On the basis of sequentially reconstructed 3D vapor structure, the 3D curvatures and velocity distributions on the surface of the two-phase flow are calculated and demonstrated in this section. In past 2D flow or flame studies, 2D curvatures were first extracted on captured images, then proved to be correlated with the flow or flame essence. For example, the flame curvature influences the local heat release rate (Kosaka et al., 2020) and flame displacement speed(Sinibaldi et al., 1998; Tsuchimoto et al., 2009). However, since most practical flows or flames are inherently 3D with asymmetric surface distribution, the 3D curvature measurement is a necessity to characterize the flame nature. Recently, multiple calculation methods of 3D curvature were investigated and developed, such as the central difference method in quad-plane PIV measurements (Kerl et al., 2013), the three-point finite-differencing scheme in (Ma et al., 2016b; Yu et al., 2020), the local polynomial fitting method in flame surface measurements (Wiseman et al., 2017), and so on. By implementing these methods, 3D curvatures were calculated and discussed in some studies to reveal interesting correlations with the flow or flame essence. For example, Chi et al. finds that the response of 3D flame propagation speed to the 3D mean curvature is positively correlated (Chi et al., 2021). Enlighted by past efforts, this work investigates the 3D curvature distribution on the surface of reconstructed two-phase jet flow, aiming at finding quantitative relationship with the flow dynamics, represented by the surface velocity.

## 4.1 3D curvatures on two-phase flow surface

Two different 3D curvatures, mean curvature (*K**Mean*) and Gaussian curvature (*K**Gauss*), are calculated based on the tomographic reconstructions of the vapor surface found in Section 3. For arbitrary point on the vapor surface, infinite number of 2D curvatures can be decided by slicing the surface with a 2D plane at the location of the point. Among all 2D curvatures, the maximum and the minimum are named principal curvatures. The average and the product of two principal curvatures are called mean curvature (*K**Mean*) and Gaussian curvature (*K**Gauss*), respectively. Both *K**Mean* and *K**Gauss* can be only obtained by 3D measurements, and are usually used to describe the instantaneous surface shape and forecast the flow development (Chi et al., 2021). The process of *K**Mean* and *K**Gauss* calculation is divided into 4 steps. First, a point cloud of vapor surface is extracted according to the tomographic reconstruction. Second, the normal vectors of all points included in the point cloud are respectively calculated. Third, the first and the second fundamental forms of the surface fitted by the point cloud are calculated through the normal vectors obtained in the second step. Finally, the distributions of *K**Mean* and *K**Gauss* are derived from the first and the second fundamental forms.

Specifically, in the first step the point cloud is extracted from the 3D reconstruction using the ISO-surface function in MATLAB, as shown in Fig. 7. The point cloud denotes all surface points of the reconstructed vapor. For an arbitrary point *P* in the cloud, the closest *n* points around *P* (namely *Q*1, *Q*2, …, *Q**n*) are marked out, as shown in Fig. 8. Hence, the curved surface s is fitted through points *Q*1 - *Q**n* to enclose point *P*, expressed as Eq. 4:

$${s}={s}\left(u,v\right) \left(4\right)$$

where *u* and *v* are coordinates on the point cloud, which are a set of linearly independent vectors. Eq. 4 determines the intrinsic geometry of the surface at *P*. Moreover, the derivative of s is shown in Eq. 5:

$$\text{d}{s}={{r}}_{u}\text{d}u+{{r}}_{v}\text{d}v \left(5\right)$$

where r*u* and r*v* are the differential coefficient of s on coordinate *u* and *v*, respectively. Hence, the coefficients in the first fundamental forms *E*, *F* and *G* of the surface s can be calculated by Eq. 6:

$$E={{r}}_{u}\bullet {{r}}_{u} ;F={{r}}_{u}\bullet {{r}}_{v} ;G={{r}}_{v}\bullet {{r}}_{v} \left(6\right)$$

