Method of Measurements

The purpose of this study is understanding the relation among air entrainment, bubbles size and velocity components. If the void fraction exceeds 10%, the most reliable probes are the intrusive phase detection probes. In this study, dual-tip resistivity void probes (DVP) were developed in-house and used to measure high void fractions and large scale bubble size as shown in Figure 1.

Figure 1: Photograph of dual-tip resistivity void probe (DVP)

The needle type probe was originally developed by Neal and Bankoff (1965) and its design have since been refined by many researchers. The DVP used in this study consists of two stainless resistivity void probes and measures the electric voltage change of the probes. The diameter of the probe is 0.12 mm and the head was sharpen to an angle of 21 degrees. Each probe was insulated except at the head. Two probes are combined into one and these heads were shifted 0.5 mm - 1.0 mm to detect the phase shift of voltage change (see Figure 1). Phase detection intrusive probes have been in development over a long-time. There are two different methods to measure double-tip void probes (see Chanson, 2004). One is direct individual bubble event analysis and another is a cross correlation analysis (Chanson, 2002). Although the direct individual bubble event analysis has more complicated post-processing than cross-correlation analysis, the direct individual bubble event analysis has reliability and robustness under the strong turbulent flows. Therefore, we adopt individual bubble analysis method in here.

The basic principal and calibration of DVP are summarized as follows. We define the distance $Y$ between two heads as shown in Figure 1. The distance $Y$ can be measured by microscope. If a bubble penetrates the DVP, the velocity of the bubble moving at speed $v_b$ can be estimated by the time lag between two heads $\Delta t_s$ and the head distance $Y$

$$v_b = \frac{Y}{\Delta t_s},$$ (1)

where $Y$ is known parameter and $\Delta t_s$ can be measured by the signal delay between the two heads. Then, the bubble velocity $v_b$ and the time lag of passing bubble through the head $\Delta t_g$ give bubble chord length $d$

$$d = v_b \Delta t_g.$$ (2)

It is recommended that distance of probe heads $Y$ should be larger than characteristic space length of the bubbly flow and be smaller than mean bubble diameter (Hibiki et al., 1998)

$$\frac{N_s v}{f} \le Y \le d_m,$$ (3)

where $N_s$ is the sampling data per one bubble, $f$ is the sampling frequency of data, $d_m$ is the mean bubble chord length. Generally, the sampling frequency is 5k-50kHz and should be small enough but minimum resolution also depends on physical sensor size. The estimated error of the bubble chord length $d$ measurement will be discussed later.

On the other hand, the time-averaged void fraction can be calculated by the wetting/drying time ratio of the tip.

$$\alpha = \frac{ \displaystyle{\sum_{i\in liquid} } t_g^i }{ \displaystyle{\sum_{i \in liquid} t_l^i }}$$ (4)

where $t_g^i$ is the drying time of the tip and $t_l^i$ is the time of tip in the liquid. The tip periodically located above the free surface for wave crest measurements. Therefore the synchronized wave gage was used for exact free surface determination together. For long-time integration of the recorded data, Eq.(4) becomes the time-averaged void fraction, $\overline {\alpha }$. If the integration time length is enough smaller than the wave period, the void fraction defined by Eq.(4) can be regarded as instantaneous or short time-averaged void fraction. The probe response depends on the wetting and drying time of the tip, and response time of the probe and electronics. Thus, the response time of the probe was calibrated by below method.

The electronic response time of the probe and head distance $Y$ depend on each DVP individually. Therefore, the precise calibration is required for each probe. We made a DVP calibration system using two LED lasers measuring a rising air slug in a tube (Figure 2). Two LED lasers were set vertically at the middle of the tube, and the DVP was located just above the upper side of the LED laser. The air slug was discharged by the PC controlled electromagnetic valve at the bottom of the tube. Then, the slug rise velocity $V$ was estimated by the time lag between the two LED lasers. Using the calibration system, an accurate threshold value of voltage change of the DVP for the air-water interface was determined comparing $V$ and $v_b$.

Figure 2: Illustrations of DVP calibration system (left:photograph of calibration system, center and right: process of DVP calibration with LED lasers)

To check the validity of the DVP and its calibration, the DVP measured bubble chord length was compared with visually measured bubble chord lengths. Figure 3 shows the relative error computed by the ratio of bubble chord length measured by the DVP to the imaging technique using a high-speed camera (PHOTORON FASTCAM-1280PCI, 1280$\times$1024 pixel, 500frame/s). The image resolution was about 0.01 mm. The DVP over estimates bubble chord lengths by about 2-3% in comparison with the imaging technique. The DVP over estimated the bubble chord length by an average error of 4.6%. The DVP measures the chord length which is intersects of the bubble sphere instead of bubble diameter. Thus, the statistical characteristics of the measured chord length are different from the bubble diameter. For example, a particular size bubble has one diameter but has different chord lengths which depend on the relative location of the measurement point and bubble. This relation also depends on the fluctuations of the bubble trajectory due to turbulence. Therefore, the statistical relationship between the chord length and the diameter was examined by Monte Carlo simulations of advected bubbles in constant flow. More than one million bubbles with arbitrary bubble size spectra ( $\propto \mbox{diameter}^{-1} \sim \mbox{diameter}^{-3}$) were advected by the constant current in the virtual domain and the statistical relationship between the chord length and the diameter was computed. The turbulence effects were not taken into consideration to simplify the problem. Figure 4 shows the log plotted chord length and diameter spectra, respectively. The chord length and diameter in the figure denote $d$ and $D$, respectively. The solid and dashed lines are the spectra of chord length $d$ and diameter $D$ for given bubble diameter spectra $D^{-1}$ and $D^{-3}$, respectively. Obviously, the measured bubble chord length is smaller than the diameter. The ensemble mean of the ratio of bubble chord length and diameter was 0.81, although this value fluctuated slightly by the shape of bubble size spectra. Thus, the diameter statistics can be estimated by the measured chord length statistics through the numerical simulation. Furthermore, the shape of the bubble chord length spectra is not different from the bubble diameter spectra for $D^{-1}$. In particular, the similarity of the bubble chord length and diameter spectra is important to compare the measured data to the conventional bubble theories that will be discussed in a later section.

Figure 3: Relative error of DVP measured bubble

Figure 4: Relationship between bubble chord length spectra and diameter spectra by Monte Carlo simulation (solid and dashed lines are the chord length spectra and marks are the bubble diameter spectra for given $D^{-1}$ and $D^{-3}$ spectra, respectively)

The instantaneous water velocity was measure by acoustic Doppler velocimeter (ADV, Sontek). The sampling frequency of the ADV was 25Hz. ADV recorded data included spike noises due to the Doppler signal aliasing, air bubble effects and so on (i.e. Voulgaris and Trowbridge, 1998; Elgar et al., 2001). A major problem is that the spike noise looks similar to turbulent components in the velocity data. Therefore, several despiking algorithms have been proposed to remove spike noise from ADV recorded velocity data. Generally, Fourier low-pass filtering, moving averaging, and acceleration threshold (first differential of velocity) methods are used for removing spiking noises from ADV data. The 3D phase space method proposed by Goring and Nikora (2002) is an excellent method despiking of ADV data due to the efficiency and no requirement for empirical coefficients. The modification of the 3D phase space method was evaluated by Wahl (2003) and application of it to bubbly flow was investigated by Mori et al. (2007). We use the true 3D phase space method with correlation and SNR filter to exclude spike noise from raw data.