Large-eddy simulation of bubble plume in stratified crossflow

Characteristics and material transport of a bubble-driven plume in stably stratified water with uniform crossflow are studied using an Eulerian-Eulerian large-eddy simulation model. Four laboratory-scale plume conditions with different bubble rise velocities ($w_r=3$, 6, 12, and $20\mathrm{cm}/\mathrm{s}$) are considered, and their characteristics under three weak crossflow conditions ($U_c=0.5$, 1, and $2\mathrm{cm}/\mathrm{s}$) are studied. The interaction between the rising bubbles and the stratified water generates a double-plume structure, consisting of a rising plume of bubble/water mixture and a falling plume of dense water peeled from the rising plume due to the stratification effect. The presence of crossflow forces the rising plume to incline and causes the falling plume to form on the downstream side (with respect to the crossflow direction) of the rising plume, resulting in reduced contact area between the two plumes. Consequently, the turbulent mixing of the vertical momentum between the rising and falling plumes is reduced, causing the magnitude of the vertical velocity in both plumes to increase when the crossflow velocity is increased. The material transport from the plume to the horizontal intrusion layer (traced using dye) also exhibits strong dependence on the crossflow velocity. Faster crossflow results in narrower lateral extension and wider vertical extension of the intrusion layer due to the interaction between the peeling process and the crossflow. For cases with the two smaller $w_r$ (3 and $6\mathrm{cm}/\mathrm{s}$), the plume can form a distinct peeling event that dominates the material transport process. As a result, the mean intrusion layer height $h_t$ shows only small variation for these two plume conditions when $U_c$ is increased. In contrast, the plumes with $w_r=12$ and $20\mathrm{cm}/\mathrm{s}$ exhibit noticeable decrease of $h_t$ ($9\%$ and $24\%$, respectively) when $U_c$ is increased from $0.5$ to $2\mathrm{cm}/\mathrm{s}$. Statistical analysis of the streamwise dye flux shows that the decrease of $h_t$ with increased $U_c$ for the cases with $w_r=12$ and $20\mathrm{cm}/\mathrm{s}$ is mainly due to the crossflow-enhanced mean flux of dye from the rising plume at the low elevation.

Figure 1

Schematics of plume structure types in stratified water without crossflow based on the plume classification defined by Asaeda and Imberger [4] and Socolofsky et al. [12].

Figure 2

Schematics of plume structures in stratified water with (a) strong and (b) weak crossflows. $U_c$ denotes the crossflow velocity, $h_s$ denotes the height where the bubble and water plumes separate [19], and $h_t$ denotes the trap height of the detrained water [11, 19].

Figure 3

Three-dimensional instantaneous plume for case W6U2.

Figure 4

Instantaneous plume velocity and scalar fields on the $(x,z)$-plane across the source location for case Wr6-Uc0 ($w_r=6\mathrm{cm}/\mathrm{s}$, $U_c=0\mathrm{cm}/\mathrm{s}$): (a) bubble concentration ${\tilde{C}}_b$; (b) dye concentration ${\tilde{C}}_{\mathrm{dye}}$; (c) streamwise velocity $\tilde{u}$; (d) vertical velocity $\tilde{w}$. In (b)–(d) the solid lines are the isolines of ${\tilde{C}}_b=0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the shape and location of the bubble column.

Figure 5

Instantaneous plume velocity and scalar fields on the $(x,z)$-plane across the source location for case Wr6-Uc05 ($w_r=6\mathrm{cm}/\mathrm{s}$, $U_c=0.5\mathrm{cm}/\mathrm{s}$): (a) bubble concentration ${\tilde{C}}_b$; (b) dye concentration ${\tilde{C}}_{\mathrm{dye}}$; (c) streamwise velocity $\tilde{u}$; (d) vertical velocity $\tilde{w}$. In (b)–(d) the solid lines are the isolines of ${\tilde{C}}_b=0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the shape and location of the bubble column.

Figure 6

Instantaneous plume velocity and scalar fields on the $(x,z)$-plane across the source location for case Wr6-Uc2 ($w_r=6\mathrm{cm}/\mathrm{s}$, $U_c=2\mathrm{cm}/\mathrm{s}$): (a) bubble concentration ${\tilde{C}}_b$; (b) dye concentration ${\tilde{C}}_{\mathrm{dye}}$; (c) streamwise velocity $\tilde{u}$; (d) vertical velocity $\tilde{w}$. In (b)–(d) the solid lines are the isolines of ${\tilde{C}}_b=0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the shape and location of the bubble column.

Figure 7

Instantaneous plume velocity and scalar fields on the $(x,z)$-plane across the source for case Wr20-Uc05 ($w_r=20\mathrm{cm}/\mathrm{s}$, $U_c=0.5\mathrm{cm}/\mathrm{s}$): (a) bubble concentration $C_b$; (b) dye concentration $C_{\mathrm{dye}}$; (c) streamwise velocity $u$; (d) vertical velocity $w$. In (b)–(d) the solid lines are the isolines of $C_b=0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the shape and location of the bubble column.

