3.1. Governing equation

To determine the motion of a deformable bubble in turbulence rigorously using the generalized Kelvin–Kirchhoff equation (1.1), the force ($\boldsymbol {F}$) and torque ($\boldsymbol {\varGamma }$) from the surrounding flow should be calculated by integrating the hydrodynamic stresses over the entire bubble surface at each time instant along its trajectory. This calculation remains impractical, however, due to the challenges of measuring the full three-dimensional flow velocity very close to the bubble surface at a sufficiently high resolution. Instead of directly measuring the total force, this paper expands the force on the right-hand side of (1.1a) by following the formulation provided by Magnaudet & Eames (Reference Magnaudet and Eames2000) for spherical bubbles under the point-particle assumption, without considering the contribution of rotation. Furthermore, the original equation is adapted for deformable non-spherical bubbles by maintaining the tensorial form of the added-mass coefficient,

(3.1)\begin{align} V_{b} \rho_{b} \frac{\textrm{d} \boldsymbol{u}_{b}}{\textrm{d} t}&= \rho_{l} V_{b} \boldsymbol{C}_{A}\left(\frac{{\rm D} \boldsymbol{u}_{l}}{{\rm D} t}-\frac{\textrm{d} \boldsymbol{u}_{b}}{\textrm{d} t}\right) + \frac{\rho_l}{2}A_b C_D(\boldsymbol{u}_l-\boldsymbol{u}_b)|\boldsymbol{u}_l- \boldsymbol{u}_b| - V_b \boldsymbol{\nabla} P_w + \boldsymbol{F}_h \nonumber\\ &\quad + \rho_l C_L(\boldsymbol{u}_l-\boldsymbol{u}_b)\times (\boldsymbol{\nabla} \times \boldsymbol{u}_l) + V_b(\rho_l-\rho_b)g\boldsymbol{\hat{e}}_z, \end{align}

where $\textrm {d}/\textrm {d}t$ and $\textrm {D}/\textrm {D}t$ represent the material derivative along the bubble and tracer Lagrangian trajectories, respectively. The five terms on the right-hand side of (3.1) represent the drag ($F_{D}$), pressure ($F_{P}$), history ($F_{h}$), lift ($F_{L}$) and the buoyancy ($F_{G}$) forces, respectively. Here, $\boldsymbol {\nabla } P_w$ is the pressure gradient of flows around the bubble, $\boldsymbol {u}_l$ is the velocity of the unperturbed ambient flow taken at the centre of the bubble if the bubble were absent, and $\boldsymbol {u}_s = \boldsymbol {u}_b - \boldsymbol {u}_l$ is the bubble slip velocity. The equation also contains other constants, including the gravitational constant $g$ and the density of water and gas, i.e. $\rho _l$ and $\rho _b$. Here $V_b$ is the bubble volume and $A_b$ is the projected area of a volume-equivalent sphere. In addition, $C_D$ and $C_L$ are the drag and lift coefficients, respectively, and $\boldsymbol {C}_A$ is the added-mass coefficient tensor. In the previous work by Salibindla et al. (Reference Salibindla, Masuk, Tan and Ni2020), the $C_D$ and $C_L$ for bubbles with contaminated interfaces in intense turbulent environments were determined based on the mean bubble and flow velocities, and they are

(3.2)\begin{gather} C_D = {\text{max}}(24/Re_b(1+0.15Re_b^{0.687}),{\text{min}}(f(Eo), f(Eo)/We^{1/3})), \end{gather}

