Hydrodynamics and shape reconstruction of single rising air bubbles in water using high-speed tomographic particle tracking velocimetry and 3D geometric reconstruction
Time-resolved tomographic particle tracking velocimetry (TR-3D-PTV), also called 4D-PTV, is used here to obtain the instantaneous 3D liquid flow field information in the wake of a single rising bubble in water. Simultaneously, the bubble shape, size and velocity are determined by tomographic reconstruction of the 3D bubble shape. Both, tracer particles for PTV and bubbles, are imaged in a shadow mode with background illumination. The Lagrangian method used in this paper, especially combined with the shake the box algorithm, has big advantages compared to particle image velocimetry, in situations, where only low particle per pixel values can be obtained. In this research, single air bubbles of different sizes, with diameters of around 2.4 mm, 4.0 mm, 6.0 mm and 9.6 mm, were injected into stagnant de-ionized water. Their shape was reconstructed in 3D, and an equivalent bubble diameter was determined from this reconstruction. Compared to conventionally used 2D shadow imaging, this diameter is about 13% smaller. The 3D bubble trajectory can be analysed and decomposed into a sinusoidal function curve lateral projection and an ellipsoidal shape vertical projection. As the bubble diameter increases, the radius of the spiral trajectory is decreasing as well as the amplitude of vertical sinusoidal oscillation. The wake structure in the liquid behind the bubbles is also changing with bubble size: from simple vortex pairs for smaller bubbles to an intertwined structure of several twisted vortices for the bigger ones.
Graphical abstract
Three-dimensional bubble reconstruction (grey surface) and liquid stream lines coloured with velocity magnitude around an ascending air bubble in de-ionized water.
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Notes
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1 Introduction
The motion of a buoyancy-driven bubble rising through a liquid has been studied since many centuries and is still an interesting subject for numerous researchers because of its wide range of applications in industry. In engineering applications, bubble column reactors have been widely investigated due to their simple and easy to build structure, while still inducing complex flow pattern that leads to good mass transfer effects. Numerical simulation of such devices is still a complex task, because of the wide scale range, from the individual bubble to the whole column. To cover all these, many different models have to be implemented and validated. Therefore, profound knowledge also of the dynamics of single rising bubbles and the interaction between the gas and liquid phases is crucial to comprehend the fundamental transport processes and optimize the design of bubble columns.
The dynamics of a rising bubble in a stagnant liquid is normally described by the following five non-dimensional parameters: Galilei number (Ga), Bond number (Bo) or also called Eötvös number (Eo), Reynolds number (Re), Weber number (We) and Morton number (Mo). Three types of maps, describing bubble shape regimes, bubble rising behaviour and bubble rising path oscillation regimes, are generally referred: (1) The bubble shape regime map, also called “Grace diagram” (Grace 1973) combining Re, Eo and Mo numbers, depicts the bubble shapes; (2) a modified bubble shape regime map by Tripathi et al. (2015) using Eo and Ga number has the main advantage that it is independent of the bubble velocity and (3) a recently proposed rising bubble oscillation path regime map by Cano-Lozano et al. (2016) selecting Ga and Eo number and showing the bubble rising path types. Researchers have proposed valuable insights on the bubble wake vortex structure in terms of rising path instability (Fernandes et al. 2007; Horowitz and Williamson 2010; Lee and Park 2022; Mougin and Magnaudet 2001). Path instability means here path oscillations of the bubble trajectories laterally following zigzag, flattened spiral or spiral paths. The transition from straight to oscillating trajectories is induced by the symmetry breakage of the wake vortex behind the bubble (Cano-Lozano et al. 2016; Lee and Park 2022; Tripathi et al. 2015). For the pure water and air system, Wu and Gharib (2002) have found that 1.5 mm is the critical bubble size where the axisymmetric standing wake and a straight path exist. This size provides a reference for selecting the bubble diameter for an oscillating path. With the increase in bubble diameter, the vortex structure behind the bubble becomes more asymmetric and complex. This demands higher requirements for measuring methods to fully visualize the 3D dynamic vortex structure in the liquid flow behind the bubble.
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For a zigzag path, a regular production and discharge of alternate oppositely oriented hairpin-like vortex structures is the main reason for the zigzagging motion (Brücker 1999). Besides, the asymmetric shape deformation of the bubbles is induced by the non-uniform differential pressure of the leading and wake vortices. Magnaudet and Mougin (2007) proposed an empirical criterion based on the maximum vorticity generated at the bubble surface, to determine the onset of the flow path instability, after discussing the critical parameters of aspect ratio and Reynolds number. An air bubble with volume-equivalent diameter of 5.2 mm (Re = 598) rising in water was simulated by Gaudlitz and Adams (2009) to show the bubble shape change associated with the shedding of hairpin vortices in the zigzag stage. They concluded that the transition from zigzag to a spiral path was induced by a twisting vortex chain. This transition from zigzag to spiral was also attributed to the asymmetric wake vortex in combination with a variation in bubble surface at the bubble equator (Zhang and Ni 2017). Compared to the zigzag path, the spiral path has an additional orientation oscillation, driven by the combination of torque and a lateral force produced by the 3D wake structure (Mougin and Magnaudet 2006). In the DNS research of Cano-Lozano et al. (2016), the authors deduced that the flattened spiral regime is a transitional state before converging towards a planar zigzagging regime or a spiral regime. Furthermore, the wake vortex structure of this flattened path, combining vortex structures of zigzag and helical (twisted mode) paths, was observed in the simulation results. On each side of the primary vortex, a vortex street of normally four superimposed vortices, having the form of single-side loops, is formed. To validate and compare these results, experimental measurements of such bubble paths are necessary in a temporally and spatially resolved manner. In the interest of the current work, we have referred to a limited number of cases in the domain of the flattened spiral regime with different bubble sizes to show the bubble dynamics and vortex structures in simultaneous experimental measurements of the bubble’s shape, trajectory, velocity and the liquid flow field.
