Experiments in Fluids

, Volume 39, Issue 2, pp 407–419

Extended glare point velocimetry and sizing for bubbly flows

Authors

    • Von Karman Institute for Fluid Dynamics
    • Department of Flow, Heat and Combustion MechanicsGhent University
  • J. P. A. J. van Beeck
    • Von Karman Institute for Fluid Dynamics
  • M. L. Riethmuller
    • Von Karman Institute for Fluid Dynamics
Originals

DOI: 10.1007/s00348-005-1004-6

Cite this article as:
Dehaeck, S., van Beeck, J.P.A.J. & Riethmuller, M.L. Exp Fluids (2005) 39: 407. doi:10.1007/s00348-005-1004-6

Abstract

A novel measuring technique for bubbly flows, named glare point velocimetry and sizing (GPVS), was developed in order to measure both bubble size and velocity with high accuracy in a 2D plane. This is accomplished by observing glare points in-focus under an observation angle of 96°. When a second laser-sheet is added, even higher accuracies are obtained and the relative refractive index of the bubble can be measured. It also allows non-spherical bubbles to be rejected and arbitrary angles to be used (e.g. 90°). The accuracy of the size and refractive index measurements was found to be within 1.3%.

1 Introduction

One of the reasons why bubbly flows are difficult to simulate correctly is that the forces exerted on the bubbles are mainly coming from the surrounding fluid and not from the inertia of the bubble itself. Therefore, a thorough understanding of the interaction between the bubbles and the fluid is needed. To this end a 2D experimental technique is necessary, which is able to give accurate information concerning both speed and diameter of the bubbles as well as the speed of the surrounding fluid.

There exist a number of techniques today which give only part of the required information. Backlighting is one of these. It is a well known technique that simply records the shadow images of diffusively illuminated bubbles. Although this technique can give the size and velocity of the bubble, it cannot give its precise location along the optical axis of the camera since the position and dimension of the detection volume is uncertain. This makes accurate gas-water interface studies almost impossible and measurements of the void fraction are crippled because of it.

Another more recent technique in droplet diagnostics is interferometric laser imaging for droplet sizing (ILIDS) (Koenig et al. 1986; Glover et al. 1995), which is also known under various different names such as IPI (interferometric particle imaging) in Damaschke et al. (2002a) or PPIA (planar particle image analysis) in Hess (1998). This technique is based on the interference pattern observed when reflection and refraction spots on the droplet surface (glare points) are seen out-of-focus. The main advantage of this technique is its ability to measure microscopic particles while maintaining a relatively large field of view. However, observing an interference pattern that can be 10 to 100 times larger than the actual bubble leads to a severe restriction on the allowable amount of bubbles per mm3 since only a minimal overlap of the images is allowed. This concentration limit was investigated for different optical configurations in Damaschke et al. (2002b). The configurations examined by these authors also demonstrate a second drawback of the technique: its low dynamic range (dmax/dmin≈10). The concentration limit can be improved if the glare points are observed out-of-focus in only one dimension through the use of a cylindrical lens, as proposed by Akasaka et al. (2002). Possibilities to increase the dynamic range were investigated by several authors. Damaschke et al. (2002a) suggested the use of a second laser sheet under a small inclination angle (15–40°) in their global phase Doppler technique (GPD). If the size ranges of the ILIDS- and the GPD-mode are made to overlap, an extension of the dynamic range can be accomplished (≈100). Another possibility to overcome both the concentration limit and the limited dynamic range is the use of holographic techniques as suggested by Burke et al. (2002). After identifying different particles in the reconstructed in-focus image, each particle is processed individually through the application of a mask. In this way, an isolated ILIDS pattern can be reconstructed for the smallest particles from the holographic recording, which is then processed to yield the droplet size. This masking procedure distinctly increases the allowable number density. Larger droplets are reconstructed in-focus and their glare point pattern gives the diameter of the particle. The largest diameter that can be measured in this way is of the same order of magnitude as the field of view of the camera, leading to a considerable increase in dynamic range. Apart from size measurements, velocity measurements are probably equally important in investigating bubbly flows. Typically, such measurements with ILIDS would involve cross-correlating two pulsed out-of-focus images (Maeda et al. 2002). Another approach was presented by Dantec Dynamics (2003) and Damaschke et al. (2002a). They observed the glare points in-focus with a second camera to yield more accurate (PTV-like) velocity information, in spite of a more cumbersome set-up and associated calibration procedure. Although most of the development on ILIDS has been performed on droplets, Niwa et al. (2000) showed how it could be applied to bubbly flows if an observation angle of 45° is used.

The technique that is introduced in this paper is based on the same principles as ILIDS but instead of observing the glare points out-of-focus to observe their interference pattern, they will be observed in-focus in our technique. This principle was mentioned originally by Hess (1998) for sprays, together with ILIDS. At the time however, the limited resolution of the CCD-cameras rendered the ILIDS-experiments more accurate, thus placing the in-focus technique in the background. In the present study we will show that the in-focus technique has several advantages over the out-of-focus technique and is actually quite complementary to it. In addition, three completely different configurations are proposed for bubbly flows as compared to Niwa et al. (2000) and the resulting decrease in measuring uncertainty for these cases will be theoretically proven. These novel configurations will also enable us to reject misreadings and determine the relative refractive index, even in the presence of non-spherical particles.

2 Basic GPVS

A rigorous theoretical basis for the intensity and location of glare points was given by van de Hulst and Wang (1991). With this theory, one is able to predict the expected intensity profile measured by a camera. However, since the measurement principle is not based on the detailed intensity information but only on the location of the most intense glare points, simple raytracing calculations are sufficiently accurate for bubbles with a size parameter πd/λ>10–20 (i.e. d>2 μm for λ=532 nm), providing that the refractive index of the bubbles differs significantly from its surroundings and that no diffraction is considered (van de Hulst 1957).