(3.3)\begin{gather}C_L = \begin{cases} 2.671We^{3/5} & We < 0.1, \\ 1.25 - 1.608 We^{3/5} & 0.1 < We < 0.9,\\ -0.23 & 0.9 < We, \end{cases} , \end{gather}

where $f(Eo) = 8Eo/3(Eo+4)$ is a function of the Eötvös number, i.e. $Eo=(\rho _l-\rho _b) g D^2 /\sigma$. The bubble Reynolds number and Weber number are defined as $Re_b= |u_b - u_l|D/\nu _l$ and $We=2.13\rho (\epsilon D)^{2/3}D/\sigma$, respectively (2.13 is the Kolmogorov constant for the second-order longitudinal structure function). Here $We$ determines the turbulence-induced deformation. The bubble size ($D$) was determined from the three-dimensional shape reconstruction for each time instant along a bubble trajectory, and instantaneous local flows and bubble velocity were used to calculate $Re_b$ and $We$. The local Weber number is determined using $We={\rho (\widetilde {\lambda _3}D)^2D}/{\sigma }$, where $\widetilde {\lambda _3}$ is the smallest eigenvalue of the coarse-grained velocity gradient tensor ($\tilde {{\mathsf{A}}}_{{\mathsf{ij}}}$) calculated around each bubble. As shown in a previous work by Masuk et al. (Reference Masuk, Salibindla and Ni2020b), this provides a reliable method to evaluate the instantaneous $We$. These dimensionless numbers were then substituted in (3.2) and (3.3) to determine the instantaneous drag and lift coefficients, respectively.

Equation (3.1) ignores the contribution from the bubble rotation. For a deformable bubble moving through water, two mechanisms enable bubbles to reorient: rotation or deformation. In a quiescent medium, bubbles can reorient via rotation driven by the unbalanced-wake-induced force. For finite-sized bubbles deforming in intense turbulence, the reorientation was found to be dominated by deformation along a different direction (Masuk et al. Reference Masuk, Qi, Salibindla and Ni2020a). Therefore, in accordance to this recent finding, the rotation of the bubble ($\boldsymbol {\varOmega }$) throughout the rest of this paper is assumed to be zero. As a result, the hydrodynamic torque, the added moment of inertia and corresponding equation (1.1b) are ignored for simplification.

After replacing the slip acceleration and the added-mass tensor $\boldsymbol {A}$ with $\boldsymbol {a}_s = \textrm {d}\boldsymbol {u}_b/\textrm {d}t - \textrm {D}\boldsymbol {u}_l/\textrm {D}t$ and $\boldsymbol {A} = \rho _l V_b \boldsymbol {C}_A$, respectively, (3.1) can be rearranged to isolate the bubble acceleration, which yields

(3.4)\begin{align} \frac{\textrm{d}\boldsymbol{u}_b}{\textrm{d}t}&= \left[V_b(\rho_b\boldsymbol{I}+\boldsymbol{C}_A \rho_l)\right]^{{-}1}\boldsymbol{\cdot}\left[\rho_lV_b\boldsymbol{C}_A\boldsymbol{\cdot}\frac{\textrm{D}\boldsymbol{u}_l}{\textrm{D}t}+ V_b(\rho_l-\rho_b)g\boldsymbol{\hat{e}}_z\right.\nonumber\\ &\quad \left. +\frac{\rho_l}{2}A_bC_D(\boldsymbol{u}_l-\boldsymbol{u}_b)|\boldsymbol{u}_l- \boldsymbol{u}_b|+\rho_lC_L(\boldsymbol{u}_l-\boldsymbol{u}_b)\times (\boldsymbol{\nabla} \times \boldsymbol{u}_l) - V_b \boldsymbol{\nabla} P_w \vphantom{\frac{\textrm{D}\boldsymbol{u}_l}{\textrm{D}t}}\right], \end{align}

