## Abstract

The hydraulic transient and thermodynamic characteristics of water flows impacting an air pocket at the vertical end of an elbow pipe are investigated. The CLSVOF (coupled level-set and volume of fluid) and URANS (unsteady Reynolds-averaged Navier–Stokes) equations with energy equation are implemented, where the RNG (Re-Normalization Group) *k*–*ε* turbulence model is adopted and the pressure–density equation is introduced to allow for the water-phase compressibility. All numerical predictions are consistent with the experimental in the literature. The evolution characteristics and mechanism of the water–air interface are analyzed based on the Froude number and dimensionless water–air mixing degree. For air-pocket Type I and Type II with the water–air mixing degree threshold of 10%, based on the first law of engineering thermodynamics and related process laws and basic theory of statistics, the applicability of the ideal gas model in the thermodynamic process of air pocket, the polytropic index regulation in the evolution process and its association with relevant typical thermodynamic processes are systematically analyzed. The polytropic index of air-pocket Type I fluctuates a little with the averaged median of 1.35 in different transient periods under different initial pressure conditions, while it fluctuates largely with the averaged median of 1.26 and 1.21 under the low and the high initial pressure conditions, respectively.

## HIGHLIGHTS

The CLSVOF model can better capture the pressure peak and water–air interface fragments than the VOF model.

The thermodynamic process was close to adiabatic for air-pocket Type I and close to a compromised one between adiabatic and isothermal for air-pocket Type II.

The recommended polytropic indexes were 1.32 for low initial pressure conditions and 1.29 for high initial pressure conditions.

### Graphical Abstract

## NOTATION

*u*_{i}velocity of the air–water mixture

*P*pressure of the air–water mixture

*α*volume fraction

*ρ*density

*μ*dynamic viscosity

*f*_{i}body force

*E*total energy

*T*temperature

*k*_{eff},*k*,*k*_{t}thermal conductivity coefficient

*τ*_{ij}shear stress

*F*_{sfi}surface tension force

*σ*surface tension coefficient

*κ*curvature of the free surface

unit normal vector of the free interface

gradient of the volume fraction of air

*δ*smoothed Dirac Delta function

*φ*level-set function

*c*wave speed

*K*bulk modulus of water elasticity

*h*heat transfer coefficient

*q*heat flux

*V*volume

*η*amount of substance

*R*gas constant

*n*polytropic index

- Δ
*u* internal energy increment

*w*process work

*C*_{v}specific heat capacity at constant volume

*ζ*adiabatic index

Subscripts

## INTRODUCTION

Pipeline systems are widely used in the fields of energy engineering, such as nuclear and thermal power station cooling systems, hydropower plants, and pump stations. In the pipelines, the phenomenon of two-phase flows with entrapped air pockets frequently occurs due to the incomplete air removal during the pipe filling process, the release of the dissolved air, and the air-core vortex in the intakes of pump stations (Pozos *et al.* 2015), where the air pockets in pipelines obstruct the main flow path, thus increasing the energy consumption and pumping cost (Alexander *et al.* 2020). In addition, the air pocket in pipelines often triggers transient flows, and its high compressibility may accelerate the velocity of the water column, and aggravate the transient pressure experienced during water hummer effects (Fuertes-Miquel *et al.* 2019; Ferreira *et al.* 2021). The investigations of the air pockets during the transient process in the pipeline are of critical importance to optimize the design of hydraulic system and equipment selection, which thus guarantee the safety of pipeline systems.

Numerical techniques are important tools to investigate the entrapped air pocket during transient processes. The 1D models including the rigid water column model (RWCM) and the elastic water model (EWM) are computationally efficient and thus widely used to analyze the pressure variation in transient flows. The 1D-RWCM assumes a rigid water column and ideal gas with a constant polytropic process, while the compressibility of the water and the elasticity of the pipe wall are introduced into the 1D-EWM (Izquierdo *et al.* 1999; Zhou & Hicks 2002; Liu *et al.* 2011). In 1D models, the influence of air pocket on the transient process is analyzed only by means of an isothermal, adiabatic, or constant polytopic index (Lee 2005; Zhou *et al.* 2013; Martins *et al.* 2015). In fact, the polytropic index of the air pocket is variable, but the relevant research on this issue is relatively insufficient. And Zhou *et al.* (2018a) argued that the polytropic process of the entrapped air pocket is closely related to the transient characteristics, which is difficult to reveal without detailed modelling of the heat transfer processes in the air pocket. Besides, the water–air interface is assumed to be perpendicular to the pipe axis in 1D models, which cannot well capture the water–air interaction.

