Thermally induced rarefied gas flow in a three-dimensional enclosure with square cross-section

Rarefied gas flow in a three-dimensional enclosure induced by nonuniform temperature distribution is numerically investigated. The enclosure has a square channel-like geometry with alternatively heated closed ends and lateral walls with a linear temperature distribution. A recently proposed implicit discrete velocity method with a memory reduction technique is used to numerically simulate the problem based on the nonlinear Shakhov kinetic equation. The Knudsen number dependencies of the vortices pattern, slip velocity at the planar walls and edges, and heat transfer are investigated. The influences of the temperature ratio imposed at the ends of the enclosure and the geometric aspect ratio are also evaluated. The overall flow pattern shows similarities with those observed in two-dimensional configurations in literature. However, features due to the three-dimensionality are observed with vortices that are not identified in previous studies on similar two-dimensional enclosures at high Knudsen and small aspect ratios.

Figure 1

Schematic of the square channel with differently heated closed ends and non-isothermal lateral walls. The left and right end sides (denoted by $OABC$ and $PDEF$) are kept at higher and lower uniform temperatures $T_h$ and $T_c$, respectively. The temperature of the lateral sides varies linearly with respect to the Y coordinate ($y$) from $T_h$ at the left to $T_c$ at the right. The dotted line $GH$ denotes the symmetry line of the lateral side $OADP$, which will be referenced in numerical results section of the body text.

Figure 2

Streamlines and sliced contour plots of $V$-velocity at various Knudsen numbers. Only one quarter of the domain ($x\in [0,0.5],z\in [0.5,1]$) is shown due to the twofold symmetry in $X$ and $Z$ directions. In each subfigure, the sliced contour plots are at positions of $y=0.5$, 2.5, and 4.5. In each contour plots, the red solid line(s) indicate(s) the contour level of $V=0$. The streamlines are only shown in the diagonal plane. All cases are based on the base temperature and geometry configuration, i.e., $T_c/T_h=0.5$ and $L_y/L_x=5$. (a) Kn = 0.01; (b) Kn = 0.1; (c) Kn = 0.2; (d) Kn = 0.5; (e) Kn = 1; (f) Kn = 10.

Figure 3

$V$-velocity (a), pressure (b), and temperature (c) profiles on the geometric center line of the channel along the $Y$ direction. : $\text{Kn}=0.01$, : $\text{Kn}=0.1$, : $\text{Kn}=0.2$, : $\text{Kn}=0.5$, : $\text{Kn}=1$, : $\text{Kn}=10$.

Figure 4

$V$-velocity profiles on the $X$-direction center lines of the slices shown in Fig. 2, i.e., slices at positions of $y=0.5$ (a), $y=2.5$ (b), and $y=4.5$ (c). : $\text{Kn}=0.01$, : $\text{Kn}=0.1$, : $\text{Kn}=0.2$, : $\text{Kn}=0.5$, : $\text{Kn}=1$, : $\text{Kn}=10$.

Figure 5

$V$-slip-velocity profiles on the $Y$-direction center line of the lateral walls (a) and the lateral edges (b). See the lines $GH$ and $OP$ in Fig. 1 for the profiles' location. The dashed black lines without symbols indicate the zero-velocity level. : $\text{Kn}=0.01$, : $\text{Kn}=0.1$, : $\text{Kn}=0.2$, : $\text{Kn}=0.5$, : $\text{Kn}=1$, : $\text{Kn}=10$.

Figure 6

Streamlines and sliced contour plots of $V$-velocity at different Knudsen numbers and temperature ratios. Only one quarter of the domain ($x\in [0,0.5],z\in [0.5,1]$) is shown due to the two-fold symmetry in $X$ and $Z$ directions. In each subfigure, the contour plots are on slices at positions of $y=0.5$, 2.5, and 4.5. In each contour plots, the red solid line(s) indicate(s) the contour level of 0. The streamlines are only shown in the two diagonal planes. The geometric aspect ratio $L_y/L_x$ is set as the base configuration, i.e., $L_y/L_x=5$. (a) Kn = 0.1, $T_c/T_h$ = 0.8; (b) Kn = 0.1, $T_c/T_h$ = 0.9; (c) Kn = 0.5, $T_c/T_h$ = 0.8; (d) Kn = 0.5, $T_c/T_h$ = 0.9; (e) Kn = 1, $T_c/T_h$ = 0.8; (f) Kn = 1, $T_c/T_h$ = 0.9; (g) Kn = 10, $T_c/T_h$ = 0.8; (h) Kn = 10, $T_c/T_h$ = 0.9.