where $[V_b(\rho _b\boldsymbol {I}+\boldsymbol {C}_A \rho _l)]^{-1}$ represents the inverse of the total mass tensor that includes both the actual mass $V_b\rho _b\boldsymbol {I}$ and the added mass tensor $V_b\boldsymbol {C}_A \rho _l$. Because the determinant of the total mass tensor is always non-zero, the tensor is invertible. The right-hand side of (3.4) essentially calculates the dot product of the inverse of the total mass tensor with the total hydrodynamic force (vector) applied on the bubble, which results in the bubble acceleration vector $\textrm {d}\boldsymbol {u}_b/\textrm {d}t$ matching the left-hand side. Within the total hydrodynamic force, we ignore the history forces acting on the bubble based on the previous works by Takagi & Matsumoto (Reference Takagi and Matsumoto1996) and Magnaudet & Eames (Reference Magnaudet and Eames2000), who suggested that the effects of the history force are typically negligible for bubbles with a large-$Re_b$ ($Re_b > 50$). For finite-sized bubbles considered in this paper, the $Re_b$ ranges between 200–4000, significantly greater than 50. Therefore, the history force is negligible for bubbles considered in our studies. Most quantities on the right-hand side of (3.4) can be directly determined for any given bubble trajectory (an example is shown in figure 1), including the instantaneous velocities ($\boldsymbol {u}_b$, $\boldsymbol {u}_l$) and accelerations ($\boldsymbol {a}_b = \textrm {d}\boldsymbol {u}_b/\textrm {d}t$, $\boldsymbol {a}_l = \textrm {D}\boldsymbol {u}_l/\textrm {D}t$) of the bubbles and surrounding flow. The velocity gradients ($\boldsymbol {\nabla } \times \boldsymbol {u}_l$) can also be estimated based on the discussions in § 2.2. Instantaneous $C_D$ and $C_L$ can be calculated from (3.2) and (3.3), respectively, based on $Re_b$, $Eo$ and $We$ calculated from instantaneous bubble and flow information. As aforementioned in § 2.2, $\boldsymbol {u}_l$, $\boldsymbol {a}_l$ and $\boldsymbol {\nabla } \times \boldsymbol {u}_l$ can be determined using the velocities of several tracer particles located within a distance of two times the bubble diameter away from the centre of the bubble. If $\boldsymbol {C}_A$ is known, the entire right-hand side of (3.4) can be calculated for each bubble trajectory.

3.2. Constant $C_A$

Equation (3.4) provides a method to estimate the added-mass coefficient since both sides of the equation can be measured simultaneously and independently from the two phases, one side from the bubble statistics and the other side from the flow statistics, with the only unknown being the added-mass coefficient. Because this equation should apply for bubbles of any shape, the first step is to test bubbles with weak deformation since their added-mass coefficient should be close to that of a sphere. The $\boldsymbol {C}_A$ for a spherical particle with three axes of symmetry can be simplified to a constant ($C_A$). Lamb (Reference Lamb1924) (article 92) analytically solved the added-mass coefficient of a sphere accelerating through fluid following a rectilinear path and found that $C_A = 0.5$ in the direction of motion. This finding was followed by experiments (Sridhar & Katz Reference Sridhar and Katz1995; Friedman & Katz Reference Friedman and Katz2002; Kendoush, Sulaymon & Mohammed Reference Kendoush, Sulaymon and Mohammed2007) and numerical simulations (Magnaudet Reference Magnaudet1997; Sankaranarayanan et al. Reference Sankaranarayanan, Shan, Kevrekidis and Sundaresan2002; Maliska & Paladino Reference Maliska and Paladino2006), where the value of $C_A = 0.5$ was consistently reported or used in different flow conditions. This work can be used to examine whether our experiments produce consistent results with this expected value as the bubble shape approaches the spherical limit.