With the development of computer technology and numerical algorithms, the multiphase theory is a useful tool for further understanding the dynamic behavior and thermodynamic characteristics of water–air transient processes, among which the volume of fluid (VOF) model is effective to simulate immiscible fluids with identifiable interfaces (Hirt & Nichols 1981), which has been used in describing the transient flow characteristics with entrapped air pockets both two-dimensional (2D) and three-dimensional (3D) based on computational fluid dynamics (CFD). Liu & Zhou (2009) adopted the 2D-VOF model combined with the standard *k*–*ε* model to investigate the transient of rapid filling water impacting the air pocket at the dead-end of a horizontal pipeline under conditions of incompressible water and ideal gas. The change in pressure was analyzed and compared with the 1D-RWCM. For the case of an initial small air pocket, when the first pressure peak of the air pocket appears, the water phase has not hit the pipe end, and the maximum pressure predicted by the two models is close and equal to the pressure of the air pocket. In the case of an initial large air pocket, when the first pressure peak of the air pocket appears, the water phase has hit the pipe end. The maximum pressure calculated by the 1D-RWCM is the pressure of the air pocket, and the maximum system pressure calculated by the 2D-VOF model is not the pressure of the air pocket and is greater than it. Zhou *et al.* (2011) experimentally and numerically investigated the pipe filling process of an undulated pipe system containing entrapped air pocket by 1D-EWM, 2D-VOF, and 3D-VOF models based on the standard *k*–*ε* model with incompressible water and air, respectively. The pressure fluctuations obtained by the 2D-VOF and 3D-VOF models are fairly close to the experimental data in the cases of both large air pockets and small air pockets while those by the 1D-EWM deviate severely from the experimental data after the first pressure peak, especially in the case of a small air pocket. Martins *et al.* (2017) investigated the rapidly filling process in a pipe containing incompressible water and entrapped air by the 3D-VOF model together with the energy equation and the realizable *k*–*ε* model. The air–water interface is clear at the maximum compression of the air pocket for the small air pocket with low initial driving pressure while the air–water interface is fragmentized for the large air pocket with high initial driving pressure. The comparative analyses show that for the first pressure peak, the predictions by the 1D-EWM are the most accurate, those by the 3D-VOF model are the second, and those by the 2D-VOF model are the worst; after the first peak, the predictions by the 3D-VOF model are the most accurate, those by the 2D-VOF model are the second, and those by the 1D-EWM are the worst. Based on the 3D-VOF and the RNG *k*–*ε*, Zhou *et al.* (2018b) further took into consideration the thermal conduction and convection in a horizontal pipe with an entrapped air pocket at the dead-end. The numerical time-dependent pressure near the dead-end during the transient of rapidly filling water impacting the air pocket shows fairly good consistency with the experimental data in the case of a large air pocket under low initial driving pressure while the accuracy of pressure prediction is relatively low in the case of small air pocket under low initial driving pressure. The VOF model can well capture the regular water–air interfaces observed in the experiments. However, the interface fragmentations and air bubbles dispersed in the water observed in experiments are not captured. The main reason is that although this can naturally guarantee mass conservation, the VOF tracks the volume fraction of a particular phase in each cell rather than the water–air interface itself, therefore the spatially discontinuous volume fraction gradient-based free-surface deformation is unsatisfactory (Scardovelli & Zaleski 1999).

Correspondingly, a level-set (LS) method can accurately capture the topologically complex interface due to its superiority in obtaining the surface curvature and the surface normal (Sussman *et al.* 1994). However, the error caused by the lack of maintenance of the distance constraints will accumulate and cause large errors in mass and momentum solutions (Sussman *et al.* 1998). To overcome the deficiencies of those two methods, Sussman & Puckett (2000) proposed a CLSVOF model combing the complementary advantages of the VOF model and the LS method. The CLSVOF model is applied to the computation of a 4-mm air bubble rising to the water–air interface and then bursting due to the stiff surface tension effects. The comparisons with the experiments show that the proposed model has mass conservation properties comparable with the VOF model and also can accurately compute surface tension driven flows. Zi *et al.* (2021, 2022) adopted the CLSVOF model to track the water–air interface of the air-core generated at the intake pipe in pump sumps and revealed the air-core evolution characteristics and mechanisms. The numerical results are in good consistency with the experiments.

In this paper, the 3D-CLSVOF model is adopted to track the water–air interface during the transient of water impacting the air pocket. The thermodynamic characteristics of the air pocket are investigated by considering the heat conduction and convection between the fluids and the pipe walls. The pressure–density equation is introduced to take into account the water compressibility. The numerical results are verified by the experimental data from the reference (Zhou *et al.* 2018b). The accuracy of the results by the CLSVOF model for the simulation of the two-phase transient flow of water impacting the air pocket is compared with that of the original VOF model. The variation feature of the air pocket's physical quantities deviates from the ideal gas equation of state inside the region near the water–air interface. Furthermore, the polytropic index is determined by the equation of the polytropic process based on CFD results by the CLSVOF model to reveal the thermodynamic characteristics of the air pocket during the transient.

## GOVERNING EQUATIONS AND MATHEMATICAL MODELS

### Governing equations for two-phase flow

*p*is the pressure of the air–water mixture, respectively; is the surface tension force; is the body force considering only the gravitational acceleration;

*E*is total energy;

*T*is temperature; is the effective thermal conductivity coefficient, , and here

*k*is the heat conductivity of the fluids and is the turbulent thermal conductivity coefficient which depends on the turbulent model; is the shear stress of the mixture; and are the density and the dynamic viscosity of the mixture, respectively, and dependent on the air volume fraction as follows:where and are the density of water and air, respectively; and are the dynamic viscosity of water and air, respectively.

