Experimental and CFD investigations of choked cavitation characteristics of the gap flow in the valve lintel of navigation locks ＊

Bo Wu , Ya-an Hu , Xin Wang Xiu-jun Yan

1. Key Laboratory of Transport Technology in Navigation Building Construction, Nanjing Hydraulic Research Institute, Nanjing 210029, China

2. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Nanjing 210029, China

3. School of Water Resources and Hydropower Engineering, Wuhan University, Wuhan 430072, China

Abstract: The cavitation is ubiquitous in the water delivery system of high hydraulic head navigation locks. This paper studies the choked cavitation characteristics of the gap flows in the valve lintel of the navigation locks and analyzes the critical self-aeration conditions. The cavitation gap flow in the valve lintel is experimentally and numerically investigated. A visualized 1:1 full-scale slicing model is designed, with a high-speed camera, the details of the cavitation flow is captured without the reduced scale effect.Moreover, the numerical simulations are conducted to reveal the flow structures in the gap. The experimental results show that the flow pattern of the gap flow in the valve lintel could be separated into four models, namely, the incipient (1) the developing, (2), the intensive, (3), and the choked (4) cavitation models. The numerical simulation results are consistent with the experimental data. The choked cavitation conditions are crucial to the gap flow in the valve lintel. When the choked cavitation occurs, the gap is entirely occupied by two cavitation cloud sheets. The gap pressure then decreases sharply to the saturated water vapor pressure at the operating temperature. This water vapor pressure is the ultimate negative pressure in the gap that remains unchanged with the continuous decrease of the downstream pressure. The volumetric flow rate reaches a peak, then remains constant, with the further decrease of the pressure ratio or the cavitation number. At the choking point, the volumetric flow rate is proportional to the root mean square of the difference between the upstream pressure (absolute pressure) and the saturated pressure of the water. Moreover, the pressure ratio is linearly correlated with the downstream cavitation number with a slope of (1+ ζ c).

Key words: Cavitation, cavitation control by aeration, self-aerated flows, critical self-aerated conditions, choked cavitation

Introduction

The valve lintel gap is inevitable between the valve panel and the corridor in the navigation lock filling and the discharging valve’s valve lintel. This gap is narrow, and the hydraulic head acting upon it is the actual working head of the lock valve[1]. When a high-speed water flows pass through the lintel gap, the pressure in the gap reduces considerably according to the Bernoulli’s law, and a strong cavitation will then occur subsequently not only to erode the flow passage but also to increase the pulsation of the valve opening and closing forces[1].

The cavitation control by the aeration is one of the most effective methods. The Venturi aeration is a technique that can be used to achieve the self-aeration rather than the forced aeration, without the need of extra air compressor or control system. As the valve lintel gap is unavoidable, our starting point is to design the gap in the shape similar to a Venturi tube and set a series of perforates at the throat part along the valve’s width direction, and use the aeration characteristics of the Venturi effect to achieve the self-aeration in the valve lintel gap. Figure 1 shows how the aerated jet flow in the valve lintel gap addresses the cavitation problems of the gap. The aerated jet flow in the gap of the valve lintel not only addresses the gap cavitation, but also could solve the hemline cavitation when it reaches the main flow and mixes with it. These features are the essence of the self-aerated technology of the valve lintel (SATVL).As an original technology, due to its simple geometry,the absence of moving parts, the high reliability, and the long life, the SATVL is widely used to address the cavitation problems in filling and emptying valves of navigation locks during the last three decades in China,such as the double-line and five-grade continuous navigation locks of the Three Gorges Project.

Fig. 1 (Color online) Sketch and photos of the self-aerated gap flow in the valve lintel. (a) Self-aerated flow in the valve lintel gap. (b) Aerated flow behind the filling valve

Despite the widespread use of the SATVL, its role and interactions among various designs and operating parameters in the resulting cavitation and aeration behavior are not adequately studied. With the gap profile in the valve lintel similar to that of a Venturi tube, the cavitating venturis (CVs) and orifice plates are commonly used in hydrodynamic cavitation reactors[2-3]and they become an important direction of experimental and numerical investigations in literature[1,4]. The related research results in these studies might provide a theoretical reference to the practical application of the SATVL. Yan and Thorpe[5]pioneered the investigation of the cavitation flows,particularly, those of the choked cavitation flows and the choked cavitation conditions, and the choked cavitation number was proposed to describe the choked cavitation flows. He pointed out that the CVs could provide a constant mass flow rate of liquid while operating under the choked cavitation conditions, and the choked flow regime could be used to control the mass flow rate. This idea was supported by Long et al.[6]and it was confirmed that once the choked cavitation occurs, the flow rate remains almost constant regardless of variations in the outlet pressure.The same cavitation mechanism was observed in the orifice plate by Cioncolini et al.[7]experimentally during the choked flow, wherein the micro-orifice plate discharge was proportional to the square root of the upstream pressure, and the choked cavitation number was related to the diameter and the thickness of the micro-orifice.

