^{1}, J. C. Serrano-García

^{2}, R. Zenit

^{2,a)}and J. A. Hernández-Cordero

^{2}

### Abstract

An experimental study was carried out to determinate the power spectral density (PSD) of mono-dispersed bubbly flows in a vertical channel using flying hot-film anemometry. To improve bubble detection, optical fibers were installed in close proximity to the anemometer sensing element; in this way, the collisions of bubbles with the probe can be detected and removed from the signal. Measurements were performed with gas fractions up to 6%. The PSD distributions were found to decay with a power of −3, in agreement with previous studies, but for a much wider range of Reynolds and Weber numbers. Our measurements indicate that the power decay does not depend strongly on the nature of hydrodynamic interactions among bubbles.

S.M.-D. acknowledges the support of the DGAPA-UNAM through its postdoctoral scholarship program. J.C.S.-G. is grateful to DGEP-UNAM for financially supporting his graduate studies.

I. INTRODUCTION

II. EXPERIMENTAL ARRAY AND MEASUREMENT TECHNIQUES

A. Single bubble measurements

B. Measurement of the bubble velocity

C. Measurement of the liquid velocity

1. Optical fibers as phase identifiers

D. Liquid velocity signal processing

III. RESULTS

A. Liquid velocity variance

B. Mean fluid spectra

C. Characteristic length scales of fluctuating motion

D. Scaling of fluid spectra

IV. FINAL COMMENTS AND CONCLUSIONS

### Key Topics

- Turbulent flows
- 29.0
- Optical fibers
- 18.0
- Position sensitive detectors
- 14.0
- Velocimetry
- 12.0
- Velocity measurement
- 12.0

##### G01F

## Figures

Terminal velocity of single bubbles as a function of equivalent diameter. The symbols denote experiments conducted in different liquids: (○), water; (⋄) water-glycerin 30%; (□) water-glycerin 50%. The filled and empty symbols refer to small and large bubbles, respectively. The lines show the predictions of Ref. 18 for the three liquids used here: (–), Mo = 2.6 × 10−8; (- - -), Mo = 3.9 × 10−6; (-·- ·), Mo = 3.1 × 10−5.

Terminal velocity of single bubbles as a function of equivalent diameter. The symbols denote experiments conducted in different liquids: (○), water; (⋄) water-glycerin 30%; (□) water-glycerin 50%. The filled and empty symbols refer to small and large bubbles, respectively. The lines show the predictions of Ref. 18 for the three liquids used here: (–), Mo = 2.6 × 10−8; (- - -), Mo = 3.9 × 10−6; (-·- ·), Mo = 3.1 × 10−5.

Optical fiber setup.

Optical fiber setup.

Bubble collision sequence with both hot-film probe and optical fiber tip. For clarity, in these images the optical probes are shown in the side of the hot-film sensor.

Bubble collision sequence with both hot-film probe and optical fiber tip. For clarity, in these images the optical probes are shown in the side of the hot-film sensor.

Effect of the signal length on the shape of the PSD. The signal length, n, is defined by the number of data points of the total sample, acquired at 15 000 samples per second. Each line represents the average power spectral distribution obtained for different values of n.

Effect of the signal length on the shape of the PSD. The signal length, n, is defined by the number of data points of the total sample, acquired at 15 000 samples per second. Each line represents the average power spectral distribution obtained for different values of n.

(a) Power spectral density: Spectral distribution obtained with and without length validation criteria, thick and thin lines, respectively. (b) Power decay: Rate of decay of the PSD, n PSD , as a function of frequency with and without length validation criteria, thick and thin lines respectively. The horizontal lines show the classical −3 (dashed line) and −5/3 (dashed-dotted line) values of n. For the case shown here d b = 4.7, α = 5%, Re = 342, and We = 3.2.

(a) Power spectral density: Spectral distribution obtained with and without length validation criteria, thick and thin lines, respectively. (b) Power decay: Rate of decay of the PSD, n PSD , as a function of frequency with and without length validation criteria, thick and thin lines respectively. The horizontal lines show the classical −3 (dashed line) and −5/3 (dashed-dotted line) values of n. For the case shown here d b = 4.7, α = 5%, Re = 342, and We = 3.2.

Typical mono-dispersed bubbly flows. Images obtained for different liquids for a gas volume fraction of α = 0.02. The top and bottom rows show images obtained for the small and large capillary array, respectively. Water, (a) and (d); water-glycerin (70-30), (b) and (e); water-glycerin (50-50), (c) and (f). The size of each image is approximately 5 × 5 cm2.

Typical mono-dispersed bubbly flows. Images obtained for different liquids for a gas volume fraction of α = 0.02. The top and bottom rows show images obtained for the small and large capillary array, respectively. Water, (a) and (d); water-glycerin (70-30), (b) and (e); water-glycerin (50-50), (c) and (f). The size of each image is approximately 5 × 5 cm2.

Normalized bubble velocity as a function of gas volume fraction, for monodisperse flows. The symbols denote experiments conducted in different liquids: (○), water; (⋄) water-glycerin 30%; (□) water-glycerin 50%. The filled and empty symbols refer to small and large bubbles, respectively. The dashed and dashed-dotted lines represents the fits from Eqs. (6) and (7) , respectively; the thick and thin lines show the fits for the highest (A = 0.49 and B = 0.1) and smallest Reynolds number experiments (A = 0.25 and B = 0.15), respectively. For clarity, only one set of data shows typical error bars.