Now in order to determine the best observation angle for our technique, the resulting intensity (actually the Fresnel coefficients) after N interactions between the light ray and the bubble surface was plotted versus the scattering angle for parallel polarised laser light in Fig. 1. This shows that for an observation angle of 96° the externally reflected light (N=1) and the internally reflected light (N=3) are equally intense. At the same time, the higher order interactions have an intensity which is two orders of magnitude lower. This is exactly what we need to distinguish both peaks from the background noise and, in addition, this angle is close to 90°, which reduces the projection error on the velocity determination.
Fig. 1

Intensity versus observation angle after N interactions with the bubble for parallel polarised light

A sketch was made of the set-up at this observation angle in Fig. 2. Here we see how both glare points lie more than one bubble radius apart from each other. The basic principle of GPVS is to multiply the distance Lint between these two sharp intensity peaks with a calculated conversion factor αint to derive the bubble diameter. This conversion factor can be derived from raytracing through the following formulas:
$$\theta _{{{\text{out}}}} = 2\tau _{1} - 2(N - 1){\tau }\ifmmode{'}\else$'$\fi_{1} $$
(1)
$$n_{{{\text{particle}}}} \cos ({\tau }\ifmmode{'}\else$'$\fi_{1} ) = n_{{{\text{medium}}}} \cos (\tau _{1} )$$
(2)
$$\nu = 4{\tau }\ifmmode{'}\else$'$\fi_{1} - \tau _{1} - \frac{\pi }{2}$$
(3)
$$\gamma = \frac{\pi }{2} - \frac{{{\left| {\theta _{{{\text{out}}}} } \right|}}}{2}$$
(4)
$${\left| {{\text{AC}}} \right|} = \frac{{R\sin {\left( {{\left| {\theta _{{{\text{out}}}} } \right|} - \nu } \right)}}}{{\sin {\left( {{\left| {\theta _{{{\text{out}}}} } \right|}} \right)}}} + \frac{R}{{2\cos (\gamma )}}$$
(5)
$$L_{{\operatorname{int} }} = {\left| {{\text{AC}}} \right|}\sin {\left( {{\left| {\theta _{{{\text{out}}}} } \right|}} \right)}$$
(6)
$$\alpha _{{\operatorname{int} }} = \frac{{2R}}{{L_{{\operatorname{int} }} }}$$
(7)
$$\alpha _{{\operatorname{int} }} = \frac{2}{{\sin {\left( {{\left| {\theta _{{{\text{out}}}} } \right|} - \nu } \right)} + \cos {\left( {\frac{{{\left| {\theta _{{{\text{out}}}} } \right|}}}{2}} \right)}}}$$
(8)
Fig. 2

Sketch of a typical set-up for basic GPVS

Here, n represents the refractive index, N is the amount of interactions (3 in this case) and the other symbols are depicted in Fig. 3. If we impose the observation angle θout, a value for τ1 and τ1 can be obtained numerically. Inserting this value in the angles ν and γ allows us to calculate the distances |AC| and Lint as a function of the radius of the bubble. From this, we can extract the multiplication factor αint to convert the measured distance Lint into the diameter of the bubble. This derivation assumes the bubble is spherical and, as we can see in the formulas, αint depends on the observation angle and the relative refractive index but not on the absolute diameter of the bubble. However, for non-spherical bubbles the derivation of an analytical formula for the conversion factor is not straightforward. Therefore, we chose to obtain this factor (for both the spherical as the non-spherical case) numerically from a raytracing program. Since this program is 2D, the light ray is not allowed to leave the incident plane. Hence, the non-sphericity can only be studied in the case when two principal axes lie within this plane. Although this simple model cannot give us complete information about the influence of non-sphericity, it will nevertheless show some important trends. For a spherical air bubble in water (nrel=1/1.33) the value that was found at 96° is αint=1.8517. Dependencies of the non-dimensionalised conversion factor on the observation angle, relative refractive index and non-sphericity are shown in Fig. 4 and can be summarised as follows:
  • Observation angle: A misalignment of 1° leads to a maximum error of 1.0% in αint and therefore a careful adjustment of the recording camera is recommended. Note also that a finite opening angle of the laser-sheet will result in a similar misalignment-error.

  • Relative refractive index: For an error in nrel of 0.01 the error in αint equals 0.5%.

  • Non-sphericity: First, we note that the error when measuring a non-spherical bubble (aspect ratio = 0.9), assuming that it is spherical, is not necessarily large. If the tilt angle is close to 45°, the correct spherical equivalent diameter is obtained. Nevertheless, for other tilt angles this relative error can reach 16.8%.

Fig. 3

Raytracing variables

Fig. 4

Dependency of the non-dimensionalised conversion factor on a the observation angle, b the relative refractive index, c the tilt angle for an aspect ratio of 0.9

3 Extended GPVS

3.1 Higher accuracy

Although basic GPVS is already very promising, a substantial improvement can be obtained by using the set-up shown in Fig. 5. Here, a second laser-sheet is used with the opposite sense from the first, which creates additional glare points under a supplementary angle of 84°. From Fig. 1 we find that only one very bright external reflection glare point will be created by the second laser-sheet, with a second, much less intense refraction peak. The trick now is to make sure that this extra reflection peak has the same intensity as the two basic glare points coming from the first laser-sheet. Thus, two distances can be measured: the distance between the two externally reflected glare points (denoted by Lext in Fig.  5) and the traditional distance (Lint). Extended GPVS now uses Lext to determine the bubble size. The conversion factor in this case can be obtained theoretically in a similar way as for basic GPVS and leads to the following formula:
$$\alpha _{{{\text{ext}}}} = \frac{2}{{\sin {\left( {\frac{{{\left| {\theta _{{{\text{out}}}} } \right|}}}{2}} \right)} + \cos {\left( {\frac{{{\left| {\theta _{{{\text{out}}}} } \right|}}}{2}} \right)}}}$$
(9)
Fig. 5