The measuring techniques regarding bubble flow characteristics have developed during the last decades from simple photographic techniques to high-speed tomographic optical measurement methods in the last years. Therein, the most widely used optical measurement techniques in bubbly flows are particle image velocimetry (PIV) and particle tracking velocimetry (PTV), often combined with shadow imaging for the bubble parameters (Kováts et al. 2020; Rzehak et al. 2017; Zähringer and Kováts 2021). Both, PTV and PIV, are non-intrusive velocity measurement methods, which allow to extract the velocity information from recorded images of tracer particles in the flow field. The dynamical behaviour in the wake of a single and a pair of spherical-cap bubbles was investigated at low resolution by Komasawa et al. (1980). PIV and LIF (laser-induced fluorescence) were utilized by Fujiwara et al. (2000) to show the influence of the lift force on the bubble and the motion induced to the surrounding flow field in a 100 × 100 × 1000 mm3 vertical measuring tank. Through a Schlieren optical technique, De Vries et al. (2002) and also recently Veldhuis et al. (2008) visualized the wake vortex structure and motion behind rising bubbles. The progress of camera sensor development (temporal and spatial resolution) and algorithms promoted the further measuring technique development. A PIV system with 1000 fps in a vertical 30-cm height tank was used to examine the streamwise vorticity in the wake of bubbles and reconstruct the vortex structure by Zenit and Magnaudet (2009). The results show that the wake is formed by two counter-rotating vortex tubes of streamwise vorticity, but it becomes not clear how the initial zigzag path evolves to a spiral path flow. With such classical 2D-PIV, e.g. (Rüttinger et al. 2018), velocity information of the out-of-plane component cannot be obtained. Recently, the distinct vortex structures behind rising bubbles in the zigzag path were captured by She et al. (2021) using tomographic PIV with shadow image reconstruction, showing a typical double-threaded and four-ring vortex structure in the generation stage and regular zigzag stages, respectively.
In the current study, the focus is laid on the simultaneous measurement of 4D liquid velocity fields and bubble trajectories as well as 3D bubble shape reconstructions, using 4D particle tracking velocimetry (PTV) and shadow image reconstruction of the bubble shape. The velocity results, obtained with the Shake-the-Box (STB) algorithm (Schanz et al. 2016; Schneiders and Scarano 2016), are transferred to the Eulerian system with Fine-Scale-Reconstruction “Vortex-in-Cell #” (Jeon et al. 2018) to obtain time-resolved, 3D velocity fields in the liquid. The evolution of the vortex structure in the wake of the single rising bubbles in the flattened spiral regime can thus be reconstructed. Vortices are identified by the Q-criterion calculated from the velocity gradients. The different path and vorticity evolutions, depending on the bubble size, have been analysed like that.
2 Experimental setup and methods
2.1 Experimental setup
The present 4D shadow imaging experiments were conducted in a transparent decagonal acrylic glass tank filled with de-ionized water with an effective conductivity of under 2 mS (pure water, see Clift diagram in Fig. 10), as shown in Fig. 1. For the decagonal tank, the diameter of the inscribed circle is 200 mm, which is large enough to avoid disturbance of the side walls (Lee and Park 2017). The height of the tank was 345 mm, and the measurement volume (x = 47 mm, y = 67 mm and z = 40 mm) is located 200 mm above the nozzle tip. The de-ionized water was seeded with polyamide particles with a mean diameter of 50 µm. Bubbles were produced by a stainless steel nozzle installed in the centre of the bottom plate. The inner diameters of the nozzles (between 0.13 mm and 1 mm) were selected to produce different sizes of bubbles, approximately 2.4 mm, 4.0 mm, 6.0 mm and 9.6 mm. The gas was introduced by a syringe pump with a flow rate of 0.2 ml/min. The time interval in between two consecutive bubbles was set, that the preceding bubble would not affect the next following bubble. The seeding particles and bubbles were illuminated by two blue LED Flashlight 300 (LaVision, \(\lambda\) = 450 ± 10 nm) arrays from behind the tank to record their shadows. Semi-transparent paper was used to cover the surface of the LED modules and the back walls of the tank to achieve a homogeneous light distribution. The measurements were carried out at room temperature (20 °C) and ambient pressure. The density of water used in the following is ρ = 998.2 kg/m3, the dynamic viscosity is \(\mu =1.002 \cdot {10}^{-3} \, \mathrm{Pa} \cdot \mathrm{s}\) and the surface tension is σ = 72.86 mN/m.