### Compressibility of water

*et al.*2021):where is the speed of sound of water;

*K*is the bulk modulus of water elasticity.

### Heat transfer model

### Turbulence model

The turbulence model used is the RNG *k*–*ε* model (Re-Normalization Group), which was developed by Yakhot *et al.* (1992) based on RNG methods and accounts for the effects of smaller scales of motion.

## NUMERICAL TECHNIQUES AND PHYSICAL MODEL

### Solution methodology

The finite volume method is used to discretize the governing equations. The pressure–velocity coupling method adopts the PISO (Pressure-Implicit with Splitting of Operators) algorithm. The spatial discrete format of pressure is PRESTO (PREssure STaggering Operation). The explicit solver is used to solve the transport equation of volume fraction. The spatial discretization scheme adopted for volume fraction is Geo-reconstruct. The spatial discretization schemes of density and momentum are both the second-order upwind scheme. The spatial discretization scheme for all the turbulent kinetic energy, turbulent dissipation rate, and energy equations is the first-order upwind scheme. The level-set function is discretized by the third-order MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) scheme. The models and the corresponding solvers are implemented in ANSYS FLUENT. The density and the speed of sound are introduced by the user-defined function (UDF) technology.

### Computational domain and boundary conditions

All above-mentioned models and numerical techniques are verified by the experimental data for transient characteristics of the two-phase flow of water impacting air pocket from the reference (Zhou *et al.* 2018b). The computational domain shown in Figure 1 reproduces the physical model including a pressurized upstream tank, a horizontal–vertical transparent organic glass pipe with an inner diameter of 40 mm and a total length of 8.862 m, and a quarter-turn ball valve. The pressurized upstream tank with sealed air in its upper part is used to adjust the upstream pressure in order to investigate the air–water interactions along with their associated pressure and temperature oscillations. In this study, the initial upstream pressure is set as 0.08 and 0.12 MPa for two cases, respectively (0.08 MPa for Case 1 and 0.12 MPa for Case 2). The pipeline behind the ball valve is filled with air at atmospheric pressure. The length of the air pocket for both cases is 1.058 m. By quickly opening the ball valve, the hydraulic transient event of water flow impacting the entrapped air pocket is generated. The pressure monitoring point P1 is located immediately near the dead end.

The whole computational domains are modeled using hexahedral structured meshes. The grid in the near-wall region is refined to meet the requirements of the near-wall flow solution of the RNG *k*–*ε* model. The grids of the valve and the pipeline behind the valve, where water–air interaction mainly occurs, are also refined to ensure the accuracy of the solution. On both sides of the mesh interface, the size of the grid is gradually reduced along the direction close to the interface to ensure interpolation accuracy. Based on the above meshing principles, three sets of grids with mesh numbers of 178964, 279644, and 406958 are adopted for both the two cases to validate the corresponding mesh independence. All sets of grids are run under the CLSVOF model. The time step of 0.0001 s is adopted. The differences between the predicted maximum pressure of three sets of grids and the experimental one at the P1 monitoring point are respectively 6.9, 5.1, and 4.6% for Case 1 and 6.7, 4.4, and 4.1% for Case 2. From the perspective of numerical calculation efficiency and accuracy, the grid with a number of 279644 is adopted for both cases. The final range of *y*^{+} values for the two cases are 13.5–23.3 and 14.3–20.7, respectively.

## RESULTS AND DISCUSSION

### Oscillation characteristics of pressure and air pocket

The time-dependent pressure variations at P1 are shown in Figure 2. It can be seen from Figure 2 that the numerical results show good consistency with the experimental data including the amplitude and pressure evolution, especially the first two pressure change cycles. The transient pressures predicted by the two models are almost identical. Compared to the experimental data, the error for the first peak pressure by the original VOF model is 6.0% for Case 1 and 5.0% for Case 2, while the errors of the first peak pressure by the CLSVOF model are 5.1 and 4.4% for Case 1 and Case 2, respectively. The comparisons show that the results obtained by the CLSVOF model are relatively more accurate than those by the original VOF model from the perspective of the pressure prediction.

The maximum pressures at the first peak at P1 obtained by the CLSVOF model are 206.9 kPa (at 0.51 s) for Case 1 and 343.5 kPa (at 0.45 s) for Case 2, and the minimum pressures at the first valley are 30.0 kPa for Case 1 and 45.6 kPa for Case 2. The differences between the first and the second peak pressure are 28.5% of the corresponding maximum pressure for Case 1 and 34.4% of the corresponding maximum pressure for Case 2, which indicates that the higher initial upstream pressure causes severer transient impacting effect and most mechanical energy dissipates and converts to thermal energy during this stage, which induce the temperature increase. The discrepancy between the cyclic period of 0.51 s for Case 1 and 0.45 s for Case 2 is probably caused by the difference between the initial pressures and the air pocket evolutions.