Besides, the normal vector \({n}\) of surface s is obtained by Eq. 7:

$${n}= \frac{{{r}}_{u}\times {{r}}_{v}}{\left|{{r}}_{u}\times {{r}}_{v}\right|} \left(7\right)$$

The second fundamental forms *L*, *M* and *N* can be expressed by Eq. 8:

$$L={n}\bullet {{r}}_{uu} ; M={n}\bullet {{r}}_{uv} ; N={n}\bullet {{r}}_{vv} \left(8\right)$$

where r*uu* represents the partial derivative of r*u* on *u* coordinate, similarly for r*uv* and r*vv*. According to Eqs. 6 and 8, the *K**Mean* and *K**Gauss* can be calculated by Eq. 9–10, respectively (Chern, 1945):

$${K}_{Mean}=\frac{EN-2FM+GL}{2\left(EG-{F}^{2}\right)} \left(9\right)$$

$${K}_{Gauss}=\frac{LN-{M}^{2}}{EG-{F}^{2}} \left(10\right)$$

Mean and Gaussian curvatures are used to distinguish local features of the surface. By calculating the mean curvature, a surface point can be classified as a concave point (*K**Mean* < 0) or a convex point (*K**Mean* > 0). Similarly for Gaussian curvature, a surface point can be defined as elliptic point (*K**Gauss* > 0) or a hyperbolic point (*K**Gauss* < 0). Combining both curvatures, parabolic points (*K**Gauss* = 0 and *K**Mean* ≠ 0) and planar points (*K**Gauss* = 0 and *K**Mean* = 0) are defined.

Figure 9 presents the *K**Mean* and *K**Gauss* distributions of the reconstructed two-phase flow in Fig. 7. As shown in Fig. 9a, the portions of vapor outside the cylinder (47 mm < *Z* < 60 mm, in red) and inside the cylinder (15 mm < *Z* < 45 mm, in purple) are respectively studied. The curvature results between the studied regions are corrupted by the cylinder edge, thus expelled from the discussion. Generally, the mean curvatures on the outside vapor portion have moderate value (-0.14 < *K**Mean* < 0.16) compared to those on the inside portion (-0.29 < *K**Mean* < 0.3). One possible reason is that kinetic energy of vapor keeps reducing it emerges from the jet outlet. After the vaper develops outside the cylinder, it has experienced sufficient expansion and tends to homogenously distributed in space, leading to even reduced absolute value of *K**Mean* (i.e., the surface tends to be flat). On the contrary, higher flow velocity and more robust air entrainment tend to generate wrinkled surface on the two-phase flow inside the cylinder, leading to noticeably increased *K**Mean*. In this comparison, it is noteworthy that the sign of *K**Mean* is related with the definition of surface coordinates, so the absolute value of *K**Mean* is more concerned. Similar observation can be found in Fig. 9b for the Gaussian curvatures. A quasi-uniform *K**Gauss* distribution is observed on the outer vapor portion, with very slight variation of -0.005 < *K**Gauss* < 0.0015. Different from mean curvature, Gaussian curvature demonstrates the intrinsic character of surface. Therefore, the sign of *K**Gauss* is independent from the surface coordinates. That is to say, the points on the outer surface are primarily elliptic. Such distribution is in accordance with the Fick's law validated by past works (Fick, 1855; Porteous, 1994). Contrarily, the Gaussian curvatures inside the cylinder are distributed in wider range (-0.05 < *K**Gauss* < 0.047) due to strong mixing effect between the air and water droplets. Such impact also causes elliptic and parabolic points alternately distributed in a narrow region (70 mm < *X* < 80 mm, 30 mm < *Y* < 35 mm, 23 mm < *Z* < 33 mm, as marked out in Fig. 9b).