Figure 8

Instantaneous plume velocity and scalar fields on the $(x,z)$-plane across the source for case Wr20-Uc2 ($w_r=20\mathrm{cm}/\mathrm{s}$, $U_c=2\mathrm{cm}/\mathrm{s}$): (a) bubble concentration ${\tilde{C}}_b$; (b) dye concentration ${\tilde{C}}_{\mathrm{dye}}$; (c) streamwise velocity $\tilde{u}$; (d) vertical velocity $\tilde{w}$. In (b)–(d) the solid lines are the isolines of ${\tilde{C}}_b=0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the shape and location of the bubble column.

Figure 9

Time-averaged plumes on the $(x,z)$-plane across the source for $w_r=6\mathrm{cm}/\mathrm{s}$: (a) Wr6-Uc0; (b) Wr6-Uc05; (c) Wr6-Uc1; (d) Wr6-Uc2. The color contours are for the time-averaged dye concentration; the solid lines are the isolines for the time-averaged bubble mass concentration of $0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the location and shape of the bubble column, and the dashed lines are the isolines of $5\%max\left\{{\overline{\tilde{C}}}_{\mathrm{dye}}\right\}$, which are used to define the mean peel height.

Figure 10

Time-averaged plumes on the $(x,z)$-plane across the source for $U_c=2\mathrm{cm}/\mathrm{s}$: (a) Wr3-Uc2; (b) Wr6-Uc2; (c) Wr12-Uc2; (d) Wr20-Uc2. The color contours are for the time-averaged dye concentration; the solid lines are the isolines for the time-averaged bubble mass concentration of $0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the location and shape of the bubble column, and the dashed lines are the isolines of $5\%max\left\{{\overline{\tilde{C}}}_{\mathrm{dye}}\right\}$, which are used to define the mean peel height.

Figure 11

Time-averaged 3D plumes for $W_r=6\mathrm{cm}/\mathrm{s}$: (a) Wr6-Uc0; (b) Wr6-Uc05; (c) Wr6-Uc1; (d) Wr6-Uc2. The red isosurfaces are for the mean vertical velocity $\overline{\tilde{w}}=1.5\mathrm{cm}/\mathrm{s}$, the blue isosurfaces are for the mean vertical velocity $\overline{\tilde{w}}=-1.5\mathrm{cm}/\mathrm{s}$, and the brown isosurfaces are for ${\overline{\tilde{C}}}_{\mathrm{dye}}=4{\mathrm{mg}/\mathrm{m}}^3$.

Figure 12

Time-averaged vertical velocity $\overline{\tilde{w}}$ on the $(x,z)$-plane at $y=0\text{m}$ for cases with $w_r=6\mathrm{cm}/\mathrm{s}$: (a) Wr6-Uc0; (b) Wr6-Uc05; (c) Wr6-Uc1; (d) Wr6-Uc2. Several representative contour lines of $\overline{\tilde{w}}$ are plotted on top of the smooth 2D color plot, among which the solid lines are for $\overline{\tilde{w}}>0$ and the dashed lines are for $\overline{\tilde{w}}<0$.

Figure 13

Time-averaged vertical velocity $\overline{\tilde{w}}$ on the $(x,y)$-plane at $z=0.25\text{m}$ for cases with $w_r=6\mathrm{cm}/\mathrm{s}$: (a) Wr6-Uc0; (b) Wr6-Uc05; (c) Wr6-Uc1; (d) Wr6-Uc2. Several representative contour lines of $\overline{\tilde{w}}$ are plotted on top of the smooth 2D color plot, among which the solid lines are for $\overline{\tilde{w}}>0$ and the dashed lines are for $\overline{\tilde{w}}<0$.

Figure 14

Time-averaged SGS shear stress $\overline{{\tau }_{13}^d}=-\overline{{\nu }_{sgs}(\partial \tilde{w}/\partial x+\partial \tilde{u}/\partial z)}$ (i.e., the color contours) on the $(x,z)$-plane across the source for the cases with $w_r=6\mathrm{cm}/\mathrm{s}$: (a) Wr6-Uc0; (b) Wr6-Uc05; (c) Wr6-Uc1; (d) Wr6-Uc2. The solid lines are the isolines of $\overline{\tilde{w}}=1\mathrm{cm}/\mathrm{s}$, and the dashed lines are the isolines of $\overline{\tilde{w}}=-1\mathrm{cm}/\mathrm{s}$, which are used to indicate the locations of the rising and falling plumes, respectively.

