## 2 Measurement Principles of Four-Sensor Probe
### Deduction of Local Flow Parameters from Electrical Signals  
Before calculation of the local flow parameters, the following assumptions should be made: (1) The probe containing the four sensors is very small in size in comparison to the bubble diameter, indicating that all measured bubbles are so-called large bubbles. Small bubbles can be detectable by one or a few sensors of the four-sensor probe, but they are neglected during the signal pre-processing based on the fact that a few rising or falling edges are missing, or no edge is missing but the eight-instant does not fit Equation (1). (2) The magnitudes of interface velocity and its direction remain unchanged when an interface passes by the sensors of probe. This is true as long as the probe size is small compared to the bubbles. The void fraction equals the ratio of the duration of the sensor contact with air to the total measurement duration. It should be noted that as there are four sensors in the probe, thus the final void fraction can be determined by their average value, as shown by Equation (2). \[\overline{V_{f}}=\frac{1}{4}\Bigg{(}\sum_{i=0,1,2,3}\frac{\sum_{ bubbles}\left(t_{i}^{\prime\prime}-t_{i}^{\prime}\right)}{t_{total}}\Bigg{)} \tag{2}\]  
where \(t_{total}\) denotes the total measurement time. With the eight-instant (edge) of each effective bubble and known size and positions of the four sensors in the probe, the local flow parameters contributed by each effective bubble can be obtained. The measurement principles of other local flow parameters are schematically presented in Figure 5, which shows the relationships between different vectors. Figure 4: A group of signals produced by one bubble. Local time-averaged IAC was predicted by [PERSON] [57] to be related to the interface velocity projected in the normal direction of particular plane:  
\[\vec{a}=\frac{1}{t_{total}}\left(\sum_{l}\frac{1}{|\mathbf{V}_{l}\cdot\mathbf{n}_{l}|}\right) \tag{3}\]  
where \(l\), \(\mathbf{V}_{l}\) and \(\mathbf{n}_{l}\) denote the \(l\)th interface, the vector of interface velocity and the unitary vector normal to the \(l\)th interface, respectively, at a particular measurement point. The short line above \(a\) indicates a time-averaged value. It is worth noting that, as mentioned in Section 2.3, since the neglected ineffective bubbles also contribute the IAC, Equation (3) must be corrected by multiplication factor, as shown in Equation (4):  
\[\vec{a}^{\prime}=\vec{a}f=\vec{a}(\frac{N_{\text{eff}}+N_{\text{ineff}}}{N_{ \text{eff}}}) \tag{4}\]  
where \(N_{\text{eff}}\) and \(N_{\text{ineff}}\) are the number of effective and ineffective bubbles, respectively. This correction to IAC has taken the contribution of each ineffective bubbles to be equal to the average contribution of effective bubbles. A bubble is treated as ineffective mainly due to its small size and continually deforming feature. However, because of its higher surface to volume ratio, its contribution to IAC is usually larger than a bubble of large size and regular shape. Therefore, the result from Equation (4) is still expected to be lower than the true IAC. As the methods proposed by previous researchers who have employed trigonometric functions to determine the direction of interface are not conducive to comprehensible and fast calculation, a distinct and brief vector-based calculation is proposed and performed to obtain the interface direction here in this section. The cross-product of two vectors is also a vector and its direction is perpendicular to the plane formed by the original two vectors, hence the normal vector of one interface can be determined by three velocities measurable by the probe, as shown in Figure 5c and mathematically by Equation (5):  
\[\mathbf{n}_{l}=\frac{(\mathbf{V}_{2}-\mathbf{V}_{1})\times(\mathbf{V}_{2}-\mathbf{V}_{3})}{\left| (\mathbf{V}_{2}-\mathbf{V}_{1})\times(\mathbf{V}_{2}-\mathbf{V}_{3})\right|} \tag{5}\]  
where \(\mathbf{V}_{1}\), \(\mathbf{V}_{2}\) and \(\mathbf{V}_{3}\) are the three measured velocities of a particular interface, respectively.