## 2 Measurement Principles of Four-Sensor Probe
### Signal Pre-Processing  
After data collection, the signals were processed by a MATLAB program consisting of mainly two functions, the pre-processing and deduction of local flow parameters. The signal pre-processing was designed to obtain the time instants when each of the four sensors penetrate or recede a bubble.  
In an ideal case, the signals should be square wave (see Figure 3d) with the high-level representing sensor contact with water and the low-level representing passing-by of a particular bubble. However, mainly due to the delaying of data collection system and electromagnetic interference, the practically collected signal demonstrates noising inclining and fluctuating features, as shown in Figure 3a. Average filtering was applied to attenuate the noise signal firstly, with the result shown in Figure 3b. Then, a threshold voltage was chosen for signal binarization, as shown in Figure 3c. The signal inversion so that the high level corresponds to air is shown in Figure 3d. The threshold value adopted should be slightly higher than the noise voltage, to avoid the influence of noise while ensuring the accuracy of instants when sensor penetrates or recedes a bubble. After that, the resulted square wave was extracted to separate the rising edge corresponding to bubble approaching a sensor and falling edge corresponding to bubble leaving a senor, as shown in Figure 3e.  
The above procedures shown in Figure 3a-e are the same for \(s_{0}\) to \(s_{3}\), and thus eight sets of signal are obtained. Since one bubble has two interfaces passing-by one sensor, eight rising and falling instants are produced in total (four rising edges and four falling edges), as shown in Figure 4, where the eight instants produced by one bubble are noted by \(t_{i^{\prime}}\) and \(t_{i^{\prime\prime}}\) (\(i\) = 0, 1, 2, 3) with'noting rising instant and '' noting falling instant. To make sure each of the eight-instant group belong to the same bubble (effective bubble), the collected signals are screened by cross-checking every bubble using the method described by Equation (1):  
\[\max(\left|t_{i^{\prime\prime}}-t_{j^{\prime\prime}}\right|,\left|t_{i^{ \prime}}-t_{j^{\prime}}\right|)<\min(\left(t_{i^{\prime\prime}}-t_{i^{\prime}} \right),\left(t_{j^{\prime\prime}}-t_{j^{\prime}}\right)),i,j=0,\ 1,\ 2,\ 3\ and\ i\
eq j \tag{1}\]  
Figure 3: A piece of signal during pre-processing. (**a**) Original signal; (**b**) Signal filtering processing; (**c**) Binarization processing; (**d**) Inversion processing; (**e**) Separation processing.  
which guarantees the time delay when bubble approaches (or leaves) one sensor and another must be smaller than the retaining duration of one bubble. This is in the first place required by the assumption that the four-sensor probe is much smaller than the measured bubbles.  
The above procedures allow omitting two kinds of ineffective bubbles. The first kind is those bubbles for which eight-instant signal is incomplete, which means that only part of the four sensors penetrate them. This means bubbles that slide or bounce off the probe. Another kind of ineffective bubbles are those whose eight-edge group does not fit the conditions of Equation (1). These bubbles can be either very small or highly deformed and are also considered as ineffective bubble. Neglecting ineffective bubbles when calculating the local flow parameters might cause error because these bubbles still have contribution to IAC. To counteract the error, the ineffective bubbles were kept counted in the program and the IAC was corrected using the average contribution of effective bubbles. The details are discussed in Section 2.4.