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Abstract
We study the Lagrangian velocity and acceleration statistics of light particles (microbubbles in water) in homogeneous isotropic turbulence. Microbubbles 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 TaylorReynolds numbers (Reλ) ranging from 160 to 265. We reconstruct the bubble trajectories by employing threedimensional particle tracking velocimetry. It is found that the probability density functions (PDFs) of the microbubble acceleration show a highly nonGaussian behavior with flatness values in the range 23 to 30. The acceleration flatness values show an increasing trend with Reλ, consistent with previous experiments [G. Voth, A. La Porta, A. M. Crawford, J. Alexander, and E. Bodenschatz, “Measurement of particle accelerations in fully developed turbulence,” J. Fluid Mech.469, 121 (2002)] and numerics [T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura, and A. Uno, “Smallscale statistics in highresolution direct numerical simulation of turbulence: Reynolds number dependence of onepoint velocity gradient statistics,” J. Fluid Mech.592, 335 (2007)]. These acceleration PDFs show a higher intermittency compared to tracers [S. Ayyalasomayajula, Z. Warhaft, and L. R. Collins, “Modeling inertial particle acceleration statistics in isotropic turbulence,” Phys. Fluids.20, 095104 (2008)] and heavy particles [S. Ayyalasomayajula, A. Gylfason, L. R. Collins, E. Bodenschatz, and Z. Warhaft, “Lagrangian measurements of inertial particle accelerations in grid generated wind tunnel turbulence,” Phys. Rev. Lett.97, 144507 (2006)] in wind tunnel experiments. In addition, the microbubble acceleration autocorrelation function decorrelates slower with increasing Reλ. We also compare our results with experiments in von Kármán flows and pointparticle direct numerical simulations with periodic boundary conditions.
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 FOMIPP 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 ICTRiCFDdatabase (http://cfd.cineca.it). Finally, we thank GertWim Bruggert, Martin Bos, and Bas Benschop for assistance with the experimental setup.
I. INTRODUCTION
II. EXPERIMENTS AND DATA ANALYSIS
A. Experimental setup
B. Smoothing method for particle trajectories
III. RESULTS
A. PDFs of microbubble velocity
B. PDFs of microbubble acceleration
C. Flatness of the microbubble acceleration
D. Autocorrelation functions
IV. CONCLUSION
Key Topics
 Turbulent flows
 24.0
 Lagrangian mechanics
 15.0
 Isotropic turbulence
 11.0
 Acceleration measurement
 10.0
 Probability density functions
 8.0
B81B
F15D
G01P1/00
Figures
The Twente Water Tunnel: an experimental facility for studying twophase turbulent flows. The picture shows the measurement section and on top the active grid, which allows homogeneous and isotropic turbulent flows upto Reλ = 300, and the 4camera particle tracking velocimetry (PTV) system to detect the positions of particles in threedimensions. For illumination we use a high energy, highrepetition rate laser. Microbubbles with a diameter ≈340 μm are generated above the active grid using a ceramic porous plate and are advected downwards into the measurement volume.
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The Twente Water Tunnel: an experimental facility for studying twophase turbulent flows. The picture shows the measurement section and on top the active grid, which allows homogeneous and isotropic turbulent flows upto Reλ = 300, and the 4camera particle tracking velocimetry (PTV) system to detect the positions of particles in threedimensions. For illumination we use a high energy, highrepetition rate laser. Microbubbles with a diameter ≈340 μm are generated above the active grid using a ceramic porous plate and are advected downwards into the measurement volume.
Flow parameters characterization from hotwire measurements. Compensated secondorder longitudinal structure function D LL (r) calculated in order to estimate dissipation.
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Flow parameters characterization from hotwire measurements. Compensated secondorder longitudinal structure function D LL (r) calculated in order to estimate dissipation.
The rms of the vertical component of the microbubble (a) velocity and (b) acceleration, and (c) the acceleration flatness at Reλ = 195 as a function of N for polynomial smoothing. The arrows in the figures indicate the chosen value (N = 50) for the smoothing of the trajectories.