Figure 3 shows two example time traces of the bubble's vertical acceleration in conjunction with their respective aspect ratio. For both cases, the bubble acceleration measured from the bubble trajectory is shown as black circles, while the red dashed lines indicate the calculations based on the right-hand side of (3.4) by using $C_A=0.5$. For the weakly deformed case (figure 3a), it is evident that the time trace of the calculated $a_b$ is close to that of the measured results. This agreement suggests that, despite the non-negligible measurement uncertainties of the second-order quantities (e.g. velocity gradients and acceleration), this framework seems to successfully predict $\boldsymbol {a}_b$. The assumption of using the added mass coefficient close to 0.5 seems to be reasonable for bubbles close to the spherical shape. Furthermore, compared with the weakly deformed case, applying $C_A=0.5$ seems to underpredict the acceleration fluctuations of the strongly deformed bubble, as shown in figure 3(b). This indicates that the acceleration fluctuation and added-mass coefficient may be coupled, and both may change as a function of the bubble deformation. Strictly speaking, (3.4) should only be applied to microbubbles with their sizes below the Kolmogorov scale immersed in a surrounding linear flow. Here, we assume that this equation may also work for finite-sized bubbles with their surrounding flow coarse-grained at the bubble size.

Figure 3.

The time series of the vertical acceleration of (a) a weakly deformed bubble and (b) a strongly deformed bubble. In conjunction with the measured results, the time series of the bubble acceleration calculated from (3.4) using $C_A=0.5$, accompanied by the aspect ratio time trace, is also shown.

Before discussing the dependence of $C_A$ on the geometry of finite-sized deformable bubbles, we aim at identifying a statistical quantity that can be used to constrain $C_A$. Since $C_A$ is closely related to the acceleration fluctuation, the p.d.f.s of one component of the calculated accelerations using two different values of $C_A$ are shown in figure 4(a). For both $C_A$, the calculated p.d.f.s exhibit a stretched exponential form, which is consistent with our observations and other previous measurements of fluid acceleration in turbulence (Voth et al. Reference Voth, La Porta, Crawford, Alexander and Bodenschatz2002; Volk et al. Reference Volk, Calzavarini, Verhille, Lohse, Mordant, Pinton and Toschi2008; Prakash et al. Reference Prakash, Tagawa, Calzavarini, Mercado, Toschi, Lohse and Sun2012). Applying $C_A=0.5$ appears to produce a p.d.f. that agrees with the measured results. Since $C_A=0.5$ was derived for rigid spherical particles, this agreement suggests that a large portion of our bubbles are not significantly deformed, which is consistent with the fact that 81 % of the bubbles have an aspect ratio smaller than $\chi = 2$. Despite the overall good agreement, some minor deviation is observed at the tails for large acceleration, which indicates that $C_A=0.5$ may not work well for bubbles experiencing strong acceleration. This observation will be further discussed later in this section. Moreover, the p.d.f. calculated by employing a constant $C_A=0.3$ seems to spread wider about the central value, thereby overpredicting the fluctuation of the bubble acceleration. This is expected since a smaller $C_A$ suggests a reduced effective bubble inertia; under the same hydrodynamic forces, bubbles tend to experience greater acceleration.

Figure 4.

The distribution of (a) the horizontal bubble acceleration and (b) the horizontal bubble velocity, obtained from the measured data (red square) and calculated from (3.4) with two different $C_A$ values of 0.3 (open circle) and 0.5 (open square).

The same calculation is repeated for the p.d.f. of the bubble velocity to examine its sensitivity to $C_A$. Like the acceleration, the bubble velocity can be determined from their trajectories or by integrating the calculated bubble acceleration from (3.4). The p.d.f.s of one component of the horizontal bubble velocity calculated from such an integration using both $C_A = 0.3$ and 0.5 are shown in figure 4(b). Both p.d.f.s agree with each other and the measured results, suggesting that the velocity p.d.f.s is less sensitive to the value of $C_A$. This is expected since many events with strong acceleration are probably smoothed during integration and thus not reflected in the velocity p.d.f.. Similar findings were also reported by Calzavarini et al. (Reference Calzavarini, Volk, Lévêque, Pinton and Toschi2012) for neutrally buoyant finite-sized particles, which suggested that the acceleration statistics are dominated by the inertial effects while the velocity statistics are dominated by the drag and history effects.

