We study the Lagrangian velocity and acceleration statistics of light particles (micro-bubbles in water) in homogeneous isotropic turbulence. Micro-bubbles with a diameter *d*_{b} = 340 μm and Stokes number from 0.02 to 0.09 are dispersed in a turbulent water tunnel operated at Taylor-Reynolds numbers (Re_{λ}) ranging from 160 to 265. We reconstruct the bubble trajectories by employing three-dimensional particle tracking velocimetry. It is found that the probability density functions (PDFs) of the micro-bubble acceleration show a highly non-Gaussian behavior with flatness values in the range 23 to 30. The acceleration flatness values show an increasing trend with Re_{λ}, consistent with previous experiments [G. Voth, A. La Porta, A. M. Crawford, J. Alexander, and E. Bodenschatz, “

_{λ}. We also compare our results with experiments in von Kármán flows and point-particle direct numerical simulations with periodic boundary conditions.

## I. INTRODUCTION

Multi-phase flows where the carrier fluid transports particles under turbulent conditions are ubiquitous. A thorough understanding of the dynamics of particles (light, neutral, or heavy) in turbulent flows is therefore crucial. In most of these flows, the particles have a finite size and their density is different from that of the carrier fluid. Thus, the particle's dynamic behavior is expected to be different compared to neutral fluid tracers. The two relevant dimensionless parameters are the density ratio β = 3ρ_{f }/(ρ_{f } + 2ρ_{p}), where ρ_{f } and ρ_{p} are the fluid and particle density, and the Stokes number, which is the ratio of the particle's response time τ_{p} to the Kolmogorov time scale τ_{η}, defined as St = τ_{p}/τ_{η} = *a*^{2}/3βντ_{η}, where *a* is the particle radius and ν the kinematic viscosity of the carrier fluid.

The Lagrangian approach is naturally suited to study particles in turbulence and has recently attracted much attention (see Ref. 5). Pioneering Lagrangian particle tracking experiments in fully developed turbulence used silicon strip detectors to measure three-dimensional trajectories of tracer particles (β = 1) with high spatial and temporal resolution in a von Kármán flow at high Taylor-Reynolds numbers, Re_{λ} up to 970.^{1,6,7} The particle acceleration probability density functions (PDFs) were found to be highly intermittent with flatness values around 55, and could be fitted with either stretched exponentials or log-normal distributions. The high intermittency of the fluid particle acceleration PDFs was also observed in numerical simulations.^{2,8–10} The normalized acceleration PDFs showed a weak dependence on the Re_{λ}, and this was more prominently seen in the flatness values, which have been found to increase with Re_{λ} in both experiments^{1} and numerics.^{2}

More recent experimental investigations have focused on studying particles with different density than the carrier fluid.^{11–14} Heavy particles (β = 0) in turbulence were studied using water droplets in wind tunnel experiments at Re_{λ} = 250.^{4} By following the particle motion with a moving camera, their trajectories were obtained in two-dimensions. It was observed that the normalized acceleration PDF was less intermittent than for tracers, with narrower tails. Numerical simulations^{15} have confirmed that the acceleration PDF of heavy particles is indeed slightly narrower than that of fluid tracers.

The dynamics of light particles in turbulence (β = 3) have been investigated both numerically and experimentally. Direct numerical simulations (DNS) in the point-particle limit^{9} showed very high intermittency in the PDFs of the individual forces acting on bubbles. A direct comparison of the statistics of light, neutral, and heavy particles was done by Volk *et al.*^{12} using data from both experiments and point-particle DNS. In their experiments an extended laser Doppler velocimetry technique (extended LDV) was used to measure the particle velocity in a von Kármán flow at high Re_{λ} = 850. The experimental PDFs of the normalized acceleration for light, neutral, and heavy particles did not reveal a visible difference within the experimental accuracy. In contrast, their numerical results showed that the acceleration PDF of light particles was more intermittent than that of tracers, and heavy particles showed less intermittency than tracers. They also found both in numerics and experiments, that the acceleration autocorrelation function of light particles decorrelates faster than those of neutrally buoyant and heavy particles.