Figure 15

Time-averaged Reynolds shear stress $\overline{u^{\prime }{w}^{\prime }}$ (i.e., the color contours) on the $(x,z)$-plane across the source for the cases with $w_r=6\mathrm{cm}/\mathrm{s}$: (a) Wr6-Uc0; (b) Wr6-Uc05; (c) Wr6-Uc1; (d) Wr6-Uc2. The solid lines are the isolines of $\overline{\tilde{w}}=1\mathrm{cm}/\mathrm{s}$, and the dashed lines are the isolines of $\overline{\tilde{w}}=-1\mathrm{cm}/\mathrm{s}$, which are used to indicate the locations of the rising and falling plumes, respectively.

Figure 16

Time-averaged 3D plumes for $U_c=2\mathrm{cm}/\mathrm{s}$: (a) Wr3-Uc2; (b) Wr6-Uc2; (c) Wr12-Uc2; (d) Wr20-Uc2. The red isosurfaces are for the mean vertical velocity $\overline{\tilde{w}}=1.5\mathrm{cm}/\mathrm{s}$, the blue isosurfaces are for the mean vertical velocity $\overline{\tilde{w}}=-1.5\mathrm{cm}/\mathrm{s}$, and the brown isosurfaces are for ${\overline{\tilde{C}}}_{\mathrm{dye}}=4{\mathrm{mg}/\mathrm{m}}^3$.

Figure 17

Time-averaged vertical velocity field on the $(x,z)$-plane at $y=0\text{m}$ for cases with $U_c=2\mathrm{cm}/\mathrm{s}$: (a) Wr3-Uc2; (b) Wr6-Uc2; (c) Wr12-Uc2; (d) Wr20-Uc2.

Figure 18

Time-averaged SGS shear stress $\overline{{\tau }_{13}^d}=-\overline{{\nu }_{sgs}(\partial \tilde{w}/\partial x+\partial \tilde{u}/\partial z)}$ (i.e., the color contours) on the $(x,z)$-plane across the source for the cases with $U_c=2\mathrm{cm}/\mathrm{s}$: (a) Wr3-Uc2; (b) Wr6-Uc2; (c) Wr12-Uc2; (d) Wr20-Uc2. The solid lines are the isolines of $\overline{\tilde{w}}=1\mathrm{cm}/\mathrm{s}$, and the dashed lines are the isolines of $\overline{\tilde{w}}=-1\mathrm{cm}/\mathrm{s}$, which are used to indicate the locations of the rising and falling plumes, respectively.

Figure 19

Time-averaged Reynolds shear stress $\overline{u^{\prime }{w}^{\prime }}$ (i.e., the color contours) on the $(x,z)$-plane across the source for the cases with $U_c=2\mathrm{cm}/\mathrm{s}$: (a) Wr3-Uc2; (b) Wr6-Uc2; (c) Wr12-Uc2; (d) Wr20-Uc2. The solid lines are the isolines of $\overline{\tilde{w}}=1\mathrm{cm}/\mathrm{s}$, and the dashed lines are the isolines of $\overline{\tilde{w}}=-1\mathrm{cm}/\mathrm{s}$, which are used to indicate the locations of the rising and falling plumes, respectively.

Figure 20

Nondimensional mean plume trap height $H_T=h_t/{(B_0/N^3)}^{1/4}$ as a function of the nondimensional bubble rise velocity $W_N=w_r/{\left(B_{0}N\right)}^{1/4}$. Results from the current LES are denoted by solid symbols: $\times$, cases with crossflow $U_c=0.5\mathrm{cm}/\mathrm{s}$; $•$, cases with crossflow $U_c=1.0\mathrm{cm}/\mathrm{s}$; $+$, cases with crossflow $U_c=2.0\mathrm{cm}/\mathrm{s}$. The data from various laboratory experiments for plume in stratified water without crossflow are plotted for comparison: $\circ$, Asaeda and Imberger [4]; $▹$, Lemckert and Imberger [6]; $\diamond$, Socolofsky [67]; and $\square$, Socolofsky and Adams [9]. The empirical parametrization from Socolofsky and Adams [9] is denoted by $---$.

Figure 21

Time-averaged streamwise dye entrainment flux $\overline{u^{\prime }{C}_{\mathrm{dye}}^{\prime }}$ (i.e., the color contours) on the $(x,z)$-plane across the source for the cases with $w_r=6\mathrm{cm}/\mathrm{s}$: (a) Wr6-Uc0; (b) Wr6-Uc05; (c) Wr6-Uc1; (d) Wr6-Uc2. The solid lines are the isolines of $\overline{\tilde{w}}=1\mathrm{cm}/\mathrm{s}$, and the dashed lines are the isolines of $\overline{\tilde{w}}=-1\mathrm{cm}/\mathrm{s}$, which are used to indicate the locations of the rising and falling plumes, respectively.