Typical set-up for extended GPVS

For a spherical bubble this factor only depends on the observation angle and equals 1.4162 at 96°. Using this distance to calculate the bubble diameter enables us to achieve a higher accuracy because it is:
  • Virtually insensitive to misalignment: Dependency on the observation angle is even smaller in this case (a relative error in αext of 0.05% versus 1.00% for basic GPVS for a misalignment of 1°), as can be seen in Fig. 4a.

  • Insensitive to the refractive index: No refracted light is used (Fig. 4b).

  • Less sensitive to non-sphericity: For an aspect ratio of 0.9 the maximum relative error in αext equals 11.4% instead of 16.8% for basic GPVS (Fig. 4c).

3.2 Extra information

What is probably even more interesting in this configuration is that we can use the relative location of the internal glare point to give additional information concerning the bubble. This is possible because this relative position, characterised by (Lint/Lext)=(αext/αint), is independent of the bubble size. In fact, this relative position gives us information concerning the observation angle, the relative refractive index and the non-sphericity of the bubble (aspect ratio and tilt angle) in the observation plane. This is shown in Fig. 6. Since there is only one measurement available, it is impossible to extract information concerning one of these parameters without making assumptions concerning the other three. When bubbles smaller than 1 mm are being measured it is common practice to assume that they are spherical, thus eliminating two variables. However, there is a possibility to relax this restriction of sphericity and perform global measurements in flows with non-spherical bubbles. If the average distance ratio over a large number of randomly oriented bubbles is taken, the influence of the non-spherical bubbles is eliminated completely through the superposition of many different tilt angles (this principle is also used in global rainbow thermometry (van Beeck et al. 1999)). The proof for this statement is also shown in Fig. 6c, where the solid curve represents the distance ratio as a function of the aspect ratio averaged over all possible tilt angles (but still restricted to the planar case). Since the averaged distance ratio for every aspect ratio equals the spherical case, the superposition of a large number of bubbles with random aspect ratio and tilt angle will approach the spherical case asymptotically, thus proving our statement. In general, the following procedure is suggested for the extraction of additional information from a flow with non-spherical particles:
  • Verification of the alignment: The average distance ratio of a large number of bubbles is calculated and inserted in Fig. 6a. This gives us a better estimate of the real observation angle, which can be used to better align the system or obtain a better conversion factor accordingly.

  • Extract average temperature: If we assume that the alignment of the camera is perfect, we can extract from the averaged ratio an estimation of the relative refractive index from Fig. 6b, thus yielding the average temperature of the considered bubbles (or the surrounding water) even in the extreme case where there are no spherical bubbles present in the flow.

  • Rejection of non-spherical bubbles and misreadings: For specific combinations of aspect ratio and tilt angle, the error on the calculated spherical equivalent diameter can be quite large as seen in Fig. 4c. In these cases the distance ratio will also be significantly different from the spherical (or averaged) ratio. Thus, it is possible to eliminate these cases by specifying a certain range on the acceptable distance ratio. Misreadings coming from spurious reflections on the surface of the bubble can also be detected in this way. In general, such reflections will have a markedly lower intensity and their appearance will almost certainly result in a different glare point pattern than expected, allowing them to be detected and dealt with. Next to spurious reflections, multiple scattering or obscuration due to the presence of bubbles between the laser-sheet and the camera will also modify the observed glare point pattern. Extended GPVS will be able to reject such measurements where this is not possible with basic GPVS or ILIDS measurements.

  • Obtain per bubble information: If the distance ratio is within bounds and the glare point spacing is sufficiently small to guarantee a spherical bubble, its relative refractive index or observation angle can be determined. This information allows us to define the appropriate conversion factor and obtain a very accurate diameter measurement.

Fig. 6

Dependency of relative position of the internal glare point on a the observation angle, b the refractive index, and c the aspect ratio for different tilt angles

3.3 Different observation angles

With the use of a second laser-sheet, there is no restriction any more on the observation angle. This means that the technique can be used in stereoscopic arrangements or, in particular, an observation angle of 90° is possible. From Fig. 1 we obtain that at 90° only the externally reflected light is important. Thus, with two equally intense laser-sheets we would obtain two bright glare points per bubble (αext=1.4142), independent of the refractive index and of the observation angle (0.007% for 1° misalignment). However, it is likely that the internally reflected glare points are no longer visible and with it goes the ability to extract extra information.

4 Experimental set-up considerations

4.1 Magnification requirements

One of the main obstacles of applying this technique in industrial applications is the necessity to observe both (or all three) the glare points separately, which requires a sufficiently large magnification factor. Therefore, a small study of what is already feasible today will be given next (which follows the derivation by Theunissen (2003)).