The statistics of 3D curvature distributions are analyzed based on all 21 sets of vapor frames recorded at 300 Hz (60 frames per set). Figure 10 presents the curvature statistical results for the set that the vapor in Fig. 7 belongs to. As shown in Fig. 10, the total recording period for vapor development is 200 ms, where the origin (0 ms) is set as the frame in which the vapor just leaves the jet. Figure 10a shows the population of classified surface points. As for the concave and convex points (distinguished by *K**Mean*), both numbers significantly increase with time during the vapor development. This is because the vapor surface area is expanded during its upward movement. It is also observed that the populations of concave and convex points grow faster after ~ 80 ms, when the vapor head leaves the cylinder. The higher growth rate is supposed to be dominated by enhanced vapor diffusion outside the cylinder. Similar trends are also observed on the population variations of the elliptic and hyperbolic points (distinguished by *K**Gauss*). More elliptic points than hyperbolic points are observed at all time sequences, while the number difference tends to decrease over time. This may be caused by the vapor diffusion process with irregular wrinkles generation on the surface (the readers are referred to Video 1 for straightforward observation). Specially, no parabolic or planar point (distinguished by *K**Mean* and *K**Gauss*) is obtained among all tested cases. To more distinctly compare the population variation of different surface points, Fig. 10b shows the development of probability density functions (PDFs) of all points, calculated based on Fig. 10a. Specifically, both proportions of concave and convex points fluctuate around 50%, since the vapor is in the state of Brownian Motion (Feynman, 1964; Chern, 1945). Comparatively, the elliptic points occupy ~ 90% of all surface points at 0 ms whereas the hyperbolic points only occupy 10%. This is because the vapor head diffuses upward in the shape of quasi-hemisphere when it just leaves the jet, leading to majority of elliptic points. From 0 to 200 ms, the proportion of elliptic points gradually decreases from 90–50%, since the initial kinetic energy is exhausted over time while the irregular Brownian Motion becomes dominant factor in the vapor diffusion process (Feynman, 1964).

## 4.2 Vapor Surface Velocity

The vapor surface velocity distribution is calculated using so-called normal-vector method (Wiseman et al., 2017) as illustrated in the 2D schematic plot Fig. 11. As shown in Fig. 11, surface s1 and s2 are consecutive frames of a vapor surface (divided by unity time). For s1, the direction of surface normal vector is defined from inside (the shaded area surrounded by red line) to outside of the vapor. Specifically, for point *P*1 on surface s1, the normal vector is denoted by n1, which then intersects surface s2 at *P*1’. Hence, the surface velocity vector on point *P*1 can be defined by V1 (green vector). The velocity is positive since the n1 and V1 are in the identical direction. In contrast, point *P*2 has its normal vector n2 and velocity vector V2 in opposite directions, leading to negative velocity value.

By implementing above method, the surface velocity distributions of the vapor are calculated. Figure 12a shows the surface velocity distribution of the vapor reconstructed in Fig. 5–7 (recorded at 116.55 ms). In details, for the portion outside cylinder in Fig. 12a, the velocity of the vapor circled by green triangle is mainly negative except the region in the lower right corner. Meanwhile, the velocities in other regions are mostly positive. For the portion inside cylinder, the velocity distribution is mostly negative or close to zero, as represented by the green rectangle. To better demonstrate the deformation trend of vapor surface as a consequence of surface velocity, Fig. 12b superimposes the reconstructed vapor surface in adjacent frames (i.e., 116.55 ms in blue and 119.88 ms in red). By comparing Fig. 12a and 12b, the surface velocity distribution overall matches the surface location reconstruction (e.g., similarity distribution marked out by green triangle and rectangle). Specifically, 99.7% of surface points with positive velocity in Fig. 12a are also painted in red in Fig. 12b, while 99.5% of negative velocity points are presented in blue. Such comparison indicates the consistency between the normal-vector method and the FCICT method.