Normalized bubble velocity as a function of gas volume fraction, for monodisperse flows. The symbols denote experiments conducted in different liquids: (○), water; (⋄) water-glycerin 30%; (□) water-glycerin 50%. The filled and empty symbols refer to small and large bubbles, respectively. The dashed and dashed-dotted lines represents the fits from Eqs. (6) and (7) , respectively; the thick and thin lines show the fits for the highest (A = 0.49 and B = 0.1) and smallest Reynolds number experiments (A = 0.25 and B = 0.15), respectively. For clarity, only one set of data shows typical error bars.

Normalized liquid velocity variance as a function of gas volume fraction, α. All symbols are the same as in Fig. 7 . The (+) and (×) symbols show experiments from Martinez-Mercado et al., 6 for which (Re,We) are (500,1.7) and (200,2.0), respectively. The lines show trends of the form T l ∼ α C : continuous line, C = 0.4; and dashed line, C = 1.0. For clarity, only the ( )-data show error bars.

Normalized liquid velocity variance as a function of gas volume fraction, α. All symbols are the same as in Fig. 7 . The (+) and (×) symbols show experiments from Martinez-Mercado et al., 6 for which (Re,We) are (500,1.7) and (200,2.0), respectively. The lines show trends of the form T l ∼ α C : continuous line, C = 0.4; and dashed line, C = 1.0. For clarity, only the ( )-data show error bars.

Variation of PSD with gas volume fraction. (a) Power spectral density: spectral distribution; (b) power decay: rate of decay of the PSD, n PSD as a function of frequency. The horizontal lines show the classical −3 (dashed line) and −5/3 (dashed-dotted line) values of n PSD . The liquid used in this case was the 70-30 water-glycerin solution; the bubble size is d b ≈ 3.4 mm.

Variation of PSD with gas volume fraction. (a) Power spectral density: spectral distribution; (b) power decay: rate of decay of the PSD, n PSD as a function of frequency. The horizontal lines show the classical −3 (dashed line) and −5/3 (dashed-dotted line) values of n PSD . The liquid used in this case was the 70-30 water-glycerin solution; the bubble size is d b ≈ 3.4 mm.

Variation of PSD with liquid viscosity and bubble diameter. (a) Power spectral density: spectral distribution; (b) power decay: rate of decay of the PSD, n PSD . The horizontal lines show the classical −3 (dashed line) and −5/3 (dashed-dotted line) values of n. A gas volume fraction of α = 0.02 was used for all the cases.

Variation of PSD with liquid viscosity and bubble diameter. (a) Power spectral density: spectral distribution; (b) power decay: rate of decay of the PSD, n PSD . The horizontal lines show the classical −3 (dashed line) and −5/3 (dashed-dotted line) values of n. A gas volume fraction of α = 0.02 was used for all the cases.

PDF of liquid velocity for all cases tested here. The black, blue and red lines show the results for water, W-G 70-30 and W-G 50-50 mixtures, respectively. The continuous and dashed lines show the results for small and large bubbles, respectively. (a) PSD and (b) power decay. The green thick line show a typical result from Ref. 14 .

PDF of liquid velocity for all cases tested here. The black, blue and red lines show the results for water, W-G 70-30 and W-G 50-50 mixtures, respectively. The continuous and dashed lines show the results for small and large bubbles, respectively. (a) PSD and (b) power decay. The green thick line show a typical result from Ref. 14 .

Normalized PDF of liquid velocity for all cases tested here. The black, blue, and red lines show the results for water, W-G 70-30 and W-G 50-50 mixtures, respectively. The continuous and dashed lines show the results for small and large bubbles, respectively. (a) and (b) show normalization proposed by Ref. 13 (Eqs. (12) and (13)) and that proposed here (Eqs. (14) and (15) ).

Normalized PDF of liquid velocity for all cases tested here. The black, blue, and red lines show the results for water, W-G 70-30 and W-G 50-50 mixtures, respectively. The continuous and dashed lines show the results for small and large bubbles, respectively. (a) and (b) show normalization proposed by Ref. 13 (Eqs. (12) and (13)) and that proposed here (Eqs. (14) and (15) ).

## Tables

Physical properties of the test liquids: water and water-glycerin (W-G) mixtures. Morton number: Mo = gμ4/ρσ3.

Physical properties of the test liquids: water and water-glycerin (W-G) mixtures. Morton number: Mo = gμ4/ρσ3.

Experimental results for isolated bubbles. For each liquid, two bubbles sizes were produced (two capillary sizes). The letters “O” and “S” correspond to the type of trajectory observed for each bubble, oscillating or straight, respectively. The Reynolds and Weber numbers are defined as Re = ρU b d eq /μ, We = , respectively. The superscript “0” refers to isolated bubble conditions.

Experimental results for isolated bubbles. For each liquid, two bubbles sizes were produced (two capillary sizes). The letters “O” and “S” correspond to the type of trajectory observed for each bubble, oscillating or straight, respectively. The Reynolds and Weber numbers are defined as Re = ρU b d eq /μ, We = , respectively. The superscript “0” refers to isolated bubble conditions.

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