For a camera placed at infinity, the amount of pixels covered by each glare point is solely determined by lens aberrations and the diffraction imposed by the aperture. However, for measuring with GPVS both glare points should not be allowed to overlap since the resulting interference pattern might render a correct measurement impossible and would surely make the sub-pixel fitting incorrect. Assuming that each glare point corresponds to a peak of approximately three pixels on the CCD-camera (which allows a Gaussian fitting) through a correct setting of the aperture, this implies that the location of the two peaks should be a minimum of 4–5 pixels apart. Knowing that the multiplication factor for basic GPVS is approximately equal to two, this would require that the image of the bubble on the CCD-camera should span roughly 10 pixels. This magnification requirement can be translated with the thin lens formulas to the following restriction:
$$ r \leqslant f + \frac{{f \times d_{{\text{b}}} }} {{10 \times w_{{{\text{pix}}}} }} $$
(10)
where r is the distance of the camera to the bubble, db is the bubble diameter, f is the focal length of the camera and wpix is the width of a pixel in the camera. In the Canon Powershot G3 (which was used in our experiments) wpix equals 7 μm. Thus, we can plot a curve of the maximum separation distance as a function of the size of the bubble and this for different focal lengths of the camera. Such a graph is shown in Fig. 7, with the curve f=36 mm corresponding to the focal length used in our experiments. The complete experimental condition is also indicated in this figure by a cross-hair, using the smallest encountered bubbles in our experiments (0.7 mm). Theoretically the camera could have been placed at a distance of 386 mm in our experiments. However, we chose to put our camera as close as possible (r=155 mm) in order to obtain a higher accuracy in the size determination. At this distance, the smallest bubble diameter that could have been measured is 0.2380 mm. Using, for instance, a lens with a focal distance of 100 mm would allow a separation of 1 m and at a separation of 155 mm the smallest detectable bubbles are 39 μm! Of course, for extended GPVS this restriction is even more severe since the distance between the closest peaks is roughly one sixth of the bubble diameter. This renders the correctly detectable bubbles to be two to three times larger than what is indicated in the figure. For our separation distance and focal length, this implies a minimum detectable bubble diameter with extended GPVS of 0.7143 mm (0.12 mm for a 100 mm lens).
Fig. 7

Magnification requirements for basic GPVS. Typical experimental conditions for ILIDS experiments were extracted from Damaschke et al. (2002b)

Next to the smallest detectable bubbles, there is also the question of accuracy. Since the localisation error of the peaks is independent of the separation between peaks, the accuracy of the technique will improve if the bubble corresponds to more pixels (i.e. a higher magnification). This means that there is a trade-off between accuracy and field of view (similar to backlighting). In our experiments, a single pixel (and more or less the uncertainty) corresponded to 1/42 of a mm, which resulted in a field of view of 5.4×4 cm with our 4 megapixel camera. The continuing improvement of the CCD-cameras concerning pixel-size and total amount of pixels will only help to enlarge the working domain of GPVS. Keep in mind however that these calculations assume that the diffraction spot can be limited to a size of 3 pixels, which might become problematic at the edges of the proposed working domain.

4.2 Number density limits

In Damaschke et al. (2002b), allowable particle number concentrations for performing ILIDS and GPD on droplets are rigorously derived using Poisson statistics. In this subsection we will apply the same derivation in order to compare the limitations of ILIDS and GPVS in the case of bubbly flows.