The maximum pressures of the whole pressure field (*P*_{smax}) and the maximum pressure inside the air pocket (*P*_{amax}) are compared with the pressure before the valve (*P*_{bv}) based on the CLSVOF model as shown in Figure 3. The first pressure peak of *P*_{smax} is slightly higher than that of *P*_{amax} with a discrepancy of 0.01 kPa for Case 1 and 0.33 kPa for Case 2, and the subsequent pressure peaks are identical. Furthermore, the pressure peaks of *P*_{smax} and *P*_{amax} are all higher than those of *P*_{bv}, but with the evolution of the two-phase flow, the oscillations of *P*_{smax}, *P*_{amax}, and *P*_{bv} gradually decrease, and all three final pressure values tend to be almost consistent.

The evolutions of the volume and the volume-averaged pressure of the air pocket for the two cases are shown in Figure 4. There is an obvious feature that the variation trend of the air-pocket volume is opposite to that of the pressure of the air pocket, and both have the same period which is 0.91 s for Case 1 and 0.74 s for Case 2. First, the air pocket is compressed as the volume-averaged pressure increases, and the volume decreases to the minimum value (41.7% of initial volume for Case 1 and 30.7% of initial volume for Case 2) as the corresponding volume-averaged pressure almost reaches the maximum. After that, the air pocket expands with its volume gradually increasing. In the subsequent compression and expansion process, the magnitude of the changes in the air pocket volume becomes smaller as it periodically oscillates. Meanwhile, the amplitude of the subsequent pressure wave decreased due to the viscous dissipation.

### Water–air interface evolution mechanism

Figure 5 shows the water–air interface evolutions obtained by the original VOF model, the CLSVOF model, and the experiment for Case 1. Overall, all the air volume fraction distribution contours by the two models show good consistency with the high-speed images. The regular and clear water–air interfaces by the two models are almost identical at 0.25 and 0.5 s. The large curvature deformation occurs at the water–air interface and forms many dispersed bubbles at 1.2 and 1.9 s. The interface fragmentations captured by the CLSVOF model are more consistent with the experimental observations than those by the original VOF model. Therefore, the CLSVOF model is employed for the later analysis of water–air interactions.

The water–air interface evolutions are illustrated in Figure 6 and the corresponding time moments of *t*_{0}–*t*_{9} for Case 1 and Case 2 in Figure 7. The iso-surface with an air volume fraction of 0.5 is taken as the leading edge of the water and the area-averaged velocity components on the corresponding iso-surface in *x*, *y*, and *z* directions are shown in Figure 8, where the velocity in the *y*-direction dominates the movement of the water phase, especially in the horizontal pipe.

From *t*_{0} to *t*_{2}, the air pocket is in the first compression stage. The transient is excited by the rapid opening valve from *t*_{0}. For Case 1, when the valve is opened, the water has a large acceleration due to the action of upstream high pressure, but at this time the velocity is small. During the accelerating process, the water velocity increases rapidly from 0 to the maximum value of 1.82 m/s at *t*_{1}, resulting in the rapid increase of Fr (Froude number, , where is the area-averaged velocity of the water–air interface, *D* is the pipe diameter, shown in Figure 9) from 0 to the maximum value of 3.6 at *t*_{1}, that is, the gravity effect decreases gradually while the inertial force effect increases gradually, leading to the inclination angle of the water–air interface (i.e. the angle between the waterfront and its upstream pipe axis) gradually increases to the largest value of 19° at *t*_{1}. Compared to Case 1, the acceleration of the water is greater with a smaller gravity effect due to the higher upstream pressure for Case 2. The greater acceleration causes the water velocity to rapidly increase from 0 to the maximum value of 2.87 m/s at *t*_{1}, with a corresponding increase in Fr from 0 to a maximum value of 5.69, which leads to a faster increase in the inclination of the water–air interface and a larger inclination angle increases to the maximum of about 40°.

For Case 1, after *t*_{1}, with the continuous advancement of water flow, the water flow reaches near the elbow at 0.37 s, and the velocity component in the *y*-direction rises to the maximum value of 1.81 m/s. After that, the water flow gradually passes from the horizontal direction to the vertical direction through the elbow. Due to the action of centrifugal force and inertial force, the water through the elbow moves upward close to the outer wall of the elbow, and the velocity component in the *z*-direction begins to increase to the maximum value of 0.9 m/s at 0.46 s. And then, under the action of gravity, the water flow velocity decreases, the air pocket is still continuously compressed, and the kinetic energy of water is continuously transformed into the internal energy of the air pocket and the potential energy of water flow as the kinetic energy of water flow is becoming smaller, which results in an increase in the static pressure of the water flow and an increase in the air-pocket pressure greater than the pressure of the water flow, until the air-pocket pressure rise to the maximum value of 206.9 kPa at *t*_{2}. At the same time, Fr decreases with the decrease of water velocity, resulting in the inertia effect decreasing and the gravity effect increasing. Compared to Case 1, the evolution process of Case 2 is similar, but due to the higher initial pressure, the maximum velocity in the *z*-direction and *y*-direction is greater, i.e. 2.68 m/s at 0.32 s and 2.18 m/s at 0.39 s, respectively. The greater water flow kinetic energy is transformed into the internal energy of the air pocket and the potential energy of the water flow and results in greater increases in the static pressure of the water flow and the air-pocket pressure, and the air-pocket pressure rises to the maximum value of 343.6 kPa at *t*_{2}.