3.3. Tensorial form of the added-mass coefficient

Based on the previous tests, we conclude that $C_A$ can be constrained by the acceleration variance. The $\sqrt {\langle a_b^2 \rangle }$ in both horizontal and vertical directions are shown in figure 5(a) as a function of the bubble aspect ratio $\chi$, where $\chi$ represents the track-averaged bubble aspect ratio. The $\sqrt {\langle a_b^2 \rangle }$ of both directions increase from approximately ${\sim }2g$ to ${\sim }10g$. This large value of $\sqrt {\langle a_b^2 \rangle }$ can be attributed to the strong ambient turbulence around bubbles with an average energy dissipation rate of $0.5\ \textrm {m}^{2}\ \textrm {s}^{-3}$ (Masuk et al. Reference Masuk, Salibindla, Tan and Ni2019b). No significant difference in acceleration between the two directions is observed since the buoyancy effect is overwhelmed by the intense turbulence. The error bars in this figure were calculated by dividing the data set into six equal subsets and calculating $\sqrt {\langle a_b^2 \rangle }$ among these them, similar to the work by Voth (Reference Voth2000).

Figure 5.

The standard deviation of the bubble acceleration along (a) the horizontal ($x$) and vertical ($z$) directions in the laboratory frame and (b) the semi-major ($\hat {\pmb {r}}_1$) and semi-minor ($\hat {\pmb {r}}_3$) in the bubble body-axis frame, normalized by the gravitational constant $g$, versus the bubble aspect ratio $\chi$. The shaded areas in panel (b) represent the range of acceleration standard deviation calculated by assuming that the added-mass coefficient $\boldsymbol {C}_A$ follows the potential flow solutions either for prolates or oblates.

Despite establishing a connection between the standard deviation of acceleration with the added-mass coefficient, we have thus far been limited to a constant $C_A$, which only works for spherical particles. For deformable bubbles, the added-mass coefficient ($\boldsymbol {C}_A$) is a full $3\times 3$ second-rank tensor, where each element must be determined.

To address this challenge, we examine (3.1) and (3.4), which are formulated using the added mass $\boldsymbol {A}$ and its coefficient $\boldsymbol {C}_A$ defined in the laboratory frame with three coordinates $(\hat {\boldsymbol {e}}_{x},\hat {\boldsymbol {e}}_{y},\hat {\boldsymbol {e}}_{z})$. In a frame ($\mathfrak {R}$) aligned with the body axes of the bubble $(\hat{\boldsymbol {r}}_{1},\hat {\boldsymbol {r}}_{2},\hat {\boldsymbol {r}}_{3})$, the added-mass coefficient tensor can be denoted as ${\boldsymbol {C}_A}^{\mathfrak {R}}$. When bubbles undergo only affine deformation, their shapes will remain spheroid with geometric symmetries. The ${\boldsymbol {C}_A}^{\mathfrak {R}}$ can then be simplified into a diagonal matrix (Newman Reference Newman1977), i.e. ${\boldsymbol {C}_A}^{\mathfrak {R}} \equiv \textrm {diag}(C_A^{r_1},C_A^{r_2},C_A^{r_3})$, with the superscripts ($r_1$, $r_2$, and $r_3$) denoting one of the three principal bubble axes. The two tensorial coefficients $\boldsymbol {C}_A$ and $\boldsymbol {C}_A^{\mathfrak {R}}$ and their corresponding frames can be connected via a rotation matrix $\boldsymbol {R}$, following $\boldsymbol {C}_A = \boldsymbol {R}^T\,{\boldsymbol {C}_A}^{\mathfrak {R}}\,\boldsymbol {R}$. Our experiments provided a full three dimensional (3-D) reconstruction of each bubble geometry, from which the bubble body-axis frame ($\mathfrak {R}$) and the rotation matrix $\boldsymbol {R}$ at each time instant can be determined.