In this work, we present an experimental study of the Lagrangian dynamics of light particles in turbulence. A majority of the previous Lagrangian particle tracking experiments focused mainly on tracer particles. Furthermore, grid-generated turbulence experiments in wind tunnels have dealt with either heavy or neutrally buoyant particles.^{3,4,11} Previous experiments with bubbles^{13} measured only one component of the velocity and acceleration in a von Kármán flow. Here, we provide results on the three components of the velocity, acceleration, and autocorrelation statistics of micro-bubbles (β = 3) in homogeneous and isotropic turbulence. The micro-bubbles are dispersed in a turbulent water tunnel at moderate Re_{λ} (160–265), and their size is comparable to the Kolmogorov length scale. In the present experiments, the particle size ratio Φ = *D*/η, where *D* is the particle diameter, and η is the Kolmogorov length scale, is in the range Φ = 0.8–1.9. The effect of finite-particle size on the Lagrangian acceleration statistics has been the subject of many recent investigations (see Refs. 1 and 16–18). It has been found that the acceleration statistics do not show significant changes: in experiments^{1,18} for Φ < 5, and in numerics^{17} for Φ < 2. In the present study, the largest Φ value is ∼1.9, and thus we conclude that finite-size effects do not play a significant role.

## II. EXPERIMENTS AND DATA ANALYSIS

### A. Experimental setup

We conduct experiments in the Twente Water Tunnel, an 8 m long vertical water tunnel designed for studying two-phase flows (see Figure 1). By means of an active grid, nearly homogeneous and isotropic turbulence with Re_{λ} up to 300 is achieved (see Refs. 19,20). A measurement section with dimensions 2 × 0.45 × 0.45 m^{3} with three glass walls provides optical access for the three-dimensional particle tracking velocimetry (PTV) system. Micro-bubbles with a mean diameter *d*_{b} = 340 ± 120 μm are generated by blowing pressurized air through a ceramic porous plate that is located in the upper part of the water tunnel. These micro-bubbles are advected downwards by the flow and pass through the measurement section.

Our three-dimensional PTV system consists of four Photron PCI-1024 high-speed cameras that are synchronized with a high-energy (100 W), high-repetition rate (up to 10 kHz) Litron laser (LDY303HE). The four cameras are focused at the center of the test section on a 40 × 40 × 40 mm^{3} measurement volume that is illuminated by expanding the laser beam with volume optics. The arrangement of the cameras and laser is such that the four cameras receive forward scattered light from the micro-bubbles. We acquire images at 10 000 frames per second (fps) with a resolution of 256 × 256 pixels, resulting in a spatial resolution of about 156 μm/pixel.

The Re_{λ} is varied by changing the mean flow speed of water in the tunnel. Table I summarizes the flow properties for the various cases considered. The flow was characterized by measurements using a cylindrical hot-film probe (Dantec R11) with a sampling rate of 10 kHz placed in the center of the imaged measurement volume. The dissipation rate ε was obtained from the Kolmogorov scaling for the second-order longitudinal structure function *D*_{LL} = *C*_{2}(ε*r*)^{2/3}, with *C*_{2} = 2.13.^{21} For each case of mean flow speed in the water tunnel, the dissipation rate is obtained from the value of the plateau region (see Figure 2), and other parameters follow.

V_{mean}
. | . | u_{rms}
. | η . | τ_{η}
. | L
. | λ . | ε . | . | . |
---|---|---|---|---|---|---|---|---|---|

m s^{−1}
. | Re_{λ}
. | m s^{−1}
. | μm . | ms . | mm . | mm . | m^{2} s^{−3}
. | St
. | N_{data}
. |

0.22 | 160 | 0.0161 | 400 | 160 | 64 | 9.9 | 39e-6 | 0.02 | 5.5 · 10^{6} |

0.33 | 175 | 0.022 | 300 | 90 | 54 | 7.8 | 123e-6 | 0.04 | 9.4 · 10^{6} |

0.45 | 195 | 0.027 | 250 | 65 | 56 | 7.0 | 237e-6 | 0.05 | 8.3 · 10^{6} |

0.57 | 225 | 0.035 | 210 | 47 | 58 | 6.4 | 450e-6 | 0.07 | 6.5 · 10^{6} |

0.65 | 265 | 0.043 | 180 | 35 | 70 | 6.0 | 786e-6 | 0.09 | 4.5 · 10^{6} |

V_{mean}
. | . | u_{rms}
. | η . | τ_{η}
. | L
. | λ . | ε . | . | . |
---|---|---|---|---|---|---|---|---|---|

m s^{−1}
. | Re_{λ}
. | m s^{−1}
. | μm . | ms . | mm . | mm . | m^{2} s^{−3}
. | St
. | N_{data}
. |