The first step in this derivation is to determine the size of the particle image. When using GPVS, the depth of field of the camera is chosen larger than the thickness of the laser-sheet. Therefore, the imaged size of a bubble does not depend on the position of the bubble inside the laser-sheet but only on the applied magnification. This leads to an image area of:
$$ A_{{\text{I}}} = \beta M^{2} d_{x} d_{y} $$
(11)
where AI is the image area, β is a coefficient depending on the shape of the image, M is the magnification and dx and dy are the physical diameters in the x and y directions, respectively. When backlighting is used, the image of the bubble is spherical (β=π/4) and dx=dy=dbubble. In GPVS however, the image of the bubble with relevant information can be approximated by a rectangle (β=1) with a height of roughly 3–5 pixels (depending on the diffraction) and a width of approximately the bubble diameter (to include extended GPVS as well). For an easy comparison with other techniques, we will assume that the height of the image is 1/5th of the diameter of the particle, which is a conservative value for particle images larger than 15 pixels in diameter. The interference pattern studied by ILIDS, on the other hand, is per definition substantially larger than the in-focus particle image. In the article of Niwa et al. (2000) the interference pattern was approximately 20 times larger than the in-focus image whereas this went from 5 to 40 in the configurations mentioned by Damaschke et al. (2002b). In this derivation we will assume that this enlargement is constant over the laser-sheet thickness and equals 20. When an optical compression with a cylindrical lens is used, the particle image is rectangular with only an enlargement in one direction. In the other direction the image size is 3–5 pixels (similar to GPVS). However, an in-focus image of a bubble with an ILIDS set-up is of the same order of magnitude (6 pixels in Niwa et al. 2000) since both glare points should overlap in the particle image for optimal ILIDS measurements. Thus, for “compressed” ILIDS we will assume that the image size in the compressed direction is the same as the imaged bubble diameter. The following results:
$$ A_{{{\text{I}},{\text{backlighting}}}} = \frac{\pi } {4}M^{2} d^{2}_{{{\text{bubble}}}} $$
(12)
$$ A_{{{\text{I}},{\text{GPVS}}}} = \frac{1} {5} \times M^{2} d^{2}_{{{\text{bubble}}}} $$
(13)
$$ A_{{{\text{I}},{\text{ILIDS}}}} = 400 \times \frac{\pi } {4}M^{2} d^{2}_{{{\text{bubble}}}} $$
(14)
$$ A_{{{\text{I}},{\text{ILIDS - compressed}}}} = 20 \times M^{2} d^{2}_{{{\text{bubble}}}} $$
(15)
A second simplification is the assumption that the perceived particle image density does not change considerably over the thickness of the laser-sheet. For GPVS this simply leads to an appropriate scaling of the density with the magnification factor. However, an additional factor is necessary for the ILIDS configuration under 45° that was suggested by Niwa et al. (2000) due to the horizontal compression. This leads to an observed concentration 1.4 times larger than for an observation angle of 96°. This yields with t the laser-sheet thickness:
$${c}\ifmmode{'}\else$'$\fi_{{v,{\text{GPVS}}}} = \frac{{c_{v} }}{{M^{2} }}$$
(16)
$$ {c}\ifmmode{'}\else$'$\fi_{{v,{\text{ILIDS - }}45^\circ }} = \frac{{{\sqrt 2 }c_{v} }} {{M^{2} }} $$
(17)
These results can be combined to yield the mean imaging density ξ and the overlap coefficient γ. The latter indicates the ratio of the probability that a certain location is covered by more than one particle to the probability that it is covered by any particle.
$$\xi = {\int {{c}\ifmmode{'}\else$'$\fi_{v} A_{I} {\text{d}}z} } = {c}\ifmmode{'}\else$'$\fi_{v} A_{I} t$$
(18)
$$\gamma = \frac{{P(N \geq 2)}}{{P(N \geq 1)}} = 1 + \xi - \xi {\left( {1 - {\text{e}}^{{ - \xi }} } \right)}^{{ - 1}} $$
(19)
Damaschke et al. (2002b) stated that this overlap coefficient should be limited to approximately 0.1 in order to ensure that the particle information can be extracted from the partially overlapping images. We will use the same conservative value here, which leads to a corresponding value for ξMAX and cv,MAX as a function of the particle diameter (see Fig. 8). This graph shows that the concentration limit is 2.15/mm3 when measuring bubbles with a diameter of 0.7 mm (as encountered in our experiments). Note that the results obtained here are only valid if there are no bubbles between the laser-sheet and the camera. Otherwise, these limits will be smaller in order to avoid multiple scattering. A direct comparison between the different techniques can also be obtained:
$$ c_{{v,{\text{MAX}},{\text{GPVS}}}} \approx 4 \times c_{{v,{\text{MAX}},{\text{backlighting}}}} $$
(20)
$$c_{{v,{\text{MAX}},{\text{GPVS}}}} \approx 1,500 \times c_{{v,{\text{MAX}},{\text{ILIDS}}}} $$
(21)
$$c_{{v,{\text{MAX}},{\text{GPVS}}}} \approx 2,200 \times c_{{v,{\text{MAX}},{\text{ILIDS-}}45^\circ }} $$
(22)
$$c_{{v,{\text{MAX}},{\text{GPVS}}}} = 100 \times c_{{v,{\text{MAX}},{\text{ILIDS-compressed}}}} $$
(23)
$$c_{{v,{\text{MAX}},{\text{GPVS}}}} \approx 150 \times c_{{v,{\text{MAX}},{\text{ILIDS-compressed-}}45^\circ }} $$
(24)
These calculations show that the improvement in allowable number concentrations for GPVS range from a factor 4 compared to backlighting to a staggering 2,200 compared to an uncompressed ILIDS-configuration under 45°.
Fig. 8

Concentration limit versus bubble diameter for different measuring techniques

5 Experiments

5.1 Goals

To verify the validity of the above numerical and theoretical statements, several experiments were performed. First, the size determination with basic GPVS was verified through simultaneous measurements with backlighting on air bubbles in a water tank. Then, velocity measurements were performed which were compared to a correlation of the terminal velocity versus the diameter taken from Clift et al. (1978). To prove the usability of extended GPVS, measurements were performed on an air bubble entrapped in a silicone block. The refractive index of the silicone block was measured with the new technique and the theoretically obtained dependency on the observation angle was experimentally verified. Finally, a simultaneous measurement with backlighting was performed on the silicone block under an observation angle of 90°.

5.2 Experimental set-up

The experimental set-up with the water tank is shown in Fig. 9a. For backlighting a normal light bulb with a diffuser plate and a colour filter (which stops red light) was used. With the use of a red continuous Helium-Neon laser, this enables us to perform backlighting and GPVS at the same time, and separate both techniques by the use of a colour CCD-camera, in our case the Canon Powershot G3. The observation angle was measured with an angular positioning device with an accuracy of 1°. Furthermore, we notice that the observation angle used in these experiments was not 96°, as was mentioned several times before, but 98°. This is simply because the parallel lines exiting the bubble under 96° are deflected 2° when exiting the water tank.
Fig. 9

a Experimental set-up for the water experiments. b Typical colour pictures (transformed in gray scales) of a simultaneous GPVS and backlighting experiment

For the extended GPVS experiments the water tank was replaced by a silicone block. The second laser-sheet that is needed for extended GPVS was simply the reflection of the original laser-sheet at the silicone-air interface while leaving the silicone block. This creates almost perfectly the desired glare point intensity. In the water tank, the reflection of the primary laser-sheet on the Plexiglas wall created the third glare point as well. This was not desired however, since it partly merged with the internal reflection spot and therefore a metal plate was inserted at the right of the water tank to reflect the primary laser-sheet in another direction.

5.3 Results

5.3.1 Size calibration basic GPVS

A typical picture is given in Fig. 9b. For the experiments only RAW-images were used, thus ensuring that the information coming from the pixels was not modified in any way. Since our camera only has 1 CCD-array, this also means that only one colour is obtained per pixel (red, green or blue). To avoid an interpolation, only red information coming from “red” pixels was used for the GPVS measurements and similarly only “green” pixels were used for backlighting (coming from the same row as the “red” pixels). Thus, the resolution in the horizontal and vertical directions is halved; but it allows a verification of the technique under realistic conditions with only a single camera, eliminating the need for a 2D-calibration between two different cameras. Another important advantage of such an approach is the fact that this comparison does not include any error made in the determination of the magnification factor of the camera since the bubble size in both cases is obtained in pixels. These red and green figures were then processed in the following way:
  • GPVS: The peaks were Gaussian fitted to obtain a sub-pixel accuracy of their location and then their separation was multiplied by the obtained conversion factor and with a supplementary factor (= sin(96°)/sin(98°) = 1.004) coming from the deflection of the parallel beams at the Plexiglas wall.