Fig. 1
Experimental setup of the four horizontally aligned cameras around the decagonal acrylic glass bubble column with background LED illumination: left: photograph of the setup and right: schema with bubble injection system and coordinate system
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The measurement volume was imaged by four high-speed Phantom VEO L640 cameras with a resolution of 2560 × 1600 pixels. These four cameras and LED illumination were synchronized, triggered, and the images were recorded with Davis 10.2 software from LaVision. A total of 10,000 frames were taken with a frame rate of 1.4 kHz, an exposure time of 712 µs and a LED illumination pulse width of 65 µs. The recording started at the moment when the first bubbles entered the tank at the bottom. Each camera was equipped with a Scheimpflug adapter and a Tokina 100-mm lens. The angle between the cameras furthest away from each other (camera 1–camera 4 in Fig. 1) was 104.5 degrees.
2.2 Calibration and volume self-calibration (VSC)
All camera’s focal planes were aligned with a two-level calibration plate (LaVision type 058–5, dimensions: 58 × 58 mm2, dot distance 5 mm, level elevation 1 mm and dot diameter 1.2 mm), which was positioned at the top centre of the tank. The tomographic reconstruction of the tracer particle positions, bubble shape and position is then based on the geometrical relation between the 3D physical space and the image space. Taking one point in physical space coordinates P = (X, Y, Z) as an example, the coordinate on the image of the ith camera is pi = (xi, yi). The geometrical relationship is (xi, yi) = Mi (X, Y, Z), where M is the mapping function, which was solved by a 3rd-order polynomial in this work. The mapping function is obtained by a 3D volume calibration, where the calibration plate is placed at the centre of the measuring volume, and images are captured with all cameras. The centre of the calibration plate defines the coordinate origin (0, 0, 0). The target is then set at Z = 2.5 mm, 5.0 mm, − 2.5 mm and − 5.0 mm behind and before the centre position (Z = 0 mm) of the plate, respectively, in order to increase the calibration accuracy. From these five positions and four camera images each, the mapping between physical space and the images is calculated with Davis. The final resolution was 39.59 pixels per millimetre (≙25 µm/pixel) with a fit error of 0.42 pixels for the 1926 × 2611 pixels dewarped images. This calibration was then improved with a following volume self-calibration (VSC) (Schanz et al. 2013a; Wieneke 2013) procedure using the tracer particle images. The final calibration was improved to an average disparity of 0.01 voxel with a maximum disparity of 0.02 voxel after several iterations. The average VSC fit error was 0.0016 pixel using a 3rd-order polynomial fitting function for the cropped dewarped image (1880 × 2600 pixels, 47.49 × 65.68 mm2, depth ± 15 mm). Furthermore, an optical transfer function (OTF) (Schanz et al. 2013a, b) is calculated in order to increase the object contrast to the background and increase the position accuracy.
3 Processing
The raw shadow images of the PIV tracer particles and bubbles (Fig. 2a) are processed with Davis 10.2 as follows and shown in Fig. 2. The images are normalized by dividing the raw images with a white image. The shadow images are inverted (Fig. 2b), and local filters are applied to remove the background and to round and sharpen the particles (Gauss filter) before velocity evaluation. Then, particles and bubbles are segmented into two image sets being treated further separately (Fig. 2c and d). The processed particle images are used to calculate the liquid flow field using the 3D-PTV method Shake-the-Box (STB), which is explained in detail in sub-Sects. 3.2 and 3.3. The further bubble processing is explained in the following sub-Sect. 3.1. Both results, the liquid flow field from the particles, and the shape and trajectories of the bubbles, are then combined in the Ansys Software EnSight 2020 R1 to evaluate the liquid velocity, vorticity and Q-criterion as a parameter for vortex occurrence, together with the bubble trajectories, shape, size and velocity.
Fig. 2
Image processing (view of camera 1): a raw image; b normalized, inverted image with local filters; c separated segmented tracer particle image and d separated segmented bubble image
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3.1 Bubble reconstruction
The segmented bubble images (Fig. 2d) are used for a tomographic reconstruction in Davis 10.2, based on the mapping function, and resulting in 3D bubbles as objects (Fig. 3a). An additional smoothing procedure with MART (multiplicative algebraic reconstruction algorithm, Gordon et al. 1970; Herman and Lent 1976; Raffel et al. 2018) and a smoothing strength of 3 over 6 iterations were applied to cure defects on the bubbles glare points and surface. The bubble shape is given by the 3D reconstruction, and its centre of mass is tracked over time, resulting in the bubble trajectory as shown in Fig. 3b. The bubble size is calculated from the sum of all voxels of the reconstructed volume as sphere equivalent diameter and is evaluated for each time step together with the bubble position and velocity.
Fig. 3
Bubble reconstruction and tracking in the adopted coordinate system. a Bubble reconstruction from segmented bubble images of the four camera views at one instant of time and b bubble trajectory obtained from the movement of the volume centre of the reconstructed bubble, coloured with the instantaneous bubble velocity (see colour bar)
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Typically, 2D shadowgraphy with one camera is used to determine bubble sizes and trajectories. Therefore, we here also compare in Sect. 4.1 the bubble sizes obtained with the aforementioned 3D approach with a conventional 2D sizing algorithm. The images from the two inner cameras (camera 2 and camera 3) are used for that, thus allowing also the comparison of the results from two different viewing angles. Those images are processed in Davis 8.4 with the implemented Particle Master, which segments bubbles from shadow images and calculates the 2D circular equivalent diameter together with long and short axis.