In the bubble body-axis frame, (3.4) can be explicitly written as three independent equations, linking each diagonal component of $\pmb {C}_A^{\mathfrak {R}}$, i.e. $C_A^{r_1}$, $C_A^{r_2}$, $C_A^{r_3}$, to the bubble acceleration along one of the three bubble principal axes, i.e. $a_{b,r_1}$, $a_{b,r_2}$, $a_{b,r_3}$, respectively. Because both the bubble acceleration ($\boldsymbol {a}_b$) and its principal axes ($\hat {\boldsymbol {r}}_{i}$, $i=1,2,3$) can be determined using our measurements, the projected acceleration $a_{b,r_i}$ can be easily calculated by using $\boldsymbol {a}_b\boldsymbol {\cdot } \hat {\boldsymbol {r}}_i$ for every bubble at every instant, which enables obtaining the bubble standard deviation projected along the three bubble principal axes.

Figure 5(b) shows the bubble acceleration standard deviation projected along the bubble semi-major ($a_{b,r_1}$) and semi-minor axes ($a_{b,r_3}$), respectively. Although the standard deviations of $a_x$ and $a_z$ in the laboratory frame appear to be very close to each other for all bubble aspect ratios considered, $\sqrt {\langle a_{b,r_1}^2\rangle }$ is systematically larger than $\sqrt {\langle a_{b,r_3}^2\rangle }$, and their differences increase as $\chi$ grows. This difference brings confidence to our experiments, because if the bubble orientation is not well reconstructed, it is likely that the acceleration standard deviations along both bubble principal axes do not show any difference, as with what is shown in the laboratory coordinates along the $x$ and $z$ axes. In addition, because the added-mass coefficient along the major axis accompanied by a smaller frontal area (end-on configuration) should be smaller, the acceleration standard deviation should be larger, which is consistent with the trend observed in figure 5(b).

The standard deviation of each acceleration component can be used to constrain one respective component of $\boldsymbol {C}_A^{\mathfrak {R}}$, for which purpose an iterative fit was put forward. Consider $C_A^{r_1}$ for example, where an initial guess of $C_A^{r_1}=0.5$ was used. For each iteration, $C_A^{r_1}$ was updated by adding its value from the previous iteration with a small step. The acceleration standard deviation was then calculated from the right-hand side of (3.4) by replacing $\boldsymbol {C}_A$ with a constant of $C_A^{r_1}$ and projecting all other terms to the direction of $\hat {\pmb {r}}_1$, and the resulting acceleration was naturally in the direction of $\hat {\pmb {r}}_{1}$. The total squared difference between the calculated acceleration standard deviation and those measured directly from bubble trajectories was used to correct $C_A^{r_1}$. The iteration continued until the difference using the two independent methods ceased decreasing, and the same procedure was repeated for each bubble aspect ratio $\chi$. For each $\chi$, in addition to the mean value, the uncertainty of $C_A$ is estimated by repeating the same iterative fit to the upper and lower bounds of $\sqrt {\langle a_{b,r_1}^2\rangle }$ in figure 5(b).

The calculated $C_A^{r_1}$ is shown as closed circles with error bars in figure 6. In the limit of spherical and isolated bubbles, $C_A$ should be equal to 0.5. Although $C_A$ seems to have a trend of getting close to 0.5 as $\chi$ approaches one, it is slightly larger than 0.5 at $\chi =1.25$. One explanation for such a result is that bubbles in our experiments are not isolated; the bubble volume concentration ($\phi$) is kept close to 2 %, striking a balance between acquiring sufficient data within an affordable period and possible contamination due to bubble–bubble interaction. At this concentration, it has been shown that the bubble–bubble interaction could entail non-negligible effects on $C_A$ with the largest reported value close to $C_A=1$ at $\phi =2$ % by Pudasaini (Reference Pudasaini2019) and the smallest value close to $C_A=0.6$ by Sankaranarayanan et al. (Reference Sankaranarayanan, Shan, Kevrekidis and Sundaresan2002).