0.22 | 160 | 0.0161 | 400 | 160 | 64 | 9.9 | 39e-6 | 0.02 | 5.5 · 10^{6} |

0.33 | 175 | 0.022 | 300 | 90 | 54 | 7.8 | 123e-6 | 0.04 | 9.4 · 10^{6} |

0.45 | 195 | 0.027 | 250 | 65 | 56 | 7.0 | 237e-6 | 0.05 | 8.3 · 10^{6} |

0.57 | 225 | 0.035 | 210 | 47 | 58 | 6.4 | 450e-6 | 0.07 | 6.5 · 10^{6} |

0.65 | 265 | 0.043 | 180 | 35 | 70 | 6.0 | 786e-6 | 0.09 | 4.5 · 10^{6} |

For the three-dimensional particle tracking, we use the open source code developed at the If U-ETH group.^{22} The error in the determination of the particle's position is within sub-pixel accuracy of 60 μm, corresponding to the tolerance of epipolar matching in three dimensions. In this paper we focus on the Lagrangian statistics of micro-bubbles, but it is also possible to study particle clustering using the three-dimensional data (see Ref. 23). Here, the raw particle trajectory is smoothed out with a polynomial fitting method (see Sec. II B). Velocities and accelerations are obtained by differentiating the particle positions in the filtered trajectory. For the Lagrangian statistics shown in the results, the number of data points (*N*_{data}) used are larger than 4.5 × 10^{6}.

### B. Smoothing method for particle trajectories

Experimental errors in the determination of the particle positions are unavoidable, and obtaining the particles’ velocity and acceleration through a time differentiation of their positions would be very sensitive to these experimental errors. Hence, a smoothing of the particle trajectory has to be carried out. This smoothing process is a trade-off between filtering out the experimental noise and retaining the turbulent features of the particle motion. Therefore, the smoothing parameters must be very carefully selected.

In the Lagrangian particle tracking community, two different methods have been used to smoothen particle trajectories in turbulent flows. One method consists of fitting the trajectory to a polynomial of second or higher order,^{1,24} whereas the other method uses a Gaussian kernel.^{4,6,16} We tested both smoothing methods, and obtained very similar results. In the present paper, we only show results obtained by smoothing the micro-bubble trajectories with a third-order polynomial (also referred as the moving cubic spline method).

The entire signal of the trajectory *x*(*t*) is low-pass filtered by fitting a third-order polynomial. Using a fitting window with the particle positions from *t* − *Ndt* until *t* + *Ndt*, where *dt* is the timestep, the filtered particle position at time *i* is calculated as follows:

The Lagrangian velocity and acceleration are obtained by differentiating the particle trajectory

The parameter *N* determines the length of the time window (*t* − *Ndt*, *t* + *Ndt*), and has to be appropriately chosen to ensure that the time fitting window is smaller than the typical turbulent time scale. We explore the effect of *N* on the root mean square (rms) of the micro-bubble velocity (Figure 3(a)) and acceleration (Figure 3(b)) to find the optimum value for the case of Re_{λ} = 195. One can observe in Figure 3(a) that the rms of the velocity saturates at around *N* = 40, for smaller values of *N* there is an exponential rise owing to the noise. Since the acceleration is a second-order derived quantity, it is more sensitive to the choice of *N* as observed in Figure 3(b). Here, we can clearly distinguish two different regions: for small values of *N* < 30 the rms again increases exponentially due to the noise, while at large values of *N* > 100 the *a*_{rms} reduces considerably as an effect of the over-smoothing. For the data presented in this work we have chosen values of *N* in the range 45–50, which correspond to normalized values of *N*/fps × τ_{η} in the range 0.03–0.14. It is important to point out that the normalized acceleration PDFs obtained by choosing *N* in this range (45–50) are similar for each Re_{λ}. The flatness of the acceleration PDF can also provide a measure to identify the optimal value of *N* (as shown in Figure 3(c)). We describe the details of the flatness calculation procedure in Sec. III C. Here, in Figure 3(c), we see that our chosen optimal value of *N* = 50 corresponds to the starting point in a region where the flatness values are decreasing with *N* as a result of over-smoothing. At *N* < 10, the decrease in flatness is an artificial effect arising from the noise. Hence, the optimal value of *N* is chosen such that we do not over-smoothen the micro-bubble trajectories.

