [(blue) circles], $d_p=175$ $\mu$m [(green) squares], and $d_p=200$ $\mu$m [(purple) crosses].

Figure 10

Average fluid vertical velocity at the particle position as a function of height in the cell for thermally and mechanically coupled “cold” particles: $d_p=25$ $\mu$m [(red) triangles], $d_p=100$ $\mu$m [(blue) circles], $d_p=175$ $\mu$m [(green) squares], and $d_p=200$ $\mu$m [(purple) crosses].

Figure 11

Vertical [(blue) circles] and horizontal [(red) triangles] mean accelerations of thermally and mechanically coupled “cold” particles as functions of the particle diameter $d_p$. The (green) squares and line show the mean of the modulus of the vertical acceleration. All accelerations are normalized by division by $g$.

Figure 12

Area- and time-averaged normalized number density of thermally and mechanically coupled “cold” particles as a function of height in the cell: $d_p=25$ $\mu$m [(red) triangles], $d_p=100$ $\mu$m [(blue) circles], $d_p=175$ $\mu$m [(green) squares], and $d_p=200$ $\mu$m [(purple) crosses].

Figure 13

Normalized particle flux, defined in Eq. (26), at the bottom plate as a function of the particle diameter for mechanically and thermally coupled cold particles, $T_p=T_c$.

Figure 14

Area- and time-averaged normalized fluid temperature at the particle location for thermally and mechanically coupled “cold” particles as a function of height in the cell: $d_p=25\mu$m [(red) triangles], $d_p=100$ $\mu$m [(blue) circles], $d_p=175$ $\mu$m [(green) squares], and $d_p=200$ $\mu$m [(purple) crosses].

Figure 15

Hot-plate Nusselt number as a function of the particle diameter $d_p$ for “hot” particles, $T_p=T_h$. The dashed line (labeled $Q$) shows results with thermal coupling only, and the solid line (labeled $f+Q$) results with combined thermal and mechanical coupling.

Figure 16

Fluid Reynolds number calculated with the volume- and time-averaged r.m.s. velocity as a function of the particle diameter $d_p$ for “hot” particles, $T_p=T_h$. The dashed line (label $Q$) shows results with thermal coupling only, and the solid line (labeled $f+Q$) results with combined thermal and mechanical coupling.

Figure 17

Convective contribution to the hot-plate Nusselt number, Eq. (20), as a function of the particle diameter $d_p$ for “hot” particles, $T_p=T_h$. The dashed line (labeled $Q$) shows results with thermal coupling only, and the solid line (labeled $f+Q$) results with combined thermal and mechanical coupling.