Hydrodynamic forces on a clean spherical bubble translating in a wall-bounded linear shear flow

The three-dimensional flow around a spherical clean bubble translating steadily in a wall-bounded linear shear flow is studied numerically. The present work is concerned with the drag and lift forces experienced by the bubble over a wide range of Reynolds number ($0.1\le \text{Re}\le {10}^3$, Re being based on the bubble diameter and relative velocity with respect to the ambient fluid), wall distance ($1.5\le L_{\text{R}}\le 8$, $L_{\text{R}}$ being the distance from the bubble center to the wall normalized by the bubble radius), and relative shear rate ($-0.5\le \text{Sr}\le 0.5$, Sr being the ratio between the velocity difference across the bubble and the relative velocity). Based on the above range of parameters, situations where the bubble is repelled from or attracted to the wall are both covered. The flow structure and vorticity field are analyzed to obtain qualitative insight into the interaction mechanisms at work. The drag and lift forces are computed as well. Their variations agree well with theoretical predictions available in the limit of low-but-finite Reynolds number and, when the fluid is at rest, in the potential flow limit. Numerical results and analytical expressions are combined to provide accurate semiempirical expressions for the drag and lift forces at arbitrary Reynolds number and separation distance.

Figure 1

Wall- and shear-induced contributions to the lift force acting on a spherical bubble in the low-to-moderate Reynolds-number regime. (a) Stagnant fluid in the presence of a wall, where only the vortical wall-induced lift exists and the bubble always leads the fluid. (b) Unbounded linear shear flow, where only the shear-induced lift exists; the force points away from the wall when the bubble lags behind the fluid, and its direction reverses if it leads the fluid. [(c), (d)] Wall-bounded linear shear flow. The bubble lags the fluid in (c), where the two effects cooperate, while it leads the fluid in (d), making the two effects combine in an antagonistic manner.

Figure 2

Wall- and shear-induced contributions to the lift force acting on a spherical bubble in the moderate-to-large Reynolds-number regime. (a) Stagnant fluid in the presence of a wall, where only the irrotational (Bernoulli) transverse mechanism exists. (b) Same as in Fig. 1. [(c), (d)] Wall-bounded linear shear flow. The bubble lags the fluid in (c), where the two effects act in an antagonistic manner, while it leads the fluid in (d), where they cooperate.

Figure 3

Sketch of the various length scales and regions of the low-Re flow disturbance around a sphere with diameter d standing a distance $\tilde{L}$ from the wall in a situation such that ${\tilde{L}}_u<{\tilde{L}}_{\omega }$, i.e., $\varepsilon <1$. The inner and outer regions correspond to distances $r$ from the sphere center such that $r<\text{min}\left({\tilde{L}}_u,{\tilde{L}}_{\omega }\right)$ and $r>\text{max}\left({\tilde{L}}_u,{\tilde{L}}_{\omega }\right)$, respectively. The wall lies in the outer region in (a), where $\tilde{L}>\text{max}\left({\tilde{L}}_u,{\tilde{L}}_{\omega }\right)$, and in the inner region in (b), where $\tilde{L}<\text{min}\left({\tilde{L}}_u,{\tilde{L}}_{\omega }\right)$.

Figure 4

Schematic of a bubble moving in a wall-bounded linear shear flow.

Figure 5

Illustration of the grid system. (a) Shape and size of the computational domain. (b) Definition of the number of nodes in the computational domain. [(c), (d)] Partial view of the grid near the bubble with $L_{\text{R}}=1.5$ and $2\delta /d=0.01$ in (c), and $L_{\text{R}}=2.0$ and $2\delta /d=0.002$ in (d).

Figure 6

Predictions for $L_{\text{R}}\mathrm{\Delta }{C}_{\text{D}u}^{\text{W-out}}$ and $C_{\text{L}u}^{\text{W-out}}$ based on (11) for $0.01\le L_u\le 300$. Open symbols: numerical integration; solid line: approximate expression (12). (a) $L_{\text{R}}\mathrm{\Delta }{C}_{\text{D}u}^{\text{W-out}}$; (b) $C_{\text{L}u}^{\text{W-out}}$.

Figure 7

Distribution of the streamwise velocity disturbance $(\mathit{u}-{\mathit{u}}_{\infty })·{\mathit{e}}_z/U_{\text{rel}}$ along the line ($y=0$, $z=0$). (a) Stagnant fluid ($\text{Sr}=0$) for different separation distances; (b) linear shear flow for $L_{\text{R}}=4$. The wall stands at position $2x/d=-2$ (resp. $-4$) for $L_{\text{R}}=2$ (resp. 4).

Figure 8

Isocontours of the normalized vorticity disturbance $d/\left(2U_{\text{rel}}\right)(\mathit{\omega }-{\mathit{\omega }}_{\infty })·{\mathit{e}}_y$ in the symmetry plane $y=0$ at $L_{\text{R}}=2$. Left column: $\text{Sr}=-0.5$ ($U_{\text{rel}}<0$); central column: $\text{Sr}=0$ ($U_{\text{rel}}<0$); right column: $\text{Sr}=0.5$ ($U_{\text{rel}}>0$). The wall stands at the bottom of each panel. The relative flow with respect to the bubble is from left to right, i.e., in the $+z$ (resp. $-z$) direction for $\text{Sr}=0.5$ (resp. $\text{Sr}=0$ and $-0.5$).