For Case 1, the water flow along the outer pipe wall of the elbow continues to move downstream along the vertical pipe under the action of an inertial force from *t*_{2}. Because the pressure of the air pocket is greater than that of the upstream water flow, the air pocket begins to expand upstream. Under the joint action of the water inertia force and the air-pocket pressure, the upper portion of the water flow inside the pipe upstream of the water–gas interface moves upstream and the inertia force drives the water flow close to the outer pipe wall to move downstream, which results in the mutual shear action of the two parts of the water flow and the formation of a vortex near the water–air interface, causing the water–air interface to stretch along the pipe axis. During the process, the velocity of water backflow upstream gradually increases and develops into full pipe backflow while the velocity of water flow downstream gradually decreases to 0 at *t*_{3}. For the water flow downstream, the kinetic energy transforms into potential energy, and the front edge reaches approximately the middle of the vertical pipe. Compared to Case 1, because the inertial force of water flow downstream in Case 2 is greater and the pressure of the air pocket is much greater than that of the water flow upstream, the water backflow velocity increases faster. At *t*_{3}, the water flow downstream impacts the dead-end of the pipe.

For Case 1, the water flow downstream returns and the potential energy of the water flow in the vertical pipe is gradually transformed into the kinetic energy of water flow after *t*_{3}. With the continuous expansion of the air pocket, the pressure energy of the air pocket is also continuously converted into the kinetic energy and pressure energy of the water flow. The return-water velocity increases and the air-pocket pressure decreases gradually. At 0.71 s, the air-pocket pressure is balanced with the upstream water flow pressure, the return-water velocity rises to the maximum 0.62 m/s, and has a large inertial force. Since then, the water flow continues to move upstream under the action of inertial force, thus the air pocket continues to expand and its pressure continues to decrease. The pressure of the return water full of pipe is higher than that of the air pocket and this pressure difference acts as a resistance to the return water, and the water flow kinetic energy gradually reduces. The air-pocket volume expands to the extreme value of 8.69 × 10^{−4} m^{3} at *t*_{4}. Compared to Case 1, the variation law for Case 2 is similar, but because of the higher upstream water flow pressure, the maximum return-water velocity of 0.47 m/s is relatively smaller, and the extreme value of 7.54 × 10^{−4} m^{3} of the air-pocket volume is relatively smaller.

*t*

_{4}, the subsequent periodic evolution of the water–air two-phase dynamic process is similar to each other. Due to the dissipation of viscosity and friction resistance, the extreme values of the pressure of the air pocket gradually decay. Compared to the regular and clear water–air interface in the first cycle, the water–air two-phase mixing degree for two cases increases significantly from

*t*

_{4}to

*t*

_{7}, i.e. subsequent one and a half cycle. The water–air mixing degree is time-dependent for any case, and there are great differences in the water–air mixing degree between the two cases shown in Figure 6(f)–6(h). In order to well describe the water–air mixing degree, a dimensionless parameter

*r*is introduced, which is defined as the ratio of the volume occupied by the mixture to the total volume occupied by the fluid in the valve and its downstream pipeline as follows:where

_{v}*V*

_{mix}is the volume occupied by the water–air mixture where air-phase volume fraction is greater than 0 and less than 1, i.e. the volume of the zone occupied by the blue 3D solid shown in Figure 10;

*V*

_{zone}is the total volume occupied by the fluid in the valve and its downstream pipe. It can be seen that

*r*ranges from 0 to 1, and the larger

_{v}*r*, the more sufficient water–air mixing in the region. When

_{v}*r*= 1, the water–air in the region is mixed well, and when

_{v}*r*= 0, the corresponding region is full of pure air or pure water. For Case 1 and Case 2, the

_{v}*r*in the valve and its downstream pipe changes with time shown in Figure 11. For Case 1, the water–air mixing degree first increases and then decreases with time. From

_{v}*t*

_{0}to

*t*

_{4},

*r*gradually increases. At this stage, Fr number plays a major role, and therefore, the water–air interface is regular and clear. From t

_{v}_{4}to t

_{7},

*r*increases sharply and reaches the maximum value of 0.18 at 0.74 s, and then decreases gradually. Compared to Case 1, the higher initial pressure difference results in larger water flow impacting velocity and then enhancing water–air mixing degree for Case 2. From

_{v}*t*

_{4}to

*t*

_{7}, the water–air mixing degree for Case 2 is overall higher than that for Case 1, reaching the maximum value of 0.25 at 1.04 s.

In the successive evolutions after *t*_{7}, the water flow velocity for two cases further decreases gradually due to attenuation and the gravity effect increases relatively gradually, that is, as Fr number decreases, the water–air interface becomes regular and clear, the water flow gathers at the lower part of the pipeline and the corresponding *r _{v}* is small (Figure 6(i) and 6(j)).

Based on the above-mentioned analyses, for the transient rapid water impacting the entrapped air pocket, the air pockets can be identified into two types: Type I, i.e. air pocket with a low water content of *r _{v}* < 10% based on the description of the relatively water–air regular and clear interface, and Type II, i.e. air pocket with a high water content of

*r*< 10% based on the description of water–air mixing degree. Type II lasts one and a half cycles after the first compression process while the corresponding evolution belongs to Type I at the other time.