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The rms of the vertical component of the microbubble (a) velocity and (b) acceleration, and (c) the acceleration flatness at Reλ = 195 as a function of N for polynomial smoothing. The arrows in the figures indicate the chosen value (N = 50) for the smoothing of the trajectories.
(a) PDFs of the three components of microbubble velocity at Reλ = 195. The three velocity component distributions are nearly Gaussian compared to the solid line that represents a Gaussian distribution. (b) PDFs of the three components of the normalized microbubble acceleration at Reλ = 195. The three components of the acceleration are strongly nonGaussian, i.e., the tails of the distribution show high intermittency.
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(a) PDFs of the three components of microbubble velocity at Reλ = 195. The three velocity component distributions are nearly Gaussian compared to the solid line that represents a Gaussian distribution. (b) PDFs of the three components of the normalized microbubble acceleration at Reλ = 195. The three components of the acceleration are strongly nonGaussian, i.e., the tails of the distribution show high intermittency.
(a) The rms values of the three components of the microbubble velocity for all Reλ, compared with the hotfilm probe measurements. (b) The rms values of the three components of the microbubble acceleration for all Reλ.
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(a) The rms values of the three components of the microbubble velocity for all Reλ, compared with the hotfilm probe measurements. (b) The rms values of the three components of the microbubble acceleration for all Reλ.
PDFs of the vertical component of the normalized microbubble acceleration. (a) Comparison with experiments under similar flow conditions (gridgenerated turbulence in a windtunnel) at Reλ = 250. Our results are shown with open symbols; stars are heavy particles ^{4} and black crosses represent tracer particles. ^{3} (b) Comparison with von Kármán flow results: fit for tracers at Reλ = 140–690 (Refs. 1 and 6 ) is the black line; bubbles at Reλ = 850 (Refs. 12 and 13 ) are shown with a blue line. (c) Comparison with DNS simulations for point particles at Reλ = 180 (from iCFDdatabase http://cfd.cineca.it): the red line indicates tracers and the black line bubbles.
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PDFs of the vertical component of the normalized microbubble acceleration. (a) Comparison with experiments under similar flow conditions (gridgenerated turbulence in a windtunnel) at Reλ = 250. Our results are shown with open symbols; stars are heavy particles ^{4} and black crosses represent tracer particles. ^{3} (b) Comparison with von Kármán flow results: fit for tracers at Reλ = 140–690 (Refs. 1 and 6 ) is the black line; bubbles at Reλ = 850 (Refs. 12 and 13 ) are shown with a blue line. (c) Comparison with DNS simulations for point particles at Reλ = 180 (from iCFDdatabase http://cfd.cineca.it): the red line indicates tracers and the black line bubbles.
(a) PDF of the vertical component of the microbubble acceleration at Reλ = 195. Open squares are the experimental data, solid line is the fitted stretched exponential function. The inset shows the plot of the fourthorder moment x ^{4}PDF(x) for experimental data and fit. (b) The flatness value of the fitted PDFs of microbubble acceleration as a function of Reλ. (c) The flatness values versus the Reynolds number. Comparison with Voth et al. ^{1} and Ishihara et al. ^{2} reveals that the present microbubble trend agrees well with the data in the literature, at least till Re λ = 225.
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(a) PDF of the vertical component of the microbubble acceleration at Reλ = 195. Open squares are the experimental data, solid line is the fitted stretched exponential function. The inset shows the plot of the fourthorder moment x ^{4}PDF(x) for experimental data and fit. (b) The flatness value of the fitted PDFs of microbubble acceleration as a function of Reλ. (c) The flatness values versus the Reynolds number. Comparison with Voth et al. ^{1} and Ishihara et al. ^{2} reveals that the present microbubble trend agrees well with the data in the literature, at least till Re λ = 225.
Autocorrelation function of the three components of the microbubble acceleration at Reλ = 195. The acceleration autocorrelation of the microbubbles is nearly isotropic. The time lag is normalized with the Kolmogorov time scale τη.
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Autocorrelation function of the three components of the microbubble acceleration at Reλ = 195. The acceleration autocorrelation of the microbubbles is nearly isotropic. The time lag is normalized with the Kolmogorov time scale τη.