## III. RESULTS

### A. PDFs of micro-bubble velocity

In this section we present results on the PDFs of micro-bubble velocity obtained by smoothing the raw trajectories. Figure 4(a) shows the PDF of the three components of the normalized micro-bubble velocity at Re_{λ} = 195. We observe that the velocity distributions of the three components closely follow a Gaussian profile. The flatness values (*F *) of these PDFs for different Re_{λ} (see Table II) are close to that of a Gaussian distribution (*F * = 3). Gaussian-type flatness values have also been reported for neutrally buoyant particles in turbulent von Kármán flows. Voth *et al.*^{1} measured velocity distributions close to Gaussian with flatness values in the range 2.8−3.2, and more recently, Volk *et al.*^{16} obtained sub-Gaussian distributions with flatness around 2.4−2.6. In Figure 5(a), we show a plot of the rms values of the micro-bubble velocity versus Re_{λ}. We observe an increasing trend with Re_{λ}, and the three components are nearly isotropic. We also compare the rms values obtained from the hot-film data and find a reasonable agreement with the 3D-PTV velocity measurements.

Re_{λ}
. | v_{x}
. | v_{y}
. | v_{z}
. |
---|---|---|---|

160 | 3.19 ± 0.13 | 3.19 ± 0.35 | 2.79 ± 0.09 |

175 | 3.09 ± 0.05 | 3.08 ± 0.33 | 2.96 ± 0.14 |

195 | 3.12 ± 0.02 | 3.07 ± 0.06 | 2.87 ± 0.10 |

225 | 3.04 ± 0.04 | 3.04 ± 0.09 | 3.25 ± 0.16 |

265 | 2.96 ± 0.05 | 3.01 ± 0.02 | 2.94 ± 0.06 |

Re_{λ}
. | v_{x}
. | v_{y}
. | v_{z}
. |
---|---|---|---|

160 | 3.19 ± 0.13 | 3.19 ± 0.35 | 2.79 ± 0.09 |

175 | 3.09 ± 0.05 | 3.08 ± 0.33 | 2.96 ± 0.14 |

195 | 3.12 ± 0.02 | 3.07 ± 0.06 | 2.87 ± 0.10 |

225 | 3.04 ± 0.04 | 3.04 ± 0.09 | 3.25 ± 0.16 |

265 | 2.96 ± 0.05 | 3.01 ± 0.02 | 2.94 ± 0.06 |

### B. PDFs of micro-bubble acceleration

Contrary to the velocity PDFs, the micro-bubble acceleration PDFs normalized with the rms (a/a_{rms}) exhibit a strong non-Gaussian behavior. Figure 4(b) shows the PDFs for all the three components of the micro-bubble acceleration at Re_{λ} = 195. We observe that the acceleration PDFs are highly intermittent with stretched tails that extend beyond 5 a_{rms}, indicating that the probability of rare high acceleration events is much higher than for a Gaussian distribution. At this Re_{λ}(= 195), the acceleration is nearly isotropic as the PDFs of the three components show a good collapse at least till 5 a_{rms}. We have observed the same trend for the higher Re_{λ}, whereas for smaller Re_{λ}, the components of the acceleration (a_{x}, a_{y}) in the plane perpendicular to the mean flow direction are not yet isotropic. These components have tails that are slightly narrower than the vertical component (a_{z}). The flow in the Twente water tunnel is not fully isotropic, as has been discussed in Poorte and Biesheuvel.^{25} This slight anisotropy is visible in the PDFs, also in the *a*_{x} component in Figure 4(b). In Figure 5(b), we show the dependence of the acceleration rms values on Re_{λ}. Again, there is an increasing trend with Re_{λ} and a visible anisotropy in the three components. In the discussion that follows, we will only present results of the vertical component *z* (mean flow direction) of the acceleration PDF.