_{v}### Deviation of the physical quantities of air pocket from the ideal gas model

*V*

_{a}are the absolute air-phase pressure and the total volume calculated by the air-phase volumetric fraction, respectively; is the amount of substance of the air phase;

*R*is the universal air-phase constant, and

*T*is the temperature of the air phase.

Figure 12 shows the time-dependent evolution of and based on the volumetric fraction ranging from 0 to 1 obtained by the CLSVOF model for Case 1 and Case 2. It can be seen that those two terms change periodically and are similar to each other. However, there is a noticeable discrepancy between the values of and , especially at the peaks and valleys. The maximum difference between and accounts for 2.8% of the corresponding for Case 1 and 3.3% of the corresponding for Case 2. This implies that the variation of the physical state quantities of the air pocket deviates from the ideal gas model.

In order to explore the reasons the deviation, the air pocket is divided into four independent portions with the air volume fraction ranges of 0.0 < *α*_{a} ≤ 0.25, 0.25 < *α*_{a} ≤ 0.50, 0.50 < *α*_{a} ≤ 0.75, and 0.75 < *α*_{a} ≤ 1.0, respectively, for Case 1 and Case 2, and then the applicability of separate EOS for these four portions is studied systematically. And only the results for Case 1 are shown here (Figure 13) because the two cases provide similar effects.

First, for 0.75 < *α*_{a} ≤ 1.0, this portion of the air-phase volume can be regarded as a pure gas. It can be seen from Figure 13(a) that and are exactly equal to each other, which means that the variation of the state quantities of the air phase exactly follows the ideal gas EOS. Second, for 0.50 < *α*_{a} ≤ 0.75 and 0.25 < *α*_{a} ≤ 0.50, and of these two potions of the air-phase volume approximately follow the ideal gas EOS, respectively, that is, and are roughly equal to each other as shown in Figure 13(b) and 13(c). The maximum difference between the values of and accounts for 27.2% of the corresponding for 0.50 < *α*_{a} ≤ 0.75 and 12.2% of the corresponding for 0.25 < *α*_{a} ≤ 0.50, respectively.

Third, for 0.0 < *α*_{a} ≤ 0.25, and are completely not equal to each other as shown in Figure 13(d). The maximum difference between the values of and accounts for 97.0% of the corresponding . Because the air-phase volume fraction of this portion is very small and its corresponding water-phase volume fraction is particularly large, the properties of the water–air mixture are mainly determined by the water, and the temperature and pressure calculated based on the mixture model thus completely deviate from the ideal gas model.

Based on the above analyses, the ideal gas EOS is applicable to the condition that the air-phase volume fraction is greater than 0.25, and on the contrary, it is not applicable at all.

### Determination of polytropic index of the air pocket and relevant analyses

#### Identifications of polytropic process and compression–expansion switch

*P*and

*V*are the absolute air-phase pressure and the total air-phase volume, respectively; the subscripts 1 and 2 denote state parameters at two time moments;

*n*is the polytropic index ranging from −∞ to +∞; and C is a constant.

For air, *n* is 1.0 for isothermal, 1.4 for adiabatic, 0 for isobaric, and ∞ for isochoric. Based on the absolute air-phase pressure and total air-phase volume obtained by the CLSVOF model for Case 1 and Case 2, *n* at different moments can be calculated by using Equation (24) and it changes with time shown in Figure 14, where Figure 14(b) and 14(d) display the enlarged views of the areas labeled by the red wireframes in Figure 14(a) and 14(c), respectively, and the time moments *t*_{0} − *t*_{10} correspond to those in Figure 7. It should be noted that the state parameters of the air phase in the fraction range of 0.0 < *α*_{a} ≤ 0.25 deviate completely from the ideal gas EOS, however, based on the CFD results, the volume of this portion of air is very small and it can be confirmed from Equation (23) that is very small with a magnitude of 10^{−4}, the influence of this air portion on *n* can be considered although it is very small.

For Case 1, it can be seen from Figure 14(a) that the polytropic index of the air pocket varies periodically during the transient. This period consists of a compression process and an expansion process and its time interval is about 0.91 s. *n* changes sharply around the switching moments of compression and expansion processes of the air pocket due to the complex energy conversion processes. The switching processes include compression–expansion switches (around the time moments of *t*_{2}, *t*_{5}, *t*_{7}, *t*_{9}) and expansion–compression switches (around the time moments of *t*_{4}, *t*_{6}, *t*_{8}, *t*_{10}), in which the energy conversion processes are similar but the energy conversion directions are opposite. According to the first law of thermodynamics, the energy-conversion regulation of these two switch processes can be well analyzed as follows (Zemansky & Dittman 1997).

*q*is the heat exchange between the air pocket and the external; Δ

*u*is the internal energy increment of the system;

*w*is the process work of the air pocket;

*C*is the specific heat capacity at constant volume;

_{v}*T*

_{1}and

*T*

_{2}are the temperatures at two moments; and is the adiabatic index.