3.2 Liquid velocity with Shake-the-Box (STB)
Volumetric Lagrangian particle tracking (LPT) using Shake-the-Box (STB) (Schanz et al. 2016; Schneiders and Scarano 2016) is an iterative process and overcomes the limitations in particle image density, existing for particle image velocimetry, by incorporating the temporal domain into the reconstruction process. The particle seeding density in this experiment ranges from 0.04 to 0.06 particles per pixel (ppp). The particle images were processed with the STB algorithm, as implemented in Davis 10.2, and with the parameters shown in Table 1.
Table 1
STB parameter
Reconstructed volume
1880 × 2600 × 1577 voxels
Threshold for 2D particle detection
0.00 counts
Allowed triangulation error
1.5 voxel
Shaking
Adding particles (outer loop)
4 iterations
Refine particle position and intensity (inner loop)
4 iterations
Shake particle position by
0.1 voxel
Remove particles of closer than
1.0 voxel
Remove weak particles if intensity <
0.1 of avg. int
Tracking
Minimum track length
4 time steps
Max. abs. change in particle shift
1.00 voxel
Max. rel. change in particle shift
20.00%
Figure 4 shows a resulting instantaneous snapshot of the tracer particle tracks over 25 time steps coloured by the velocity magnitude obtained by STB.
Fig. 4
Instantaneous snapshot of volumetric Lagrangian particles tracking of the liquid velocity (see colour bar) using the Shake-the-Box (STB) method with visualized particle track length of 25 time steps. Bubble position at that instant is marked in yellow
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3.3 Fine-scale reconstruction (FSR) with VIC#
Several derived quantities such as velocity gradients and Q-criterion can only be calculated on a regular Eulerian grid. Therefore, the Lagrangian velocity information obtained with STB has to be transformed to a Cartesian grid. To this end, the fine-scale reconstruction (FSR) algorithm introduced by Schneiders and Scarano (2016) is used, as implemented in Davis 10.2 with the data assimilation technique VIC# (Jeon et al. 2019, 2018; Sellappan et al. 2020). For our experiments, the FSR is run on a grid with 16 voxels (0.4042 mm) (Müller et al. 2022) over 40 iterations with a VIC# de-noising factor of 0.001, interpolating the flow field and resulting in a vector room with 118 × 163 × 99 vectors, corresponding to a vector resolution of about 15 vec/mm3.
For each measurement case, the number of snapshots, reconstructed to a uniform grid to calculate the flow parameters, differs, owing to the bubble size/velocity and its time passed in the measuring volume. Generally, around 400 snapshots are reconstructed to a uniform grid to calculate the flow parameters. Figure 5 shows the related instantaneous vortex structure snapshot of the fine-scale reconstruction (FSR) after processing the previous STB results from Fig. 4. The vortex morphology behind the bubble was rendered with an isosurface of the Q-criterion, and the structure was coloured by the vorticity in Z-direction. The symmetric threshold around zero of the vorticity indicates the rotation from the frontal view (X–Y plane), namely the negative and positive or clockwise and counter-clockwise rotation.
Fig. 5
Instantaneous snapshot of fine-scale reconstruction (FSR) after processing the STB results from Fig. 4. Vortices are identified by an isosurface of the Q-criterion. The surface of the isosurface is coloured by the Z-vorticity (see colourbar). Bubble position at that instant is marked in yellow
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4 Results and discussion
4.1 Bubble size, shape and rising path
In this following part, the calculation of the bubble diameter with two different methods is discussed. The tomographic measurements used a geometric reconstruction of the bubbles and calculated, based on the reconstructed volume, a sphere equivalent diameter for the bubbles. As comparison, a two-dimensional bubble sizing method (Particle Master LaVision) was used to evaluate camera views 2 and 3 of the original setup. The sphere equivalent diameter of the 2D method is calculated from the long (L) and short axis (S) of the segmented bubble area to \(\left[ {D_{{\text{sphere, 2D}}} = \left( {L^{2} S} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} } \right]\). Both results are shown in Fig. 6 in function of the bubble position. The average equivalent bubble diameter D of the four example bubbles, based on the 3D reconstruction results, is 2.4 mm, 4.0 mm, 6.0 mm and 9.6 mm, respectively. These time averaged results of the equivalent bubble diameters together with the standard deviation (StdDev), minimum and maximum values in the measurement volume and the corresponding range (= max–min) are listed in Table 2, for the three different evaluations done (3D, 2D camera 2 and 2D camera 3).