Figure 7

$V$-velocity profiles on the $Y$-direction center lines of the geometry. (a) $\text{Kn}=0.1$, (b) $\text{Kn}=0.5$, (c) $\text{Kn}=1$, and (d) $\text{Kn}=10$. In each subfigure, different lines represent different temperature ratios (: $T_c/T_h=0.5$, : $T_c/T_h=0.8$, : $T_c/T_h=0.9$). The geometric aspect ratio $L_y/L_x$ is set as the base configuration, i.e., $L_y/L_x=5$.

Figure 8

$V$-velocity profiles on the $X$-direction center lines of the slices shown in Fig. 6. (a) $\text{Kn}=0.1$, (b) $\text{Kn}=0.5$, (c) $\text{Kn}=1$, and (d) $\text{Kn}=10$. In each subfigure, different lines represent different temperature ratios (: $T_c/T_h=0.5$, : $T_c/T_h=0.8$, : $T_c/T_h=0.9$). The geometric aspect ratio $L_y/L_x$ is set as the base configuration, i.e., $L_y/L_x=5$.

Figure 9

Streamlines and sliced contour plots of $V$-velocity at different Knudsen numbers and aspect ratios. Only one quarter of the domain ($x\in [0,0.5],z\in [0.5,1]$) is shown due to the twofold symmetry in $X$ and $Z$ directions. In each subfigure, the contour plot is on the slice at the center position along the $Y$ direction. In each contour plots, the red solid line(s) indicate(s) the contour level of 0. The streamlines are only shown in the two diagonal planes. The temperature ratio is set as the base configuration, i.e., $T_c/T_h=0.5$. (a) Kn = 0.1, $L_y/L_x$ = 1; (b) Kn = 0.1, $L_y/L_x$ = 2; (c) Kn = 1, $L_y/L_x$ = 1; (d) Kn = 1, $L_y/L_x$ = 2; (e) Kn = 10, $L_y/L_x$ = 1; (f) Kn = 10, $L_y/L_x$ = 2.

Figure 10

$V$-velocity profiles along the $Y$-direction center lines of the channel at different aspect ratios (: $L_y/L_x=1$, : $L_y/L_x=2$, : $L_y/L_x=5$) and Knudsen numbers. (a) $\text{Kn}=0.1$, (b) $\text{Kn}=1$, and (c) $\text{Kn}=10$. The temperature ratio is set as the base configuration, i.e., $T_c/T_h=0.5$.

Figure 11

Streamlines and contour plots of temperature on the $Y\text{−}Z$ symmetric plane at different Knudsen numbers and aspect ratios. The temperature ratio is set as the base configuration, i.e., $T_c/T_h=0.5$. Only one half of the symmetric plane ($z\in [0,0.5]$) is shown due to the symmetry in the $Z$ direction. (a) Kn = 0.1, $L_y/L_x$ = 1; (b) Kn = 0.1, $L_y/L_x$ = 2; (c) Kn = 1, $L_y/L_x$ = 1; (d) Kn = 1, $L_y/L_x$ = 2; (e) Kn = 10, $L_y/L_x$ = 1; (f) Kn = 10, $L_y/L_x$ = 2.

Figure 12

$V$-velocity profiles along the $X$- (a) and $Y$- (b) direction center lines obtained on different velocity space grids (: nonuniform ${64}^3$, : nonuniform ${128}^3$) for the cases of $\text{Kn}=10$.

Figure 13

$V$-velocity profiles along the $X$- (a) and $Y$- (b) direction center lines obtained on different physical space grids for the cases of $\text{Kn}=0.01$ and $\text{Kn}=0.1$. : coarse grid $\text{Kn}=0.01$, : fine grid $\text{Kn}=0.01$, : coarse grid $\text{Kn}=0.1$, : fine grid $\text{Kn}=0.1$.