  • Backlighting: First the Sobel filter was applied to the picture in order to correctly detect the edges. The edge was assumed to be at the location of the maximum gradient and was Gaussian fitted accordingly, which is a fair approximation of the true edge. A more exact localisation is only possible through a rigorous theoretical derivation with the generalised Lorenz-Mie theory, which was not performed here. In addition, the same supplementary factor coming from the deflection at the Plexiglas wall had to be applied on this diameter.

For the displayed bubble this leads to a difference in the calculated diameter of 1.3% (0.2 on a total of 15.2 pixels). Considering the approximation that is used to determine the bubble edge location in backlighting, an error of 0.2 pixel in the diameter is certainly negligible. Note also that this set-up allows us to measure the bubble in two perpendicular planes allowing us to perform an accurate volume measurement.

5.3.2 Velocity calibration basic GPVS

For measuring the velocity with GPVS there are basically two different approaches: PTV-like and streak velocimetry. Although streak velocimetry is certainly inferior to the PTV-approach, it was used here in view of the resulting simple set-up. Using a shutter speed of 1/60 s and no backlighting, the two peaks transformed into two parallel lines as can be seen in Fig. 10a. The size determination was performed as explained before on a randomly selected height and the velocity was extracted from the vertical length of the parallel lines. This result was then placed in a correlation of the terminal velocity of a free rising bubble versus its diameter from Clift et al. (1978) in Fig. 10b. We notice that our velocities are systematically higher than the proposed correlation, although similar results from other authors are also included in this graph. The reason for this “overshoot” probably comes from the fact that our bubbles move in a bubble train and therefore are expected to obtain higher velocities than a single bubble in stagnant water.
Fig. 10

a Streak velocimetry performed with basic GPVS (f*=5, M=0.3011 and shutter speed = 1/60 s). b Comparison of the obtained velocities with a reproduction of the correlation from Clift et al. (1978, p. 172), along with the experiments of Aybers and Tapucu (1969), Davies and Taylor (1950), and Datta et al. (1950). The current experiments with GPVS are indicated with open triangles and drop-lines

5.3.3 Extended GPVS verification

For verification of the principles of the extended GPVS technique, we chose to reproduce the curve of Fig. 6a experimentally and determine the behaviour of the distance ratio versus the observation angle. However, the limited magnification of the camera used often resulted in partially overlapping glare points with extended GPVS. Now, in order to have a higher accuracy in the ratio determination and to be able to adjust the observation angle more easily, a larger static spherical bubble (2.23 mm) was created inside a silicone block.

Before we can compare the experimental dependency of the distance ratio on the observation angle with the theory, we first need the refractive index of the silicone. Earlier experiments performed by Ramuzat (2002) with a prism made from the same material yielded a refractive index of 1.414 ±0.033. In order to demonstrate the ability of extended GPVS to measure the relative refractive index independent of the bubble size, this value will be extracted from the relative position of the internal reflection glare point at an observation angle of 90° (since the laser light is not deflected while leaving the silicone block at this angle). Thus, we obtained an average ratio of 0.7759 (taken over five pictures). From a theoretical plot of the distance ratio versus the refractive index at an observation angle of 90° (shown in Fig. 6b), we found that the refractive index of the silicone must be 1.4325. This value only differs 1.3% from the value obtained by Ramuzat (2002) and lies well within their stated uncertainty.

Then, several pictures were taken at effective angles of 90°–110°, thus including the additional deflection of the laser while leaving the silicone block. These distance ratios were then plotted against the theoretical ratio versus the observation angle (before deflection) computed with the previously obtained refractive index of the silicone (by GPVS) in Fig. 11a. The error bars that are shown in this figure are the standard deviations from several pictures. The experiments agree with the theory within 1.5%, with higher errors appearing for larger angles. This is explained by the emergence of extra glare points at this point, which rendered a sub-pixel interpolation impossible, as could be clearly seen while processing these pictures. We would like to stress that this good agreement was obtained while maintaining a field of view of 5.4×4 cm. If a better accuracy is needed, higher magnifications are still possible.
Fig. 11

a Comparison of experimental and theoretical distance ratio versus observation angle. b Picture of extended GPVS with two laser-sheets under 90°

5.3.4 Extended GPVS under 90°

This experiment was also performed on the silicone block and the second laser-sheet was created by enhancing the reflection of the primary laser-sheet at the silicone wall with a mirror. A typical example is shown in Fig. 11b. Note that the internal reflections are still visible due to the high saturation of the external peaks. This saturation makes the image inappropriate for sizing purposes but it nicely shows the different glare points. Also notice the spurious reflection that is visible on the top, this does not lead to a measurement error since it is not on the same height as the other glare points and therefore can easily be rejected. To test the accuracy of the technique, a simultaneous measurement with backlighting was performed as well, which showed only a difference of 1% (0.48 on a diameter of 46.86 pixels).