Fig. 6
Comparison of bubble sphere equivalent diameters calculated from two different angles (cameras 2 or 3) with 2D shadow imaging (dotted and dashed lines) versus 3D shadow imaging with bubble reconstruction (continuous line). The averaged diameters of the 3D reconstruction are D1 = 2.4 mm, D2 = 4.0 mm, D3 = 6.0 mm and D4 = 9.6 mm
Table 2
Averaged 3D sphere equivalent diameter of 3D bubble reconstruction versus 2D sphere equivalent diameter of 2D shadow imaging
Method
Bubble eq. diameter ± StdDev [mm]
Minimum [mm]
Maximum [mm]
Range [mm]
Three-dimensional reconstruction
Avg ± StdDev
2.4 ± 0.1
4.0 ± 0.2
6.0 ± 0.1
9.6 ± 0.4
Max
2.53
4.28
6.28
10.09
Min
2.24
3.68
5.68
9.03
Range
0.29
0.60
0.60
1.06
Two-dimensional (cam2)
Avg ± StdDev
2.5 ± 0.1
4.1 ± 0.4
6.7 ± 0.5
11.1 ± 0.6
Max
2.59
4.95
7.17
11.61
Min
2.42
3.46
5.27
10.11
Range
0.18
1.49
1.89
1.50
Two-dimensional (cam3)
Avg ± StdDev
2.5 ± 0.1
4.3 ± 0.2
6.8 ± 0.4
10.8 ± 0.5
Max
2.63
4.73
7.45
10.91
Min
2.41
3.92
6.14
9.49
Range
0.22
0.82
1.31
1.42
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From Fig. 6, it is obvious that the bubble diameters do not change significantly in the measurement volume. This is expected, since mass transfer of air bubbles in water over a height of 67 mm is negligible. The values obtained for the three different evaluations are comparable, but the 3D evaluation gives always smaller mean diameters (see averages in Table 2). For the small bubbles, the results of the 2D and 3D method converge, since the bubbles become more and more spherical. For the bigger bubbles, the difference is around 15%.
The bubbles are non-spherical and thus moving around their three axes, thus a constant bubble diameter over the rising path is not expected from the measurements (Bröder and Sommerfeld 2007). However, the 3D reconstructed bubble diameters show lower fluctuations during rising especially for the three bigger bubble cases. This more physical behaviour is shown in Fig. 6 and on lower standard deviations and ranges between minimum and maximum sizes in Table 2. With a 2D shadow image size measurement, much bigger size changes are determined, depending on which bubble axis is oriented in the viewing direction and from which angle the camera is viewing. This is clearly visible in Fig. 6 and in Table 2 by comparing the StdDev or range which is always bigger for the 2D results.
However, both methods struggle with glare points on the bubbles, resulting in faulty segmentations shown in the red circled parts in Fig. 7 for the 3D segmentation on the left and for the 2D segmentation process on the right.
Fig. 7
Left: segmented bubble with bubble edge enlargement for 3D reconstruction. Green: final segmentation mask, black: bubble shadow. Circled in red is a faulty segmentation section due to a glare point. Right: 2D bubble sizing process. Yellow: segmented bubble, blue: equivalent ellipsoid with long and short axis and circled in red: faulty segmentation area due to a glare point
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The glare points in the 3D method lead to bad masking functions (green area in Fig. 7) and segmentation errors. These can lead in the following 3D reconstruction to holes in the reconstructed bubble surface despite application of bubble edge enlargement (see, e.g. slit in largest bubble in Fig. 11d). This artificially can reduce the bubble volume and the calculated sphere equivalent diameters.
For the 2D method, glare points can also arise due to illumination with an angle, or as also is the case here, due to a wavy bubble surface on large bubbles. Since the glare points have the same intensity as the background, they are not recognized as part of the bubble in the segmentation (see Fig. 7 right), and the bubble diameter is, therefore, underestimated. In addition, the 2D method is not able to track deformations and movements of the bubble surface out of the image plane. This causes the calculated bubble diameter to fluctuate stronger in the 2D results, especially for large deforming, oscillating bubbles.
Since the characteristic rising behaviour of bubbles, expressed through characteristic numbers, like the Ga number, Eo number and Morton number, is strongly determined by the bubble form itself rather than through an equivalent diameter, the 3D reconstruction method has its advantages for the evaluation of bubble dynamics. Nevertheless, improving the reconstruction accuracy by the use of additional cameras from more angles also on the back side of the measurement volume and containing the whole surface information of the bubble would be necessary. Bubble sizing by a 2D method requires less effort, but gives up to 13% bigger diameters for the size range investigated here.
With the equivalent diameters D determined through 3D reconstruction of the bubbles, the characteristic dimensionless numbers of the system used here can be calculated (Table 3). The Weber number We is defined as \(\mathrm{We}=\rho {v}^{2}D/\sigma\), which is a ratio of the inertial force to the surface force. The Galilei number Ga is defined as \({\text{Ga}} = \rho g^{1/2} D^{3/2} /\mu\), which is a ratio of the gravitational force to the viscous force, the Eötvös number Eo, also called Bond number Bo, is defined as \({\text{Eo}} = {\text{Bo}} = \rho gD^{2} /\sigma\), which is a ratio of the gravitational force to the surface force and, finally, the Morton number Mo is calculated by eliminating D using the ratio of Galilei and Eötvös numbers, \({\text{Mo}} = {\text{ Eo}}^{3} /{\text{Ga}}^{4} = g\mu^{4} /\rho \sigma^{3}\). Here ρ, μ and σ denote the density, dynamic viscosity and surface tension of the liquid, respectively, D is the bubble sphere equivalent diameter calculated by 3D reconstruction, g is the gravitational acceleration and v is the average bubble rising velocity. All dimensionless numbers, the bubble diameters and the average bubble velocities are listed in Table 3.