Various experimental methods were adopted to further study this complex two-phase cavitation flow,such as the visual analysis of high-speed instantaneous series images and the PIV measurements, as presumably the most effective methods available today for investigating cavitation flows[6,8-10].Recently, Long et al.[6]measured the cavity length based on instantaneous cavitation photos and reported that this length is only a function of the pressure ratio or the cavitation number, independent of inlet pressures. According to the results of experiments for aerated and non-aerated cavitations in a horizontal transparent Venturi nozzle, Tomov et al.[8]identified three different flow regimes of the cavitation, namely,the sheet, the cloud, and the “super” cavitations.

Computational fluid dynamics (CFD) is an effective method to be used to shed further insight into the cavitation flows and the vortex shedding, and to help the cavity evolution visualization, , thereby to reveal the mechanism of the cavitation flow. CFD could be used under extreme operating conditions when physical models would fail. Various models were developed to simulate the cavitation, and the most widely used among them could be placed under two categories, namely, the Reynolds-averaged Navier-Stokes (RANS) model and the large-eddy simulation (LES).

In the first category, the RANS model is frequently used to simulate the cavitation flows[11], but this model requires turbulence models to close the RANS equations[12]. Hence, the accuracy of CFD used for the cavitation flows depends on both turbulence and cavitation models. The turbulent kinetic energy(k)-the turbulent dissipation rate ()εis a widely used and tested turbulence model. Shih et al.[13]proposed the realizablek-εmodel to improve and modify the standardk-εmodel. Ashrafizadeh and Ghassemi[14]used this turbulence model to simulate the performance of small-sized CVs and found that the main sources of errors against experimental results were related to the turbulence model and the wall functions. A series of numerical simulations of CVs using this model in the literature were completed with satisfactory simulation accuracy[4,15].

In the second category, LES has the potential of obtaining higher accuracy and wider applicability than RANS and requires less computational demands than the direct numerical simulation (DNS). Compared with RANS, LES has large energy-containing structures that can be resolved directly, but with considerably higher time and computation resource consumptions[12,16-17]. LES was widely used in hydrofoil cavitation simulations to reveal vortex shedding details[16,18-21]. Nouri et al.[22]investigated the cavitation flow in CVs via LES. The results showed that LES could reveal more flow details by using more refined meshes than RANS with large eddies being calculated and small eddies being modelled.

The valve lintel gap is a special Venturi tube with an asymmetrical geometry and complicated flow passage geometry parameters. A fundamental analysis of this gap is desirable for its design and optimization.In this paper, the performance of the valve lintel gap’s cavitating flow is experimentally and numerically investigated, to reveal the mechanism of the choking cavitation in the valve lintel gap.

1. Experimental setup and numerical preparation

1.1 Test ring and the 1:1 full-scale slicing model

Figure 2 shows (a) the sketch and (b) the photo of the test ring, which is equipped with three centrifugal pumps to provide the hydraulic head and the flow rate required for the test. A pressure-limiting valve is installed at the inlet of the test device to prevent the upstream pressure from increasing sharply after the gap is blocked by foreign matter. The incoming flow is rectified by using a pressurestabilizing tank. The upstream and downstream pressures are measured by using pressure gauges and adjusted accurately via the main valve and bypass valve installed upstream and downstream, respectively.The flow rate is measured by an electromagnetic flowmeter. Table 1 lists the main equipment used in this study.

Figure 2(a) shows the test section in detail, which is a 1:1 full-scale valve lintel slicing physical model.It is made by cutting the valve lintel gap (120 mm width) in the direction of the valve width. The gap pressure is measured by using five pressure sensors whose locations are shown in Fig. 3(a). Important features of this 1:1 full-scale slicing model might be highlighted as follows:

(1) The model is manufactured with high pressure-resistant transparent plexiglass for the visualization of vapor cavity patterns in the gap.

(2) As it is a 1:1 full-scale slicing model, there will be no size scaling effect.

(3) It not only can truly reflect the cavitation characteristics of the gap flow in the valve lintel by closing the aeration valves, but also can be used to study the aeration characteristics by opening the aerated valves, especially the critical self-aerated conditions which will be studied further in future.