In Figure 6, we present the PDFs of the micro-bubble acceleration for all the Re_{λ} covered in the present study. In order to improve the statistics of the data and obtain a better convergence around the tails, we show the positive part of the averaged PDF, i.e., (*f *(*a*_{z}) + *f *(−*a*_{z}))/2, where *f *(*a*_{z}) is the PDF of the vertical component acceleration (*a*_{z}). Although the rms of acceleration increases with Re_{λ} (Figure 5(b)), the acceleration PDFs, normalized by the rms, collapse on top of each other for all Re_{λ} (see Figure 6). Here, we cannot see a clear dependence on Re_{λ}, but the flatness of these acceleration PDFs better reveals the dependence, and will be discussed in Sec. III C.

First, we compare our micro-bubble results with experimental data of heavy^{4} and tracer particles^{3} under similar flow conditions (grid-generated turbulence with a mean flow). Ayyalasomayajula *et al.*^{4} conducted experiments with heavy particles (water droplets) in a wind tunnel at St = 0.15 and Re_{λ} = 250, the corresponding results are shown in Figure 6(a). Subsequently, measurements of tracer particles were also made in the same facility with St = 0.01 and Re_{λ} = 250 (Ref. 3) (also shown in Figure 6(a)). We observe that the present micro-bubble acceleration PDF shows a higher intermittency than heavy and tracer particles at similar Re_{λ} and St.

In Figure 6(b), we compare the present data with experiments carried out in von Kármán flows. It is important to note the differences in the flow conditions between grid-generated turbulence and the turbulence generated in between counter-rotating disks. It is known that von Kármán flows have a large-scale anisotropy. Secondly, the confinement conditions are different. These differences could affect the Lagrangian dynamics.^{11} Tracer particles in von Kármán flows at Re_{λ} = 140 to 690 (Refs. 1 and 6) show a good agreement with our micro-bubble results for |*a*_{z}/*a*_{z, rms}| ≲ 15 (see the Mordant *et al.*^{6} fit in Figure 6(b)). But beyond this value our micro-bubble acceleration PDFs are slightly less intermittent. Furthermore, a comparison with the experiments of Volk *et al.*^{13} who measured micro-bubble acceleration in von Kármán flow at Re_{λ} = 850 and St = 1.85 is presented in Figure 6(b). Unexpectedly, there is a good agreement between the two experiments with very different Re_{λ}, St, and flow conditions. This could just be a coincidence that both the results agree despite these differences. To arrive at a final conclusion on this issue, more systematic experiments need to be carried out in a wider Re_{λ}– St parameter space under our experimental conditions.

Figure 6(c) shows the comparison between the present micro-bubble acceleration results with the DNS data for point-like bubbles and tracers in homogeneous and isotropic turbulence at a similar Re_{λ} = 180 and St = 0.1 (data obtained from iCFDdatabase, http://cfd.cineca.it). The simulations considered one-way coupled point particles within a periodic cubic box of size *L* = 2π and with a spatial resolution of 512^{3} (for further details on the simulation see Ref. 15). We observe that our experimental findings agree with both numerical bubbles and tracers when |*a*_{z}/*a*_{z, rms}| ≲ 15 within experimental error. The experimental PDF is closer to the numerical tracers for |*a*_{z}/*a*_{z, rms}| ≳ 15. A possible reason for the better agreement between experimental micro-bubbles and DNS tracers could be the small St numbers O(0.01) in the present study. Another possible reason is the different flow conditions, in the experiments there is a strong mean flow which is absent in the numerics. In addition, several factors such as the lift force, buoyancy forces, and particle–particle interactions are ignored in the DNS.^{26} We emphasize again that in the present work, the micro-bubble size is comparable to the Kolmogorov scale, hence, we do not expect the finite-size effects^{17} to play an important role.