6 Discussion

6.1 Comparison with backlighting

Compared to backlighting the present technique has several important benefits. As was demonstrated in Sect. 2, GPVS is able to measure in flows with a four times higher number density. However, these absolute values are only correct if there are no bubbles present between the imaged bubble and the camera. This is often not the case and therefore the practical limiting concentrations will be smaller in order to avoid multiple scattering. In this case, the improvement of GPVS over backlighting will be even larger since it is not the entire depth of the test section but only the part between the laser-sheet and the camera that is important. In this way, we can measure in higher concentrations if we stay close to the camera. This level of control is not possible with backlighting. In addition, GPVS has a better defined probe volume. This allows better measurements of void fraction and therefore GPVS measurements could provide a valuable database for comparison with numerical simulations. Another disadvantage of backlighting is the sizing error that is made when a slightly out-of-focus bubble is measured as discussed by Bongiovanni et al. (1997). This error is eliminated altogether in GPVS, since the laser-sheet thickness defines the probe volume, which is smaller than the depth of field of the camera.

Nevertheless, the main advantage of backlighting remains its ability to give quantitative results concerning non-spherical bubbles whereas extended GPVS is only able to reject such cases. On the contrary, the proposed technique is able to extract information concerning the relative refractive index which is not possible with backlighting. But perhaps the most important benefit of GPVS is its suitability to be incorporated in true two phase measurements where, next to information about the bubble, the velocity of the surrounding fluid is also measured. This is possible now because exactly the same probe volume is used for the PIV- and GPVS-measurements. To achieve such a measurement, fluorescent particles would be added to track the liquid and both phases would be separated based on their colour. This could even be done with a single colour camera.

6.2 Comparison with ILIDS

The comparison with ILIDS will be made on two different levels. First of all, we will make a direct comparison between both techniques, independent of the configuration that is used. Such a comparison is relevant since ILIDS measurements are possible with the configuration suggested here, although this has not been tested experimentally. Then we will compare the presented configuration with the one suggested by Niwa et al. (2000), where the interference pattern coming from different glare points (i.e. N=1 and N=2) is studied under an observation angle of 45°. Through this separation we will be able to underline the advantages of GPVS compared to ILIDS and also show how the presented configuration decreases the measurement uncertainties considerably, independent of whether GPVS or ILIDS is used.

The first obvious advantage of GPVS over ILIDS is its ability to measure higher number densities (Sect. 2). Theoretically this value could reach 1,500 times the concentration for regular ILIDS and 100 times the limit for “compressed” ILIDS. However, as mentioned already, multiple scattering will put an additional limit on the maximum concentration. It is very likely that the overlapping limit is more restrictive for ILIDS whereas the practical maximum concentration for GPVS will be determined by the need to avoid multiple scattering. This will of course decrease the improvement factor of GPVS over ILIDS concerning the allowable absolute number densities. Another important distinction between the two techniques can be seen in Fig. 7, where some working conditions of typical ILIDS measurements (Damaschke et al. 2002b) are shown. These are all well within the range where the resolution of the system is insufficient to resolve between the different glare points in-focus. If the glare points do not overlap in the in-focus image, there will only be a partial overlap of the defocused discs coming from each glare point separately. This situation is illustrated in images of Damaschke et al. (2002a) and is likely to negatively affect the automatic size determination and the required out-of-focus distance. This consideration shows that the working domains of ILIDS and GPVS are actually complementary, a fact that is used in the work of Burke et al. (2002), where at a given stand-off distance and focal length, the smallest particles are measured with ILIDS and the larger particles are measured with GPVS through holographic reconstruction in different image planes. This fundamental difference in working conditions results in the use of lenses with a smaller focal length or larger separation distances for the same bubble diameter; hence a larger field of view is typically obtained in ILIDS-measurements. For the configurations used by Damaschke et al. (2002b) a typical field of view is roughly 1,000 to 6,000 times larger than the measured bubble diameter. For GPVS the field of view is only limited by the amount of pixels. In a 4 megapixel camera (≈2,000×2,000 pixels) this leads to a field of view roughly 100 times larger than the average bubble size (≈20 pixels) being measured. So we notice that the field of view for ILIDS is easily 10–60 times larger than for GPVS. In comparing the dynamic range of both techniques, we notice again that the total amount of pixels has a direct influence on the dynamic range of GPVS since it simply equals (max pixels per bubble)/(min pixels per bubble) or 2000/10=200 in this case. This value is 20 times larger than for the ILIDS configurations described in Damaschke et al. (2002b).

Next to comparing the working conditions for both techniques, it is important to compare the accuracy of the velocity measurements for both techniques. With GPVS this will be performed in exactly the same way as PTV measurements, with the added benefit that the bubble diameter can help in obtaining a correct pairing. Therefore, an accuracy of 0.1 pixel is possible (Marxen et al. 2000), similar to the sizing accuracy. Thus, a direct relation exists between the size and velocity accuracy and the magnification used. In ILIDS, the velocity is measured by cross-correlating the out-of-focus discs in two different frames (Maeda et al. 2002). However, such an approach assumes that the fringe pattern inside a disc does not change between both frames and there are several reasons why this might not be the case. A first cause is the possible influence of the Gaussian beam effect. This results in a non-uniform phase distribution over the thickness of the laser-sheet. If the bubble diameter is sufficiently large compared to the thickness and if there is a substantial velocity component in the third direction, the phase difference between the incident rays might change, thus leading to a shift of the interference pattern. This shift will inevitably result in an error in the horizontal velocity component. A second reason for a shift in the interference pattern can be a change in the bubble size in between frames. Min and Gomez (1996) showed how the phase information in the ILIDS pattern could be used in order to obtain an even higher accuracy in the size determination. This was illustrated by Lorenz-Mie simulations on droplets of 60 and 60.5 μm, where they showed that a phase jump of 180° exists between both scattering profiles. Thus, the size could be measured with sub-micron accuracy! On the other hand, it also clearly demonstrates that even a very small change in diameter can lead to a shift in the interference pattern of one fringe spacing. This would lead to a bias in the horizontal velocity component of 5–10 pixels, depending on the configuration. These results show that extra care should be taken when measuring velocities of oscillating or growing/evaporating bubbles with ILIDS. A second difficulty in determining velocities with ILIDS was mentioned by Damaschke et al. (2002a), who compared in-focus and out-of-focus images taken simultaneously. They noticed that a 2D calibration was needed in order to correlate the out-of-focus bubbles with the in-focus images correctly, partly because the magnification depends on the degree the camera is placed out of focus. This extra calibration of the out-of-focus displacement will undoubtedly add to the uncertainty of the velocity measurements. The final reason why velocity measurements with ILIDS are inherently inferior to those with GPVS is connected with its necessarily larger field of view. As mentioned before, the resolution of the camera in the case of ILIDS is such that the glare points must overlap when imaged in-focus. With GPVS this resolution is necessarily 10–20 times larger than the glare point spacing. This evidently leads to a 10–20 times larger accuracy in the velocity determination of GPVS as well.