Figure 6.

The added-mass coefficient tensor $\boldsymbol {C}_A$ versus the bubble aspect ratio $\chi$. Symbols represent one diagonal element of $\boldsymbol {C}_A$ along the bubble semi-major axis. Four lines indicate different elements of $\boldsymbol {C}_A$ calculated based on the Lamb's model for prolates or oblates.

In addition, $C_A$ seems to gradually decrease as $\chi$ increases, and we seek solutions from Lamb's potential flow models to understand this trend. Applying these models, however, requires knowing whether the shape of deformed bubbles in turbulence follows prolate or oblate spheroids as well as along which direction they are accelerating. Prolate and oblate spheroids represent ideal approximations at extremes, but a typical deformed bubble is shaped somewhere in-between. If a criterion of $r_2$ smaller or larger than $(r_1 + r_3)/2$ is used to separate prolate from oblate bubbles in our results, approximately 59 % bubbles are close to prolate spheroids, slightly higher than the remaining 41 % oblate-like bubbles.

Figure 6 shows four lines calculated using the Lamb's potential flow model for prolate and oblate spheroids accelerating either end-on (along $\hat {\pmb {r}}_1$) or broadside-on (along $\hat {\pmb {r}}_3$) ((A 1) in appendix A). All four lines begin from $C_A = 0.5$ when $\chi =1$, as expected. For either prolate or oblate spheroids accelerating end-on, $C_A$ reduces systematically as $\chi$ increases. In this case, an increasing $\chi$ is equivalent to a reducing frontal area as spheroids accelerate through fluid, which results in a smaller volume of fluid that needs to be accelerated and hence a smaller $C_A$. The same principle also applies to the opposite limit, where spheroids accelerate broadside-on. In this case, as $\chi$ becomes larger, $C_A$ increases along with the frontal area. In addition, $C_A$ of oblate spheroids increases more sharply compared with that of prolate spheroids due to the difference in their broadside-projected areas. For $C_A^{r_1}$, only two lines for the end-on configurations are relevant. It is evident that the acquired $C_A^{r_1}$, constrained by $\sqrt {\langle a_{b,r_1}^2\rangle }$, agrees with these two lines. This implies that the potential flow solution can accurately calculate the added mass coefficient, even for deformable bubbles, at least for those accelerating end-on.

We notice that the error bars of $C_A^{r_1}$ are not symmetric around the mean, unlike those for the measured $\sqrt {\langle a_{b,r_1}^2\rangle }$, because $C_A^{r_1}$ is inversely proportional to the bubble standard deviation. When the lower limit of acceleration standard deviation approaches zero, the estimated $C_A^{r_1}$ moves closer to infinity. The same argument applies also to the other two components of the added mass tensor, $C_A^{r_2}$ and $C_A^{r_3}$. Although both of them can be determined following the same procedure for $C_A^{r_1}$, their acceleration standard deviations are so small that any small errors translate to large error bars in the plot of $C_A$. In fact, the error bars of $C_A^{r_3}$, especially for large $\chi$, are so large that they cover almost the entire figure. Instead of showing them in figure 6, we decided to compare them with the Lamb's model prediction using the acceleration standard deviation in figure 5(b).

Applying Lamb's model to the right-hand side of (3.4) requires us to classify bubbles into prolate or oblate spheroids. To simplify the problem, we assume all bubbles are either prolate (${\boldsymbol {C}_A}^{\mathfrak {R}} \equiv \textrm {diag}(C_A^{pe},C_A^{pb},C_A^{pb})$) or oblate (${\boldsymbol {C}_A}^{\mathfrak {R}} \equiv \textrm {diag}(C_A^{oe},C_A^{oe},C_A^{ob})$), and the exact formulations of three diagonal values can be found in appendix A. The calculated acceleration standard deviations along $\hat {\pmb {r}}_1$ and $\hat {\pmb {r}}_3$ are shown as the two shaded areas in figure 5(b) with the upper bounds of both areas showing results from all-prolate calculations and the lower bounds indicating results from all-oblate calculations.