Table 3
Experimental results and flow parameters
D (mm)
Average velocity (m/s)
Re number
We number
Eo number
Ga number
Mo number
2.4
0.32
760
3.3
0.8
367
2.56 · 10−11
4.0
0.27
1068
3.9
2.2
789
6.0
0.24
1429
4.7
4.8
1450
9.6
0.25
2400
8.3
12.4
2935
The phase diagram in Fig. 8 is illustrating these numbers, by referring to the regimes of Cano-Lozano et al. (2016), depicting the bubble flow path types. Five categories of bubble rising paths are represented herein: rectilinear, chaotic, planar zigzag, flattened spiral and spiral paths. The dotted yellow line characterizes our experimental system with \(\mathrm{Mo}=2.56{\times 10}^{-11}\). On this line, the experimental points (see Table 3) of the four aforementioned bubble cases are represented, calculated with the determined average equivalent bubble diameter. All points belong to the flattened spiral regime, an intermediate category between planar zigzag and spiral regimes.
Fig. 8
Phase diagram showing the four experimental sample measurement points on the background of path regimes and transition boundaries from Cano-Lozano et al. (2016). The black dashed line represents the critical Mo number beyond which the circular spiral regime occurs. The solid orange line represents the critical curve above that a standing eddy exists
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The same dimensionless numbers are used in Fig. 9 to characterize the experimental conditions in the Grace diagram (Grace 1973), which shows the aspect of the bubbles. All bubbles examined here are situated in the spheroid region, since they are elliptical, as described in the experimental set-up section.
Fig. 9
Grace diagram showing the four experimental sample measurement points on the background of bubble shapes from Grace (1973)
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The Clift diagram in Fig. 10 shows the measured velocities and bubble sizes of the investigated bubbles. They all lie in the pure water region of the curves, indicating completely clean water conditions.
Fig. 10
Clift diagram showing the dependency of bubble velocity from the bubble diameter with the curves from Clift et al. (2005) and Tomiyama et al. (1998). Measured velocities and bubble sizes of the investigated bubbles in this study are marked with coloured symbols
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The trajectories corresponding to the four aforementioned bubble cases are presented in Fig. 11 together with some examples of reconstructed bubble surfaces at different instants in time (from (a) to (d) Δt is 0.0214 s, 0.0214 s, 0.0428 s and 0.0428 s). All of the here investigated bubbles show an ellipsoidal shape. In the case of the largest bubble, the bubble surface is wobbling and instable; therefore, the volumetric reconstruction becomes difficult and sometimes faulty, as discussed before and visible as a slit in the reconstructed volume in Fig. 11d. The coloured trajectories between the reconstructed bubble shapes in Fig. 11 represent the bubble rising velocities. The overall velocity (Table 3) of the bubbles is decreasing gradually with increasing bubble size (from (a) to (d)).
Fig. 11
Bubble trajectories with reconstructed bubble shapes at several instants in time and bubble rising velocity (colouring of trajectory see colourbar); from a to d, the bubble diameter increases, and the time interval between each reconstructed bubble Δt is 0.0214 s, 0.0214 s, 0.0428 s and 0.0428 s, respectively
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4.2 Effect of the bubble size on flow path instability
Projected views of the bubble trajectories in the X–Z plane (top view) are presented in Fig. 12 for several exemplary cases of each bubble size class. Each colour represents a single bubble, while bubbles with a diameter around 2.4–3.0 mm are in the top left, bubbles with a diameter around 4.5 mm are in the top right, bubbles with a diameter around 6.5 mm are bottom left and with approximately 10 mm in the bottom right of Fig. 12.
Fig. 12
Projections of 3D trajectories on the X–Z plane (top view) of rising bubbles with 2.4–3.0 mm (top left), 4.0–5.0 mm (top right), 6.0–7.0 mm (bottom left) and 9.5–10.5 mm (bottom right) diameter, respectively. The different colours represent different individual rising bubbles trajectories
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As the bubble size increases, the radius of the spiral trajectory decreases, translating to a straightening of the vertical trajectory components. The spiralling trajectories consistently only occur at bubble sizes below 3.5 mm (top left) and shift to flattened spirals (top right and bottom left) and planar zigzagging (bottom right) for the biggest bubbles. With increasing diameter, the bubble paths also become more instable due to bubble oscillation.
A direct comparison of the bubble flow paths in the X–Y, Z–Y and X–Z planes of example bubbles from each size class is found in Fig. 13 for representative cases of Fig. 12. The minimum value on the Y-axis corresponds to the centroid position of the bubble when it is entering the measurement volume. On the horizontal projections (left X–Y and centre Z–Y), the sinusoidal movement of the rising bubbles becomes evident, with a decrease in amplitude when increasing the bubble size. Round about one sinus period was observable in the height of the measurement volume. On the X–Y projection (left), the inversion and inflection points of the 2.4-mm bubble trajectory are exemplarily marked with I, II and III, respectively. The vertical projection (right) shows the spiral movement of the bubbles, which was expected since the investigated bubbles are ellipsoids. As shown before, the radius of these spirals decreases and flattens with increasing bubble size. In the 3D space, the trajectories correspond to a flattened spiral path with a clockwise or counter-clockwise rotation, depending on the initial bubble formation evolution.