Fig. 2 (Color online) Sketch (a) and photo (b) of the test ring and the 1:1 full-scale slicing model. (1- Pump, 2-Electric machinery, 3-Main valve, 4-Bypass valve, 5-Pressure limiting valve, 6-Electromagnetic flowmeter, 7-Pressure stabilizing tank, 8-Pressure gauge, 9-Pressure transducer,10-Hydrophone, 11-Air flowmeter, 12-Highspeed camera, 13-Data acquisition system, 14-Floor, 15-Sump,16-Valve panel, 17-Valve lintel)

1.2 Numerical model and verification

1.2.1 The multiphase model

The mixture model is a simplified multiphase model that can be used in different ways. The continuity equation for the mixture is:

Table 1 Technical parameters of the main equipment

wherevmis the mass-averaged velocity,mρis the mixture density,nis the number of phases andkαis the volume fraction of the phasek.

The momentum equation for the mixture can be obtained by summing the momentum equations for all individual phases. It can be expressed as:

whereFis the body force,mμis the viscosity of the mixture.

wherevdr,kis the drift velocity for the secondary phasek

1.2.2Turbulence model

As the emphasis of this paper is the large-scale simulation with real operating parameters, we adopt the RANS method with an appropriate turbulence model as the numerical method. Brinkhorst[4]and Ashrafizadeh[14]revealed that compared to the other RANS type models, with the realizablek-εmodel,the best results can be obtained for the CVs. The modeled transport equations forkandεin the realizablek-εmodel are shown in reference[12,23]. In this study, we adopt the realizablek-εturbulence model, including a Standard Wall Treatment, which shows a good accuracy in capturing the mean flow of the complex structure compared to all otherk-εmodels[14,24].

1.2.3 Cavitation model

In the cavitation, the liquid-vapor mass transfer(the evaporation and the condensation) is governed by the vapor transport equation and the bubble dynamics equation[23]:

and

whereνrepresents the vapor phase,αis the vapor volume fraction,vρis the vapor density,vvis the vapor phase velocity,Re,Rcare the mass transfer source terms related to the growth and the collapse of the vapor bubble, respectively. The termsRe,Rcaccount for the mass transfer between the liquid and vapor phases in the cavitation, which are modeled based on the Rayleigh-Plesset equation for the growth of a single vapor bubble in a liquid.b?is the bubble radius (=10-6m),σis the liquid surface tension coefficient,lρis the liquid density,Pbis the bubble surface pressure andPis the local far-field pressure.

In this paper, the Zwart-Gerber-Belamri model is used for the net mass transfer calculation. The final form of this cavitation model is[23]:

whereαnucis the nucleation site volume fraction(=5×10-4),FvapandFcondare the coefficients of the evaporation and the condensation which are equal to 50 and 0.01, respectively.

1.2.4 Mesh, boundary conditions and grid independence analysis

The geometry parameters of the valve lintel used in this paper are illustrated in Fig. 3(a), and the meshes are generated by the ANSYS ICEM software[23]. The whole computational mesh and the close-up at the throat of the medium grid are shown in Figs. 3(b), 3(c), with an increased density of the mesh in the flow direction around the throat section to capture the two-phase flow with accuracy. In this study, the 2-D geometry is meshed with three different mesh sizes as given in Table 2 to carry out the grid independence analysis.

Fig. 3(a) Geometry parameters of the valve lintel (mm)

Fig. 3(b) The whole mesh of the computational domain

Fig. 3(c) Close-up of medium mesh at throat part

The following boundary conditions are applied to all simulations: the pressure inlet and outlet boundary conditions, and the no-slip boundary condition at the wall. The pressure boundary conditions in the simulations are set the same as those in experiments. The first-order upwind method is used for the spatial discretization, the pressure-velocity coupling is considered by the SIMPLE algorithm, which is used to enforce the mass conservation and to obtain the pressure field. The simulations are carried out using the ANSYS FLUENT software with a 2-D pressurebased solver and under transient conditions. For a proper time-wise resolution, a time step of 10-4s is adopted in the transient simulations, which is approximately equal to the time that the water passes through a single cell.

Table 2 Properties of the grids for mesh independence analysis

For the turbulence model used, when the wall function is introduced to describe the boundary layer behavior up to the wall, the nodes in the first layer should be placed in the fully turbulent sublayer(namely the “l(fā)og law region”). Therefore, the value ofy+must be checked. Taking the medium mesh as an example, it is found that except for a few small regions (in the throat part), wherey+is less than 30,in most parts it varies from 34 to 134. In addition, it is checked that the three grids are in the acceptable range(y+=30-300)as reported in ANSYS fluent theory guide[23].