(a) Autocorrelation function of the vertical component of the microbubble acceleration for the different Reλ measured. The correlation of the microbubble acceleration persists longer with increasing Reynolds number. (b) The decorrelation time T D /τη of the autocorrelation function for the three components of the microbubble acceleration as a function of Reλ. The decorrelation time increases with the turbulent intensity. In the inset, we show also the result of Volk et al. ^{12} at a very high Reλ = 850 (⧫), their experimental point agrees with the trend of increasing decorrelation time with turbulent intensity. The linear fit obtained with our experimental data extrapolates a value of T D /τη = 0.27 at Reλ = 850, which is slightly higher than their experimental value of T D /τη = 0.25.
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(a) Autocorrelation function of the vertical component of the microbubble acceleration for the different Reλ measured. The correlation of the microbubble acceleration persists longer with increasing Reynolds number. (b) The decorrelation time T D /τη of the autocorrelation function for the three components of the microbubble acceleration as a function of Reλ. The decorrelation time increases with the turbulent intensity. In the inset, we show also the result of Volk et al. ^{12} at a very high Reλ = 850 (⧫), their experimental point agrees with the trend of increasing decorrelation time with turbulent intensity. The linear fit obtained with our experimental data extrapolates a value of T D /τη = 0.27 at Reλ = 850, which is slightly higher than their experimental value of T D /τη = 0.25.
Tables
Summary of the flow parameters. V mean : water mean flow speed, : TaylorReynolds number, u rms : mean velocity fluctuation, η = (ν^{3}/ε)^{1/4} and τη = (ν/ε)^{1/2}: are the Kolmogorov's length scale and time scale, respectively, L: integral length scale of the flow, ε: mean energy dissipation rate, St = τ p /τη: Stokes number, and N data : number of data points used to calculate the Lagrangian statistics.
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Summary of the flow parameters. V mean : water mean flow speed, : TaylorReynolds number, u rms : mean velocity fluctuation, η = (ν^{3}/ε)^{1/4} and τη = (ν/ε)^{1/2}: are the Kolmogorov's length scale and time scale, respectively, L: integral length scale of the flow, ε: mean energy dissipation rate, St = τ p /τη: Stokes number, and N data : number of data points used to calculate the Lagrangian statistics.
Flatness values of the distribution of microbubble velocities.
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Flatness values of the distribution of microbubble velocities.
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Abstract
We study the Lagrangian velocity and acceleration statistics of light particles (microbubbles in water) in homogeneous isotropic turbulence. Microbubbles 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 TaylorReynolds numbers (Reλ) ranging from 160 to 265. We reconstruct the bubble trajectories by employing threedimensional particle tracking velocimetry. It is found that the probability density functions (PDFs) of the microbubble acceleration show a highly nonGaussian behavior with flatness values in the range 23 to 30. The acceleration flatness values show an increasing trend with Reλ, consistent with previous experiments [G. Voth, A. La Porta, A. M. Crawford, J. Alexander, and E. Bodenschatz, “Measurement of particle accelerations in fully developed turbulence,” J. Fluid Mech.469, 121 (2002)] and numerics [T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura, and A. Uno, “Smallscale statistics in highresolution direct numerical simulation of turbulence: Reynolds number dependence of onepoint velocity gradient statistics,” J. Fluid Mech.592, 335 (2007)]. These acceleration PDFs show a higher intermittency compared to tracers [S. Ayyalasomayajula, Z. Warhaft, and L. R. Collins, “Modeling inertial particle acceleration statistics in isotropic turbulence,” Phys. Fluids.20, 095104 (2008)] and heavy particles [S. Ayyalasomayajula, A. Gylfason, L. R. Collins, E. Bodenschatz, and Z. Warhaft, “Lagrangian measurements of inertial particle accelerations in grid generated wind tunnel turbulence,” Phys. Rev. Lett.97, 144507 (2006)] in wind tunnel experiments. In addition, the microbubble acceleration autocorrelation function decorrelates slower with increasing Reλ. We also compare our results with experiments in von Kármán flows and pointparticle direct numerical simulations with periodic boundary conditions.
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