Fig. 13
Projections of 3D trajectories of rising bubbles with 2.4 mm, 4.0 mm, 6.0 mm and 9.6 mm diameter, a X–Y, b Z–Y and c X–Z plane, respectively. Here, representative cases from Fig. 11 are shown with the same colour coding as in Fig. 11. Inversion and inflection points I, II and III are referenced in the text and in Fig. 14
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The bubble velocity components corresponding to the trajectories of Fig. 13 are represented in Fig. 14 in function of the measurements volume height (y-axis). The horizontal velocity components in X- and Z-direction are represented on the left, the vertical component and velocity magnitude on the right. The strongly sign-changing horizontal velocity components are obvious from these images. The exemplarily represented inversion points (I and III) from Fig. 13 are also represented in this figure and correspond to a vanishing x-velocity component. Near the inflection point (II), this velocity component is minimum. As the bubble diameter increases, the amplitude of the low-frequency horizontal velocity fluctuations in the X- and Z-direction decreases, but additional high-frequency, low-amplitude oscillations appear. The vertical bubble velocity, that is the strongest component, determines essentially the velocity magnitude. Therefore, both have more or less the same course, which is rather constant, but shows the same aforementioned trends concerning their fluctuations. Here, the inflection points (I and III) correspond to a vertical velocity maximum, the inversion point (II) to a velocity minimum. The corresponding average bubble velocities for bubbles with a sphere equivalent diameter between 2.4 and 9.6 mm are already shown in Table 3 and have values between 0.32 and 0.25 m/s, respectively. The Reynolds number, Re = ρvD/μ, calculated with these velocities and for the corresponding bubble sizes reaches from 760 up to 2400 (see Table 3).
Fig. 14
Bubble velocity components in X (top left), Y (top right) and Z (bottom left) directions and velocity magnitude |V| (bottom right) during ascending in the tank (y-direction) for the four different bubble sizes. Inversion and inflection points I, II and III are referenced in the text
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4.3 Liquid flow field around the bubbles
The liquid velocity measurements, corresponding to the aforementioned bubbles, are presented in Fig. 15. Velocity magnitudes of the liquid phase taken at the centre planes of the bubble are shown on the XY and YZ planes, coloured with the rainbow pallet (mind the different scales). The bubbles are represented as white isosurfaces with their trajectories showing the bubble velocity along the rising path in the orange to purple colour pallet. Additionally, the vortex occurrence is visualized by isosurfaces of the Q-criterion (Q = 0.0004) calculated from the flow field and coloured with the vorticity in Z-direction pointing out the rotational sense of the vortex structures. As the bubble rises, it pushes water in front and entrains the liquid in its wake. The liquid velocity around the rising bubble is slower than the bubble rising velocity. While small bubbles produce easily recognizable vortex rings, directly around the bubble and in the bubble wake, bigger bubbles produce more and more complicated vortical structures that overlap and twist. For the smaller bubbles (Fig. 15a and b), drifting and decay of the vortices can be observed as they drift away from the original bubble path (coloured trajectory line).
Fig. 15
Liquid flow field and wake vortices induced by rising bubbles from 2.4 mm (a) to 9.6 mm (d) sphere equivalent diameter. Bubbles are represented as white isosurfaces; the rainbow coloured planes show the magnitude of liquid velocity in the centre planes, cut at the bubble position (mind the different scales); vortices induced by the rising bubbles are represented as isosurfaces obtained with the Q-criterion (Q = 0.0004), they are coloured with the Z-vorticity direction (red–blue); the brown to black coloured line throughout the volume represents the bubble trajectory coloured with the instantaneous bubble velocity
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The liquid velocity is the highest close to the bubble rear, introducing a turbulent region behind the bubble. Its spreading direction is related to the bubble motion at the shedding instant. The vortex core induces a low-pressure region at the rear of the bubble that causes a “sucking” mechanism between the front and the back regions of the bubble. As the distance between the bubble and the detached vortex pairs increases, the turbulence intensity weakens, due to the turbulent energy dissipation.
With increasing bubble size, from (a) to (d), the size of the vortices and their intensity are increasing. For the small bubble (2.4 mm), the vortices follow the bubble one by one and have a rather long distance. The gap between each vortex pair decreases for the medium bubble size (4.0 mm), but the individual vortices remain distinguishable. At 6.0 mm, the vortex structures become larger, while decreasing in intensity farther away from the bubble (same tendency as in Zenit and Magnaudet 2009). As the bubble size increases to 9.6 mm, even the farthest vortex structures remain strong, creating a continuously disturbed area in the bubble’s ascending path. In summary, the turbulent area induced by the rising bubbles increases and generates more, more complicated and compared to the bubble diameter smaller vorticities, as the bubble diameter increases. Besides, the higher turbulence also induces an intertwined vortex structure on the bubble path.
Bubble path instabilities occur, resulting from the wake-induced drag and nonzero inclination of the instant bubble velocity. Therefore, the vortex structure information determines the path instability and the bubble deformation. Nevertheless, it is still not clear which of the aforementioned is the leading factor (Cano-Lozano et al. 2016; Tripathi et al. 2015), since each transient variation of the bubble shape results in a change of the bubble surface curvature, afterwards influencing on the liquid velocity surrounding the bubble surface, and vice versa. An intimate relationship between bubble shape deformation, vortex shedding and path instability thus becomes evident.
