## 2 Measurement Principles of Four-Sensor Probe
### Deduction of Local Flow Parameters from Electrical Signals
They, respectively, have the same directions with the three position vectors formed by the probe, \(\mathbf{S}_{1}\), \(\mathbf{S}_{2}\) and  
Figure 5: Measurement principle of four-sensor probe. (**a**) Position of the four sensors; (**b**) Probe approaching a bubble; (**c**) Formation of measurement velocities. \(\mathbf{S}_{3}\), and their magnitudes are obtained by dividing each position vector by the time delay from the probe signals. Mathematically, it can be expressed by:  
\[\mathbf{V}_{i}=\frac{S_{i}}{(t_{i}-t_{0})},\ \ i=1,\ 2,\ 3 \tag{6}\]  
where \(t_{i}\) (\(i=1,\,2,\,3\)) and \(t_{0}\) denote the time-instant of rising or falling edges of \(s_{i}\) and \(s_{0}\), respectively. (\(t_{i}-t_{0}\)) are the time-instant delays between \(s_{i}\) and \(s_{0}\). When \(t_{i}\) and \(t_{0}\) correspond to rising edges, \(\mathbf{n}_{l}\) and \(\mathbf{V}_{i}\) are the unitary normal vector and measured velocities of the front interface of a bubble, respectively; and, when \(t_{i}\) and \(t_{0}\) corresponds to falling edges, \(\mathbf{n}_{l}\) and \(\mathbf{V}_{i}\) are the unitary normal vector and measured velocities of the rear interface of a bubble, respectively. The interfacial measurement theorem proposed by [PERSON] [36] indicates the projections on the \(\mathbf{n}_{l}\) of the interface displacement velocity vector \(\mathbf{V}_{l}\) and the measured velocities \(\mathbf{V}_{i}\) are the same, and the theorem can be expressed as:  
\[\mathbf{V}_{l}\mathbf{n}_{l}=\mathbf{V}_{l}\mathbf{n}_{l},\ \ i=1,2,3 \tag{7}\]  
from which the velocity component into the normal direction of interface can be easily obtained. In contrast, obtaining the three components of \(\mathbf{V}_{l}\) requires more assumptions or measuring parameters, for instance a further assumption of sphere-shape bubble [51] or symmetric bubble [56]. However, the application of these assumptions is only suitable when bubbles encountered in two-phase are not highly distorted or deforming. A new method to get the whole components of \(\mathbf{V}_{l}\) is proposed in this study by using the known or legitimately assumed velocity direction, i.e., the unit vector of \(\mathbf{V}_{l}\). Although the shape and size of bubbles keep changing in two-phase flow, the direction of bubbles velocity is usually constant and thus the direction of \(\mathbf{V}_{l}\) often remains constant or constant on an averaged level at one fixed position of the flow field. As a result, by a known or assumed interface velocity direction \(\mathbf{n}_{v}\) (Figure 5), the magnitude of \(V_{l}\) can be obtained as follows:  
\[V_{l}\mathbf{n}_{v}\mathbf{n}_{l}=\mathbf{V}_{i}\mathbf{n}_{l},\ \ i=1,2,3 \tag{8}\]  
The above equation applies well when the two-phase flow is limited internal flow, where the direction of the interface velocity can be regarded as parallel to the channel axis. For arbitrary multi-dimensional two-phase flow, the equation and the resulted method still apply, as long as the local flow direction can be provided by legitimate assumption, flow simulation or prior measurement. For a fixed location, the local time-averaged magnitude of interface velocity can be obtained by averaging through numerous bubbles and denoted by \(V_{l}\). As shown by Equation (9), the local time-averaged chord length of a bubble at a fixed position is an averaged product of the \(V_{l}\) and the averaged time duration of the four signals at high voltage. \[\overline{C}=\frac{1}{N}\sum_{N}\overline{V_{l}}\frac{\sum_{i=0}^{3}\left(t_{ i}^{\prime\prime}-t_{i}^{\prime}\right)}{4} \tag{9}\]  
where \(N\) denotes the number of bubbles during the measurement duration. The direction of interface \(\mathbf{n}_{l}\) can be shown more explicitly by the angle between \(\mathbf{n}_{l}\) and the axis of the flow by Equation (10). Figure 4:0  
where \(\mathbf{n}_{axis}\) is the direction of the channel axis.