It is clear that the measured acceleration standard deviation along $\hat {\pmb {r}}_1$ falls right in the middle of the shaded area, confirming that the measured results agree with the potential flow calculation. But the measured $\sqrt {\langle a_{b,r_3}^2\rangle }$ seems to be higher than the predicted one from the potential flow theory for large aspect ratios, $\chi >2.25$. One possible explanation is that bubbles will experience necking when they undergo large deformation, which cause the geometry to deviate away from spheroids by having a small broadside projected area. As a result, the acceleration standard deviation along the bubble semi-minor axis may become slightly larger than the one predicted by the potential flow theory assuming spheroidal bubble shapes. In addition, we cannot completely rule out the possibility that bubbles with larger aspect ratios may be under the influence of rotation based on (1.1). This could be another reason for the observed difference between the measured and calculated $\sqrt {\langle a_{b,r_3}^2\rangle }$ at large $\chi$.

Finally, in addition to $\boldsymbol {C}_A$, the added-mass force is also sensitive to the relative orientation between the slip acceleration ($\pmb {a}_s$) with the bubble principal axes, because this relative orientation determines if the bubble is preferentially accelerating end-on or broadside-on and the final magnitude of the added-mass force. Figure 7 shows the p.d.f. of the alignment between the bubble semi-major axis ($\boldsymbol {r}_1$) and slip acceleration ($\boldsymbol {a}_s$), i.e. $\langle \hat {\boldsymbol {a}}_{s} \boldsymbol {\cdot } \hat {\boldsymbol {r}}_{1} \rangle$, for four different aspect ratios. If their orientations are completely random, the p.d.f. of the dot product of the two unit vectors should follow a uniform distribution. A peak at around zero would suggest a perpendicular orientation, whereas a peak close to one would indicate an alignment between the two unit vectors. It can be seen that, in figure 7, a consistent preferential alignment between the bubble semi-major axis with the slip acceleration is observed for all aspect ratios considered, suggesting that bubbles prefer to accelerate along their semi-major axis (end-on). Moreover, such a preference seems to increase with $\chi$ as strongly deformed bubbles align better with the slip acceleration. Since an end-on configuration implies a smaller added-mass coefficient and larger acceleration fluctuation, this preferential alignment explains why the acceleration standard deviation along both horizontal and vertical directions in the laboratory frame increases as a function of the bubble aspect ratio.

Figure 7.

Probability density function of the cosine of the angle between the bubble semi-major axis ($\hat {\boldsymbol {r}}_{1}$) and the slip acceleration ($\hat {\boldsymbol {a}}_{s}$) for bubbles with different aspect ratios $\chi$.

It is surprising to find that the preferential alignment seems to be opposite to what has been observed for bubbles rising in an otherwise quiescent medium, in which the bubble semi-minor axis, instead of semi-major axis, is preferentially aligned with the slip acceleration direction (Magnaudet & Eames Reference Magnaudet and Eames2000). This difference can be attributed to the relationships among bubble deformable geometry, slip velocity, turbulent strains and slip acceleration. Note that bubble deformation is not determined by the added mass force; instead, it is driven by the gradient of dynamic pressure acting on its interface, including the contribution of both the slip velocity and turbulent strains. As a result, the bubble semi-minor axis shows preferential alignment with the slip velocity direction in both a quiescent and a turbulent aqueous medium. It just happens that the slip velocity and slip acceleration align with each other in a quiescent medium, which is not the case in turbulence. In turbulence, as bubbles are deformed and elongated along its major axis, the acceleration along its major axis tends to increase, which dominates the slip acceleration. This explains the observed preferential alignment between the slip acceleration and the bubble semi-major axis.