After the 3D curvatures and surface velocity distributions are determined, their relationship is investigated. Figure 13 shows the correspondence between the average surface velocity (absolute value) and mean/Gaussian curvatures by examining all surface points of reconstructed vapor at 116.55 ms. Specifically, in Fig. 13a, the horizontal axis (*K**Mean*) is ranged from − 0.25 to 0.25 with an increment of 0.01. The vertical axis presents the average velocity of surface points whose mean curvature is contained in each *K**Mean* element. Such statistical processing is performed to reduce the influences by extreme curvature/velocity values and to reveal the major relationship. It is noticeable from Fig. 13a that the velocity and mean curvature have distinct correlation: the greater the absolute value of *K**Mean*, the greater the absolute value of velocity. Therefore, a numerical fitting is presented in Eq. (11):

$$\left|{V}\right|=\sqrt[1.5]{{{1.646K}_{Mean}}^{2}-0.029{K}_{Mean}+0.0395 } \left(11\right)$$

where \(\left|{V}\right|\) represents the average absolute surface velocity. Similar statistical processing is also performed for Gaussian curvature - surface velocity relationship shown in Fig. 13b, except that the horizontal axis (*K**Gauss*) is ranged from − 0.003 to 0.003. As illustrated by Fig. 13b, \(\left|{V}\right|\) presents monotonical increasing trend with larger *K**Gauss* for elliptic points (*K**Gauss* > 0). Corresponding numerical fitting is presented in Eq. (12). Besides, it is also noticed that no obvious correspondence is observed for hyperbolic points.

$$\left|{V}\right|=\sqrt[1.5]{{{1.603K}_{Gauss}}^{2}+0.0537{K}_{Gauss}+0.0319 } ({K}_{Gauss}>0) \left(12\right)$$

To further understand the correspondence of surface velocity and 3D curvatures in Fig. 13, we propose a qualitative explanation by considering the movement directions of different surface points of the vapor. In brief, a surface point with higher level of concavity or convexity would be more likely to possess higher velocity. As shown in the schematic plot in Fig. 14, point *S*1 is assumed to obtain high concavity or convexity during time *t*0→*t*1. At *t*0, *S*1 coincides with location *S*0, and the region around *S*0 has very small mean curvatures (simplify to 2D, represented as the green line). n is the normal vector from point *S*0. Two situations are likely to occur to form *S*1: 1) the displacement (*S*0→*S*1) is neglected whereas all surrounded points simultaneously move in the opposite direction of n (Fig. 14a); 2) the displacement (*S*0→*S*1) is large while surrounded points move slightly (Fig. 14b). Since the surface points, initially driven by the kinetic energy from the jet outlet, are increasingly affected by the Brownian Motion when the vapor gradually develops into the open space, the possibility would be reduced for a group of close points to move in the same direction. On the contrary, the possibility would increase for a single point to gain excessive velocity and leave surrounding points behind. If that is the case, the second situation (Fig. 14b) would be more likely to occur. In 3D space, increased 2D curvatures of point *S*1 would be consequently obtained from all directions, leading to larger *K**Mean*. Moreover, all 2D curvatures on *S*1 are likely to have the same sign, indicating that *S*1 is an elliptic point. This assumption is in accordance with the observation in Fig. 13b that only elliptic points are correlated with *K**Gauss*. Furthermore, statistical analysis is performed to quantify the correspondence between surface velocity and 3D curvatures. To aid the analysis, we denote the maximum absolute surface velocity, absolute mean curvature and Gaussian curvature as |V*max*|, |*K**M*,*max*| and *K**G*,*max*. By analyzing all surface points on studied cases, it is found that about five-sixths of surface points with |*K**Mean*| > 0.9|*K**M*,*max*| or *K**Gauss* > 0.8*K**G*,*max* have the surface velocity |V| > 0.7|V*max*|. Such result further proves the correspondence between V, *K**Mean* and *K**Gauss*, all calculated based on FCICT tomographic reconstructions.