Figure 9

Isocontours of the normalized streamwise vorticity component $d/\left(2\right|U_{\text{rel}}\left|\right)\mathit{\omega }·{\mathit{e}}_z$ in the cross-sectional plane $z=d$ in an unbounded linear shear flow with $\text{Sr}=0.5$. (a) $\text{Re}=100$; (b) $\text{Re}=500$. The main flow is going inwards, along the $z$ axis.

Figure 10

Isocontours of the normalized streamwise vorticity component $d/\left(2\right|U_{\text{rel}}\left|\right)\mathit{\omega }·{\mathit{e}}_z$ in the diametrical plane $z=0$ in a wall-bounded linear shear flow, with the bubble standing a distance $L_{\text{R}}=1.5$ from the wall. The main flow is going inwards, along the $z$ (resp. $-z$) direction for $\text{Sr}>0$ (resp. $\text{Sr}\le 0$); the direction of the $y$ axis is adjusted such that the $(x,y,z)$ coordinate system remains right-handed in all configurations.

Figure 11

Relative wall-induced drag correction $\mathrm{\Delta }{C}_{\text{D}u}^{\text{W}}$ for a bubble translating parallel to a wall in a fluid at rest. (a) High-Re regime; (b) entire Re range investigated numerically. Symbols: numerical data; solid lines: high-Re expression (20); dash-dotted lines: low-Re expression (A1a); dashed lines: low-to-moderate-Re expression (22); dotted lines: composite fit (23).

Figure 12

Lift coefficient on a bubble translating parallel to a wall in a fluid at rest. (a) High-Reynolds-number range ${10}^2\le \text{Re}\le {10}^3$. Comparison of numerical results (symbols) with the potential flow prediction (17) (solid lines), and the high-Re prediction (18) including the ${\text{Re}}^{-1}$ vortical contribution (dotted lines). (b) Complete Re range ${10}^{-1}\le \text{Re}\le {10}^3$. Comparison of numerical results (symbols) with the low-to-moderate-Re semiempirical model (24) (dashed lines), the high-Re prediction (18) (dotted lines), and the composite model (solid lines).

Figure 13

Transverse force on a bubble translating parallel to a wall in a fluid at rest in the range $0.6\le \text{Re}\le 300$ according to various sources. Solid line: composite model (25) involving the low-to-moderate-Re model (24)–(21); dashed line: low-but-finite-Re outer solution (12b); $\square$: experimental data from [13] and [6]; $*$: numerical data from Ref. [9].

Figure 14

Relative near-wall drag increase $\mathrm{\Delta }{C}_{\text{D}}^{\text{W}}$ for a bubble translating parallel to a wall in a linear shear flow in the low-but-finite Reynolds-number regime for $\text{Sr}=\pm 0.5$. Symbols: numerical results for $\text{Sr}=0.5$ ($\square$), and $\text{Sr}=-0.5$ ($\circ$). Dashed lines: asymptotic prediction (10a) corresponding to conditions $L_u\ll 1,L_{\omega }\ll 1$; solid lines: semiempirical model (26). Thick (resp. thin) lines correspond to positive (resp. negative) Sr.

Figure 15

Relative drag variation $\mathrm{\Delta }{C}_{\text{D}\omega }^{\text{U}}$ for a bubble translating at moderate-to-high Re in an unbounded shear flow, with reference to the drag in a uniform stream. (a) Influence of the Reynolds number; (b) influence of the shear rate. (a) Numerical results for $\text{Sr}>0$ ($\square$), and $\text{Sr}<0$ ($\circ$). (b) Numerical results for $\text{Re}=500$ ($\square$), and $\text{Re}=300$ ($\circ$). Solid line: prediction of (27); dashed line: prediction of (9a) taken from Ref. [8]; $\times$: numerical results from Ref. [8].

Figure 16

Influence of the wall-shear interaction on the relative drag increase for $\text{Re}\ge 100$. Symbols: numerical values of the difference between the relative drag variation in the presence of wall and shear ($\mathrm{\Delta }{C}_{\text{D}}^{\text{W}}$), and the sum $\mathrm{\Delta }{C}_{\text{D}u}^{\text{W}}+\mathrm{\Delta }{C}_{\text{D}\omega }^{\text{U}}$ of the relative drag variation in the presence of the wall for $\text{Sr}=0$ ($\mathrm{\Delta }{C}_{\text{D}u}^{\text{W}}$), and the relative drag variation in an unbounded shear flow ($\mathrm{\Delta }{C}_{\text{D}\omega }^{\text{U}}$). Solid (resp. dashed) lines: empirical expression (28) for $\text{Sr}>0$ (resp. $\text{Sr}<0$).