The extended technique has several advantages over both basic GPVS as ILIDS. Obviously, its possibility to give extra information concerning the relative refractive index, observation angle and non-sphericity of the bubble is a major improvement. The appearance of the third glare point makes it also possible to extract correct size information, even when spurious reflections are visible (such as depicted in Fig. 11). Observing such a pattern out-of-focus would lead to an additional diagonal frequency component. When only a 1D-FFT is performed, this will show up as an additional horizontal frequency component, leading to a wrong size measurement or, in the best case, a rejection of the measurement. The same increase in reliability with extended GPVS is observed with respect to the influence of multiple scattering (or non-spherical bubbles) where such images can be rejected based on a restriction on the distance ratios. ILIDS and basic GPVS have no extra information available to reject such readings.

Next we will compare our configuration under 96° with the configuration under 45°. As mentioned in Semidetnov and Tropea (2004), it is possible to obtain the fringe spacing directly from the glare point separation. With some manipulations we obtain: D=(λNα)/(γ) where D is the bubble diameter, λ is the wavelength, N is the amount of detected fringes, γ is the collection angle and α is the same conversion factor as used in GPVS measurements. This formula clearly shows that the errors due to misalignment, variations in refractive index and errors due to a non-sphericity are exactly the same for ILIDS measurements as for GPVS. Lowering the error in the determination of the amount of fringes by performing a Fourier transform does not affect this statement. Therefore, it is possible to calculate the uncertainties on the ILIDS measurements under 45° with the same numerical tools as for our GPVS measurements. This also allows us to compare directly the different configurations possible for bubbly flows, independent of the glare point technique used. Under 45°, externally reflected (N=1) and refracted light (N=2) have the same intensity and both glare points lie very close to one another (the conversion factor equals 8.4, i.e. four times larger than under 96°). This is very beneficial for ILIDS since it allows a magnification factor that is approximately four times higher resulting in a higher accuracy in the velocity determination. On the other hand, performing in-focus measurements would inevitably lead to a field of view that is also four times smaller, arguably eliminating this possibility in practice. However, Fig. 4a shows that an observation angle of 45° results in a higher sensitivity to misalignment of the camera. A misalignment of 1° leads to a relative error of 3.3% in the conversion factor at 45°, whereas it is only 1.0% at 96° (and 0.05% for extended GPVS). The same increase in uncertainty is observed in Fig. 4b with respect to changes in the relative refractive index (4% for an uncertainty of 0.01 compared to 0.5% for basic and 0% for extended GPVS). The assumption of sphericity (Fig. 4c) leads to a maximum relative error of 15.2%, which lies between basic (16.8%) and extended GPVS (11.4%). Next to the increase of the measuring uncertainties, using such an oblique angle also results in a strong horizontal optical compression of the flow. This limits the allowable number densities by an additional factor of 1.4 as was shown in Sect. 2. Next to a decrease in the horizontal resolution, the projection error made in the (horizontal) velocity measurement also increases distinctly, seriously affecting the velocity determination accuracy. If no extra precautions are taken, this horizontal compression will even be worse in reality since the extra deflection at the wall will result in an actual observation angle of 20°.

7 Conclusions

A new bubble measuring technique was presented based on the appearance of glare points at the surface of a bubble placed in a laser-sheet. Under 96°, GPVS gives very accurate size and velocity measurements in a 2D plane with just a single laser-sheet and a camera. By adding a second laser-sheet, higher accuracy and extra information concerning the relative refractive index or the non-sphericity can be obtained. It was also shown how the influence of non-sphericity can be decoupled from the refractive index information by averaging over a large amount of bubbles. In this way, refractive index measurements are possible, even in the presence of non-spherical particles. Additionally, a theoretical derivation showed that this technique can be used in substantially larger bubble concentrations compared to ILIDS and backlighting. The second laser-sheet also enables the use of an arbitrary observation angle. This makes it possible to include the technique in stereoscopic arrangements or in PIV-measurements under 90°. Experimentally, the size determination was shown to agree within 1.3% with backlighting. Velocity measurements were performed with streak velocimetry and these results were in line with existing correlations. Measurement of the relative refractive index of a silicone block gave a difference of 1.3% with previous refractive index measurements. In addition, the theoretical dependency of the relative glare point position on the observation angle was experimentally verified and the maximum error between experiment and theory was 1.5%. Finally, measurements under an observation angle of 90° were performed and the calculated size differed only by 1% with the result from backlighting.

Acknowledgements

This research was funded with fellowship SB-031241 granted by the Flemish Institute for the Promotion of Scientific and Technological Research in the Industry (IWT).

Copyright information

© Springer-Verlag 2005