### C. Flatness of the micro-bubble acceleration

In order to quantify the intermittency of the acceleration PDFs, statistical convergence of the data is necessary. The number of data points needed for this convergence is crucial, and previous studies have shown that it should at least be ≈ O(10^{6}) (see Refs. 1 and 4). As shown in Table I, our measurements consist of at least 4.5 × 10^{6} data points.

The intermittency of the PDFs of the micro-bubble acceleration can be quantified by studying the flatness F:

where μ_{4} is the fourth moment and σ the rms of the distribution. The flatness being a fourth-order moment is strongly determined by the tails of the distribution, and hence convergence of the PDFs is required. Even though the number of data points used to calculate the PDFs in the present work is larger than O(10^{6}), full convergence has not yet been achieved to calculate directly the flatness from the distribution itself (the largest experimental datasets consist of ≈O(10^{8}) data points,^{6} whereas for numerics this value can go up to ≈O(10^{9}) (Ref. 8). Consequently, we fit the experimental PDF to a stretched exponential distribution^{6,27} defined as

In Eq. (5), *x* = a/a_{rms} is the normalized micro-bubble acceleration, the fitting parameters are α, β, and γ and *C* is a normalization constant. For this fitting procedure and in order to improve the convergence at the tails, we consider the positive part of the averaged PDF, as mentioned above in Sec. III B.

Figure 7(a) shows the result of the fitting for the micro-bubble acceleration PDF at Re_{λ} = 195. The stretched exponential fits the experimental PDF quite well because the three fitting parameters enable a fine adjustment. In the inset of Figure 7(a), we plot the fourth-order moment (a/a_{rms})^{4} PDF(a/a_{rms}) for the experimental acceleration measurement along with the fitted curve. This type of curve allows for a good convergence test.^{28} At the tails of the distribution, convergence is nearly achieved, and the fitted curve nicely sits on top of the experimental data. We have observed a similar behavior for the other measurements at different Re_{λ}.

Next, we calculate the flatness of the fitted acceleration PDFs as a function of Re_{λ}, as shown in Figure 7(b). The flatness is determined directly from the fitted stretched exponential functional for all Re_{λ}. The error bars are obtained by finding the difference between the flatness values for half and the entire acceleration data points. Figure 7(c) shows that the flatness values of the micro-bubble acceleration PDFs increase in the Re_{λ} range 160–225 consistent with the experimental results of Voth *et al.*^{1} for tracer particles and the numerical results of Ishihara *et al.*^{2} for fluid particles. For the highest Re_{λ}, we have less statistics compared to the other cases as the mean flow speed is the fastest. This might be the reason for the decrease (underestimation) in the flatness value of our data point at Re_{λ} = 265. Clearly, from the collection of all data (Figure 7(c)) one would not expect such a decrease.

### D. Autocorrelation functions

We now present results on the Lagrangian autocorrelation function of the micro-bubble acceleration. In Figure 8 we compare the autocorrelation for the three components of the acceleration at Re_{λ} = 195, using a time lag normalized with τ_{η}. We find that the three acceleration components correlate in a similar manner. This nearly isotropic behavior was also found for the other measurements at different Re_{λ}.

Figure 9(a) shows the autocorrelation of a_{z} for different *Re*_{λ}. It is clear that the microbubble's acceleration correlates for longer times as Re_{λ} increases. The acceleration autocorrelation function drops to zero rapidly, and the zero-crossing point has small values: < 0.1τ_{η}. Voth *et al.*^{1} and Mordant *et al.*^{7} reported values of around 2.2τ_{η} in their experiments with tracers in von Kármán flows at high turbulence intensities (Re_{λ} > 690). The value of 2.2τ_{η} was first found from DNS by Yeung *et al.*^{29} Volk *et al.*^{12} performed both micro-bubble and tracer experiments in a von Kármán apparatus, and found that the decorrelation of the microbubbles is smaller than that of tracers at a given Re_{λ}. We do not yet know the exact reason for the large disparity between 2.2τ_{η} for the fluid particles compared to 0.1τ_{η} for the present micro-bubbles. One possible reason is that our flow conditions are different as we have a strong mean flow.