In the first compression–expansion switch: in the first compression before the compression–expansion switch moment, the pressurized water flow does work on the air pocket (*w* < 0), and the heat transfer from the air pocket to the water flow is intense because the temperature (Figure 15) of the air pocket is much higher than that of the surrounding water flow (*q* < 0). It can be seen that the work done by the pressurized water flow from *t*_{21} to *t*_{2} is much smaller than that from *t*_{1} to *t*_{21} due to the fact that the corresponding volume change rate from *t*_{21} to *t*_{2} is extremely small (Figure 4(a)); while the heat quantity transferred from the air pocket to the surrounding water flow from *t*_{21} to *t*_{2} is much greater than that of from *t*_{1} to *t*_{21} due to the fact that the corresponding temperature difference between the air pocket and the surrounding water flow from *t*_{21} to *t*_{2} is larger. Therefore, is much smaller than , and according to Equation (28), the *n* at *t*_{2} is much less than *n**=* 1.0 at *t*_{21}, and then, it can be concluded that only when the *n* at *t*_{2} is much less than 0 does it satisfy the above-mentioned ratiocination.

In the first expansion after the compression–expansion switch moment, the air pocket does work on the pressurized water flow (*w* > 0), but the heat transfer from the air pocket to water flow is still intense because the temperature of the air pocket is much higher than that of the surrounding water flow (*q* < 0). It can be seen that the work done by the air pocket from *t*_{2} to *t*_{22} is much smaller than that of from *t*_{22} to *t*_{3} due to the fact that the corresponding volume change rate from *t*_{2} to *t*_{22} is extremely small (Figure 4(a)); while the heat quantity transferred from the air pocket to the surrounding water flow from *t*_{2} to *t*_{22} is much greater than that of from *t*_{22} to *t*_{3} due to the fact that the corresponding temperature difference between the air pocket and the surrounding water flow from *t*_{2} to *t*_{22} is larger. Therefore, is much greater than , and according to Equation (27), the *n* at *t*_{2} is much far greater than *n**=* 1.4 at *t*_{22}.

The above theory and techniques can be used to analyze the sharp change of *n* at the other energy conversion moments.

#### Study on the range of polytropic index based on statistics

The duration of the drastic change of *n* is short and accounts for about 5% of a period and *n* mainly changes between 1.0 and 1.4 for about 95%. In practical engineering applications, *n* is either 1.0 for the isothermal process, or 1.4 for the adiabatic process, or 1.2 for the compromise of the two schemes, which are empirical. The first subscript *i* of *C _{ij}* represents different compression–expansion periods (

*i*= 1, 2, 3, 4) and its second subscript

*j*represents a compression and an expansion process in a period, respectively (

*j*= 1, 2) shown in Figure 14(b); it can be seen that

*n*fluctuates less and less with time in each

*C*.

_{ij}*n*distributions in the first four transient periods is presented in Figure 16. It can be seen from Figure 16(a) for Case 1 that the upper and lower boundaries of the box are the third quartile (

*Q*

_{3}= 1.36) and the first quartile (

*Q*

_{1}= 1.28) for

*C*

_{11}, where

*Q*

_{1}is calculated by:

in which is the sample function concerning ; is the integer portion of (*N* + 1)/4, *N* is the number of samples; and is the decimal portion of (*N* + 1)/4. Similarly, *Q*_{3} is obtained based on 3(*N* + 1)/4 instead of (*N* + 1)/4. The box length called interquartile range (IQR, IQR = *Q*_{3} − *Q*_{1}) 0.08 characterizes the concentration of *n*. The median of 1.35 characterizes the central tendency for *C*_{11} (the solid square in the box). The upper and lower whiskers, i.e. 1.39 and 1.06 for *C*_{11}, are obtained by *Q*_{3} + 3 IQR and *Q*_{1} − 3 IQR to determine the statistical outliers, which means that the values of *n* greater than 1.39 or smaller than 1.06 are judged as the statistical outliers. Similarly, the boxplot of *n* distributions for the other *C _{ij}* can be obtained also shown in Figure 16(a), where it can be seen that the median of

*n*changes greatly in the initial stage and gently in the later stage. For

*C*

_{11}and

*C*

_{31}–

*C*

_{42}, the medians of

*n*are concentrated around 1.36, which indicates that the polytropic processes of the air pocket in these processes are close to adiabatic. The IQR of

*n*for

*C*

_{12}is the largest among all the

*C*, its median is 1.27 and the median of 1.23 for

_{ij}*C*

_{21}is the smallest. The largest median of 1.38 for

*C*

_{42}indicates that the polytropic process of the air pocket is close to the adiabatic. Based on the above analyses, the averaged median for the polytropic indexes for

*C*

_{11}−

*C*

_{42}(1.32) is recommended.

For Case 2, the boxplot patterns of *n* are similar to that of Case 1 as shown in Figure 16(b), and similar to Case 1, the recommended polytropic indexes is 1.29, the averaged median for the polytropic indexes for *C*_{11} − *C*_{42}.