Figure 16 once more depicts the morphologies of the vortex structures already shown before in a zoomed manner. They have been extracted with the help of the Q-criterion (Q = 0.0001). The vortices, extracted from the 3D velocity measurement results, show different structures, depending on the bubble size. For the smallest bubble (D = 2.4 mm), five counter-rotating vortex pairs (marked with roman numbers) are shedding from the bubble rear during half a trajectory oscillation period recognizable in Fig. 16a. At D = 4.0 mm, four distinct vortex pairs are visible (Fig. 16b). This vortex structure is similar to the simulation results of Cano-Lozano et al. (2016), where secondary vortices take the form of single-side loops superimposed onto each primary vortex. The number of single-side loops decreases with increasing bubble diameter. As the bubble size increases to 6.0 mm (Fig. 16c), the vortices start to overlay and produce a more complex structure that is difficult to distinguish. No single-loop vortices exist anymore. Only the wake structure farther away from the bubble rear indicates the loop characteristics similar to that with the smaller sizes. This morphology becomes more intense and chaotic when the bubble size increases to 9.6 mm (Fig. 16d; yellow circle with zoomed representation). The vortex threads are no more separated but intersected with each other in a helically twisted shape. The wake vortices generated at the bubble rear have a strong influence on the hydrodynamic force that acts on it (Ellingsen and Risso 2001), and resulting in a nonzero axial torque on the bubble, which further deepens the asymmetry and instability of the bubble rising path. Overall, the number of individual single-side loop vortices decreases with increasing Galileo and Eötvös numbers, which is with increasing bubble diameter. With bigger diameters, these are no longer distinguishable.
Fig. 16
Comparison of vortex structures in the wake of individual bubbles with a diameter from 2.4 mm (a) to 9.6 mm (d). Bubbles are represented as grey isosurfaces, the vortices were obtained with the Q-criterion and are coloured by the Z-vorticity direction (red–blue). In part (d), a zoom on the yellow marked area of the wake is added
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5 Conclusions
In this study, the liquid flow field in the wake of single rising bubbles in water is investigated experimentally with time-resolved tomographic particle tracking velocimetry (TR-PTV). Simultaneously, the bubble shape, size, velocity and path are determined in the measurement volume by 3D tomographic reconstruction. Both methods are based on the acquisition of shadow images of the rising bubble and tracer particles in the surrounding liquid phase. The obtained Lagrangian information concerning liquid velocity was then transformed onto a Cartesian coordinate system using the FSR algorithm. The simultaneous liquid velocity field and bubble reconstruction together with their trajectories are finally combined in Ensight to allow for a complete analysis of the bubble dynamics.
Four different bubble sizes have been examined. Their equivalent sphere diameter was calculated from the 3D reconstruction of their ellipsoidal shape. Compared to 2D bubbles size measurements, the 3D size depends less on the direction of motion of the bubble and its projection in the cameras viewing direction and gives up to 13% lower equivalent diameter values. As the bubble diameter increases, the surface and shape of the bubbles changes from ellipsoidal bubbles to more complex oscillating bubbles with wavy surfaces. This makes the reconstruction from the shadow images more complicated.
All examined bubble trajectories have the form of the flattened spiral regime. The trajectory spiral diameter, oscillation amplitude and frequency decrease with increasing bubble size. The rotation orientation of the path depends on the initial conditions at bubble detachment.
The combination of the liquid velocity and the bubble trajectories by the 3D volume reconstruction allows an extensive visualization of the bubble trajectory and wake structure morphology. The vortex structure, identified by the Q-criterion, indicates that the turbulent region and its intensity induced by the rising bubbles are increasing with the bubble size. While the region further away from the bubble has a weakened turbulence intensity, due to the turbulent energy dissipation, strong vortices can be observed directly behind the bubble. Furthermore, when increasing the bubble diameter from 2.4 to 9.6 mm, more complex vortex structures develop, starting from two counter-rotating vortices to intertwined vortex structures for the big bubbles. A strongly twisted wake behind the bubble stands in context with an asymmetric and instable bubble path. The mutually dependent mechanism between the bubble shape, its path and the vortex structure behind the bubble has been made evident by these time-resolved 3D measurements, combining an analysis of the bubble and liquid dynamics. The data will be further used for statistical evaluations as well as validation data for numerical modelling and physical models. It is available to interested readers on request from the authors of the Otto-von-Guericke University Magdeburg.
Acknowledgements
The support of China Scholarship Council (CSC) for Yingjie Chang within the PhD Joint-training Program (No. 202006280505) between Xi’an Jiaotong University and University of Magdeburg “Otto-von-Guericke” is gratefully acknowledged. Liejin Guo would like to acknowledge the financial support of the National Natural Science Foundation of China by the Basic Science Center Program for Ordered Energy Conversion (No. 51888103). The fruitful discussions with Prof. Dominique Thévenin and Wei Guan are also gratefully acknowledged. The financial support of the German Research Foundation (DFG) in the frame of project number ZA 527/3-1 is gratefully acknowledged.
Declarations
Conflict of interest
I declare that the authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.
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Hydrodynamics and shape reconstruction of single rising air bubbles in water using high-speed tomographic particle tracking velocimetry and 3D geometric reconstruction
Authors
Yingjie Chang Conrad Müller Péter Kováts Liejin Guo Katharina Zähringer