We study the time at which the autocorrelation function drops to zero for different Re_{λ} by defining the decorrelation time as

where *C*_{a} is the acceleration autocorrelation function. *T*_{D} represents the characteristic time for the evolution of the micro-bubble response to changes in the flow conditions. Figure 9(b) shows the dependence of *T*_{D}/τ_{η} on Re_{λ} for the three components of the micro-bubble acceleration. We observe that *T*_{D}/τ_{η} increases with Re_{λ}, and that the autocorrelation functions are nearly isotropic as evidenced by the very similar *T*_{D}/τ_{η} values for the different components. In the inset of Figure 9(b), the decorrelation time *T*_{D}/τ_{η} as obtained by Volk *et al.*^{12} at Re_{λ} = 850 agrees well with our increasing trend of the decorrelation time with Re_{λ}. We fit our experimental data of the decorrelation time of a_{z} to a linear relation *T*_{D}/τ_{η} = 0.00038 Re_{λ} − 0.051 that is shown in the inset of Figure 9(b) as a solid line. Evaluating these relations at Re_{λ} = 850 gives *T*_{D}/τ_{η} = 0.27, which agrees well with their experimental value of *T*_{D}/τ_{η} = 0.258.^{12} More experiments are needed to fill the gap of Re_{λ}. Very recently, Volk *et al.*^{16} found an increase of *T*_{D}/τ_{η} with Re_{λ} for a fixed particle size, just as we find in our micro-bubble experiments.

## IV. CONCLUSION

We have presented experimental results on the Lagrangian statistics of micro-bubble velocity and acceleration in homogeneous isotropic turbulence. Three-dimensional PTV was employed to obtain the micro-bubble trajectories. The PDFs of micro-bubble velocity closely follow a Gaussian distribution with flatness *F * ≈ 3, independent of Re_{λ}. But the acceleration PDFs are highly non-Gaussian with intermittent tails. Although the acceleration PDFs themselves do not show a clear dependence on Re_{λ}, the flatness values reveal a clear trend. We fit the experimental acceleration PDFs to a stretched exponential function and estimate the flatness based on the fitting. The flatness values were found to be in the range of 23–30 and show an increasing trend with Re_{λ}. This trend is consistent with previous experimental^{1} and numerical^{2} results.

A comparison of our results with experiments in von Kármán flows^{1,6,12,13} suggest that the present micro-bubble acceleration PDF is similar to tracers and bubbles (in von Kármán flows) for very different *Re*_{λ}. However, there are significant differences in the flow conditions between the two experimental systems. Therefore, it is more relevant to compare our results with previous investigations in similar flow conditions, i.e., grid-generated turbulence. We find that the acceleration PDFs of our micro-bubbles are more intermittent as compared to heavy and tracer particles in wind tunnel experiments at similar St and Re_{λ}.^{3,4}

Compared to DNS simulations in the point particle limit, our micro-bubble acceleration PDFs show a reasonable agreement with both numerical tracers and bubbles, but in the tails our data has a better match with numerical tracers. One possible reason is the differences in flow conditions between the experiments and numerics. Another possibility is that the St in our experiments are small (0.02–0.09). It will be interesting to study the acceleration statistics of finite-sized bubbles at large St in turbulent flows.

We also calculate the autocorrelation function of the micro-bubble acceleration, and observed that the decorrelation time increases with Re_{λ}. This finding is consistent with other experimental investigations^{12,16} at very high Reynolds number. More experimental data are needed to fill the gap of Re_{λ} in order to further study the scaling behavior.

## ACKNOWLEDGMENTS

We thank Enrico Calzavarini, Daniel Chehata Gómez, Beat Lüthi, Federico Toschi, and Romain Volk for useful discussions. This work was supported by the Foundation for Fundamental Research on Matter (FOM) and industrial partners through the FOM-IPP Industrial Partnership Program: *Fundamentals of heterogeneous bubbly flows*. We also acknowledge support from the European Cooperation in Science and Technology (COST) Action MP0806: *Particles in Turbulence*. The source of the DNS data was from the ICTR-iCFDdatabase (http://cfd.cineca.it). Finally, we thank Gert-Wim Bruggert, Martin Bos, and Bas Benschop for assistance with the experimental setup.