#### Influence of air-pocket types on polytropic indexes

Figures 14 and 16 show that the air pockets for *C*_{11} and *C*_{31} − *C*_{42} are Type I and Type II for *C*_{12} − *C*_{22}. For Case 1, it can be seen from Figure 16(a) that the averaged median for the polytropic indexes for *C*_{31} − *C*_{42} is 1.35 for Type I, indicating that the corresponding indexes of polytropic process change around 1.35 close to the adiabatic, and the averaged median for the polytropic indexes for *C*_{12} − *C*_{22} is 1.26 for Type II. For Case 2, it can be seen from Figure 16(b) that the polytropic indexes of air-pocket types are overall similar to those for Case 1, the averaged median for the polytropic indexes for *C*_{31} − *C*_{42} is 1.35 for Type I and 1.21 for *C*_{12} − *C*_{22} for Type II.

The reason for the discrepancy between the polytropic indexes of the two air-pocket types is that the water–air mixing degree of Type II is more intense than that of Type I. The heat transfer coefficient increases rapidly with the increase of air-pocket water content. Therefore, the heat-exchange quantity between air-pocket Type I with low water content due to the low air–water mixing degree and the water flow is relatively small, which shifts the air-pocket compression–expansion process close to an adiabatic one, while the heat-exchange quantity between the air-pocket Type II with high water content due to the high mixing degree and the water flow is large, which shifts the air-pocket compression–expansion process to tend to a compromise between the adiabatic and isothermal processes.

## CONCLUSION

The CLSVOF model and URANS equations with energy equation have been performed to study the two-phase hydraulic transient and thermodynamic characteristics. First, based on the experimental data, the prediction performance of the CLSVOF model and VOF model at pressure and the water–air interface was evaluated. Then, the CLSVOF model with higher prediction accuracy was used to further study the evolution mechanism of the water–air interface, the applicability of ideal gas EOS in air pockets with different air-phase volume fractions, and the variation law of air-pocket polytropic index and its related thermodynamic processes. Some specific conclusions were obtained as follows:

- (1)
The CLSVOF model and the VOF model can both capture the clear and regular water–air interface well. But the CLSVOF model can capture water–air interface fragments and dispersed bubbles better than the VOF model due to its superiority in capturing the curvature and normal vector of the interface. The relative errors between the first peak pressure by the CLSVOF model and the experimental data were 5.1% for Case 1 and 4.4% for Case 2, while the corresponding errors by the original VOF model were 6.0% for Case 1 and 5.0% for Case 2. Therefore, the CLSVOF model is more accurate and has a higher resolution than the VOF model.

- (2)
The volume and the pressure of the air pocket have the same change period of 0.91 s under the low initial pressure condition and 0.74 s under the high initial pressure condition, which is due to the different accelerations caused by initial pressure differences. When the air-pocket pressure reaches the maximum, the air pockets are compressed to the minimum volumes, which are 41.7 and 30.7% of the initial volume under the low and the high initial pressure conditions, respectively.

- (3)
The air pockets were identified into two types based on the analyses of the water–air interface evolution process and water–air mixing degree. For Type I, the water–air mixing degree was less than 10% and the water–air interface was regular and clear. There Type I existed approximately in the first compression process and after the second expansion process, the inclination angle of the interface increased with the increase of Fr. For Type II, there were more fragmented water–air interfaces and two or more dispersed air pockets, and the water–air mixing degree was higher than 10%. There Type II existed approximately from the beginning of the first expansion to the end of the second expansion process.

- (4)
The applicability of the ideal gas EOS was investigated to demonstrate when the air-phase volume fraction was smaller than 0.25, due to the effect of water-dominated properties of the water–air mixture, the variation of the physical quantities of air pocket was found to deviate from the ideal gas EOS. The variation of the physical quantities of the air pocket was completely following the ideal gas EOS for air-phase volume fraction in the range of 0.75 <

*α*_{a}≤ 1.0 while it had a slight deviation from the ideal gas EOS for the range of 0.25 <*α*_{a}≤ 0.75 and completely deviated from the ideal gas EOS for the range of 0.0 <*α*_{a}≤ 0.25. - (5)
The investigation of the air-pocket polytropic index determined by the EOPP showed that its variation period was 0.91 s for Case 1 and 0.74 s for Case 2. The polytropic index changed sharply near the switching moments of air-pocket expansion and compression processes, the reason being that before and after the switching moment, the heat absorption or heat release direction of the air pocket remains almost unchanged while the direction of the process work by the air pocket on the water flow is reversed. The duration of the sharp change of the polytropic index accounted for about 5% of the period while the polytropic index mainly changed between 1.0 and 1.4 with a duration of 95% of the period.

- (6)
The analyses of the air-pocket thermodynamic process showed that for air-pocket Type I, its process was close to adiabatic with an average median of 1.35 of the polytropic index for both Case 1 and Case 2. For air-pocket Type II, the more water content in the air pocket accelerated the heat transfer between the air and the water flow, so the thermodynamic process was close to a compromised one with an average median of 1.26 of a polytropic index between adiabatic and isothermal for Case 1 and 1.21 for Case 2. In practical engineering applications, the selections of air-pocket polytropic indexes should fully consider the heat transfer between air and water under different water–air mixing degree conditions. The recommended polytropic indexes were 1.32 for low initial pressure conditions and 1.29 for high initial pressure conditions based on the boxplot distribution of the polytropic index and the above-mentioned analyses.

## ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52079140, 51779257).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICTS OF INTEREST STATEMENT

The authors declare there is no conflict.