Figure 17

Relative near-wall drag increase $\mathrm{\Delta }{C}_{\text{D}}^{\text{W}}(\text{Re},\text{Sr},L_{\text{R}})$ for a bubble translating parallel to a wall in a linear shear flow throughout the Re range investigated numerically. (a) $\text{Sr}=\pm 0.2$; (b) $\text{Sr}=\pm 0.5$. $\square$ and $\circ$: numerical data for $\text{Sr}>0$ and $\text{Sr}<0$, respectively. Thick (resp. thin) solid lines: high-Re expression (29) for positive (resp. negative) Sr; thick (resp. thin) dashed lines: low-Re expression (26) for positive (resp. negative) Sr; thick (thin) dash-dotted lines: composite fit (30) for positive (resp. negative) Sr.

Figure 18

Variations of the lift coefficient $C_{\text{L}}^{\text{W}}(\text{Re},\text{Sr},L_{\text{R}})$ for a bubble translating parallel to a wall in a linear shear flow in the low-but-finite Reynolds-number regime with (a) $\text{Sr}=\pm 0.2$ and (b) $\text{Sr}=\pm 0.5$. $\square$ (resp. $\circ$): numerical data for $\text{Sr}>0$ (resp. $\text{Sr}<0$). Dashed lines: asymptotic prediction (10b) corresponding to conditions $L_u\ll 1,L_{\omega }\ll 1$; dotted lines: fit (13) of the asymptotic prediction corresponding to conditions $L_u\gg 1,L_{\omega }\gg 1$; solid lines: semiempirical model (31) taking into account finite-size effects.

Figure 19

(a) Effect of the shear-wall coupling on the lift force in the high-Reynolds-number regime. (a) Symbols: $\mathrm{\Delta }{C}_{\text{L}}^{\text{W}}(\text{Re},\text{Sr},L_{\text{R}})=C_{\text{L}}^{\text{W}}(\text{Re},\text{Sr},L_{\text{R}})-C_{\text{L}\omega }^{\text{U}}(\text{Re}\gg 1)-C_{\text{L}u}^{\text{W}}(\text{Re}\gg 1)$. (b) Performance of the shear-wall coupling model (32a) and (32b) estimated through the normalized difference ${\varepsilon }_{\text{WS}}=[C_{\text{L}}^{\text{W}}(\text{Re},\text{Sr},L_{\text{R}})-C_{\text{L}}^{\text{W}}(\text{Re}\gg 1)]/\left[\right|C_{\text{L}\omega }^{\text{U}}|(\text{Re}\gg 1)+|C_{\text{L}u}^{\text{W}}\left|(\text{Re}\gg 1)\right]$.

Figure 20

Lift coefficient $C_{\text{L}}^{\text{W}}(\text{Re},\text{Sr},L_{\text{R}})$ for a bubble translating parallel to a wall in a linear shear flow throughout the Re range investigated numerically. (a) $\text{Sr}=\pm 0.2$; (b) $\text{Sr}=\pm 0.5$. $\square$ (resp. $\circ$): numerical data for $\text{Sr}>0$ (resp. $\text{Sr}<0$). Solid lines: high-Re prediction (32); dashed lines: low-Re prediction (31); dotted lines: composite expression (33); black dash-dotted line: moderate-to-high-Re behavior predicted by (9b) in an unbounded shear flow.

Figure 21

Numerical predictions ($\square$) for the wall-induced forces in a quiescent fluid at low-to-moderate Reynolds number. (a) $\mathrm{\Delta }{C}_{\text{D}u}^{\text{W}}$; (b) $C_{\text{L}u}^{\text{W}}$ (with $C_{\text{L}u}^{\text{W}}$ multiplied by $L_{\text{R}}^{-1}$ for better readability). Dashed lines: asymptotic solution (10a) and (10b) corresponding to conditions $L_u\ll 1,L_{\omega }\ll 1$; dotted lines: asymptotic solution (11a) and (11b) corresponding to conditions $L_u\gg 1,L_{\omega }\gg 1$; solid lines: semiempirical model (A1a) and (A1b).

Figure 22

Comparison of numerical predictions for $C_{\text{L}u}^{\text{W}}$ ($\square$) with the fit (15) of the asymptotic solution corresponding to conditions $L_u\gg 1,L_{\omega }\gg 1$ (dashed lines), and the irrotational solution (17) (short-dashed lines), for $1.5\le L_{\text{R}}\le 8$ and $0.1\le \text{Re}\le 500$.

Figure 23

Comparison of present numerical predictions with available numerical results and semiempirical expressions. (a) $C_{\text{D}\omega }^{\text{U}}$ compared with numerical results from Ref. [8] (LM 1998) and semiempirical correlation $\frac{\text{Re}}{16}{C}_{\text{D}0}^{\text{U}}\approx 1+{\left[\frac{8}{\text{Re}}+\frac{1}{2}\left(1+3.315{\text{Re}}^{-1/2}\right)\right]}^{-1}$ proposed in Ref. [64] (MKL 1994). (b) $C_{\text{L}\omega }^{\text{U}}$ compared with numerical results and the composite fit (A2) from Ref. [8].