Figure 22

Time-averaged streamwise dye flux by mean flow $\overline{\tilde{u}}{\overline{\tilde{C}}}_{\mathrm{dye}}$ (i.e., the color contours) on the $(x,z)$-plane across the source for the cases with $w_r=6\mathrm{cm}/\mathrm{s}$: (a) Wr6-Uc0; (b) Wr6-Uc05; (c) Wr6-Uc1; (d) Wr6-Uc2. The solid lines are the isolines of $\overline{\tilde{w}}=1\mathrm{cm}/\mathrm{s}$, and the dashed lines are the isolines of $\overline{\tilde{w}}=-1\mathrm{cm}/\mathrm{s}$, which are used to indicate the locations of the rising and falling plumes, respectively.

Figure 23

Time-averaged streamwise dye entrainment flux $\overline{u^{\prime }{C}_{\mathrm{dye}}^{\prime }}$ (i.e., the color contours) on the $(x,z)$-plane across the source for the cases with $U_c=2\mathrm{cm}/\mathrm{s}$: (a) Wr3-Uc2; (b) Wr6-Uc2; (c) Wr12-Uc2; (d) Wr20-Uc2. The solid lines are the isolines of $\overline{\tilde{w}}=1\mathrm{cm}/\mathrm{s}$, and the dashed lines are the isolines of $\overline{\tilde{w}}=-1\mathrm{cm}/\mathrm{s}$, which are used to indicate the locations of the rising and falling plumes, respectively.

Figure 24

Time-averaged streamwise dye flux by mean flow $\overline{\tilde{u}}{\overline{\tilde{C}}}_{\mathrm{dye}}$ (i.e., the color contours) on the $(x,z)$-plane across the source for the cases with $U_c=2\mathrm{cm}/\mathrm{s}$: (a) Wr3-Uc2; (b) Wr6-Uc2; (c) Wr12-Uc2; (d) Wr20-Uc2. The solid lines are the isolines of $\overline{\tilde{w}}=1\mathrm{cm}/\mathrm{s}$, and the dashed lines are the isolines of $\overline{\tilde{w}}=-1\mathrm{cm}/\mathrm{s}$, which are used to indicate the locations of the rising and falling plumes, respectively.

Figure 25

Normalized vertical distributions of mean streamwise dye flux ${\varphi }_x\left(z\right)$ at $x_0=0.5\text{m}$ (lines) and $x_0=0.6\text{m}$ (symbols): (a) cases with $U_c=0.5\mathrm{m}/\mathrm{s}$; (b) cases with $U_c=1\mathrm{m}/\mathrm{s}$; (c) cases with $U_c=2\mathrm{m}/\mathrm{s}$.

Figure 26

Normalized spanwise distributions of mean streamwise dye flux ${\psi }_x\left(y\right)$ at $x_0=0.5\text{m}$ (lines) and $x_0=0.6\text{m}$ (symbols): (a) cases with $U_c=0.5\mathrm{m}/\mathrm{s}$; (b) cases with $U_c=1\mathrm{m}/\mathrm{s}$; (c) cases with $U_c=2\mathrm{m}/\mathrm{s}$.

Figure 27

Instantaneous plumes on the $(x,z)$-plane across the source location at $t=100\text{s}$ for case Wr6-Uc2 ($w_r=6\mathrm{cm}/\mathrm{s}$, $U_c=2\mathrm{cm}/\mathrm{s}$) obtained from LES with different grid numbers $N_x\times N_y\times N_z=$: (a), (d) $240\times 192\times 217$; (b), (e) $320\times 256\times 289$; (c), (f) $400\times 320\times 361$. The top row shows the color contours of the time-averaged dye concentration ${\tilde{C}}_{\mathrm{dye}}$, and the bottom row shows the color contours of the resolved instantaneous vertical velocity $\tilde{w}$. In each panel, the solid lines are the isolines of ${\tilde{C}}_b=0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the shape and location of the bubble column.

Figure 28

Time-averaged plumes on the $(x,z)$-plane across the source location for case Wr6-Uc2 ($w_r=6\mathrm{cm}/\mathrm{s}$, $U_c=2\mathrm{cm}/\mathrm{s}$) obtained from LES with different grid numbers $N_x\times N_y\times N_z=$: (a), (d) $240\times 192\times 217$; (b), (e) $320\times 256\times 289$; (c), (f) $400\times 320\times 361$. The top row shows the color contours of the resolved instantaneous dye concentration ${\overline{\tilde{C}}}_{\mathrm{dye}}$, and the bottom row shows the color contours of the time-averaged vertical velocity $\overline{\tilde{w}}$. In each panel, the solid lines are the isolines of ${\overline{\tilde{C}}}_b=0.001\mathrm{kg}/{\mathrm{m}}^3$, which are used to indicate the shape and location of the time-averaged bubble column.