Taking the pressure profile at the valve panel and the velocity profile at the throat as references, it can be seen from Fig. 4(a) that the pressure distribution on the valve panel calculated by different grid schemes tend to be equal, the pressure drops rapidly in the convergent part because of the increase of the velocity.The pressure at the divergent part is close to the saturated vapor pressure due to the intensive cavitation. On the other hand, from the velocity distribution at the throat, as shown in Fig. 4(b), it can be observed that the fine and medium grids give very similar results, while the coarse grid does not. That indicates that the spatial discretization errors of increasing the number of grids on the calculation results can be neglected.

On the other hand, the comparison of the volumetric flow rate between the experimental result and numerical results obtained with different grids under the same operation conditions is shown in Table 3. One can see that the medium and finer grids both have the minimal deviation as compared to the experiment.

In summary, from the comparisons of the pressure, the velocity and the volumetric flow rate between the numerical results obtained with different grids and the experimental results, it is evident that the results obtained with the medium grids have desired accuracy and mesh resolution, with good agreement with the experimental results. Therefore, to balance the accuracy and the time consumption, the medium mesh is chosen for all investigations in this study.

Fig. 4(a) (Color online) Numerical pressure distribution at the valve panel and velocity distribution at the throat

Fig. 4(b) (Color online) for the three grids used in the mesh convergence study wherepu =490.50 kPa ,pd=49.05kPa (relative pressure)

Table 3 Comparison of flow rate per unit width between experimental result and numerical results obtained with different grids

1.2.5 Verification of the numerical solution

In this study, the pressure ratio (pr) is introduced to characterize the relative magnitude of the downstream and upstream pressures.It is a dimensionless quantity and is calculated as follows

wherepais the atmosphere pressure,pvis the vapor pressure corresponding to the liquid temperature,puandpdare the upstream and downstream static pressures, respectively.

With the medium grid, the volumetric flow rate per unit widthqwand the relative errors against the pressure ratioprare plotted in Fig. 5. The calculated and measured values of the gap pressure are listed in Table 4 (pu=490.50 kPa ,pd=0 kPa ). As can be seen, the relative errors of the gap pressure and the volumetric flow rate between experimental and numerical results are both less than 10%. This deviation could be attributed to the uncertainties of the reference measurement and the error due to the experimental method, and also to the shortcomings of the CFD simulation. In the numerical method, the main sources of errors are, in addition to the cavitation model itself, the turbulence model (RANS Reynolds averaging hypothesis) and the wall function[14,25], as well as the numerical algorithm[22]. But, in general, the good agreement between the simulation and the experiment indicates that the numerical method could be used for further calculations.

Fig. 5 (Color online) Comparisons between experimental and numerical results of volumetric flow rate per unit width

2. Results and analysis

In this section, the experimental and numerical results are combined to show the flow patterns and the flow structure characteristics of the gap flow in thevalve lintel. Particularly, the choked cavitation effect is highlighted.

Table 4 Comparisons between calculated and measured values of the gap pressure (kPa)

2.1 Flow pattern in four different cavitation stages

The experimental results in Fig. 6 show that in the valve lintel gap, we have incipient, developing,intensive, and choked cavitations under different operating conditions.

Cavitation bubbles first occur at the aerator(throat) because of the large shear stress and the separation of the boundary layer, and then they collapse in the diffuser due to the high pressure (Fig.6(a)). This process is so-called the incipient cavitation.Iyer[26]observed that high levels of vorticity, strain rate, and Reynolds stress occur in the shear layer region, and similar counterclockwise vortices are found in this region, to form, thereby, bubbles in the low-pressure core of vortices. As the downstream pressure decreases, the flow in the gap contains clear water and cavitation bubbles (Fig. 6(b)). These cavitation bubbles flow downstream and collapse in the convergent region. Further decreasing the downstream pressure, the boundary layer of the valve panel will separate, and the cavitation bubbles will be congregated into the cavitation cloud. However, there will still be a narrow and unstable clear water clearance in the gap (Fig. 6(c)). The phase transition between the vapor and the liquid is always in a state of dynamic balance. Finally, when the downstream pressure is lower than a certain critical value, the two cavitation cloud sheets contact each other and block the bubble flow. At this time, the clear water area is no longer there and the gap flow is choked (the choked cavitation) (Fig. 6(d)). The choked cavitation will be discussed in combination with the pressure distributions, the velocity profiles along the throat,and the volumetric flow rate characteristics.

2.2 Flow structures within the gap

To observe the choked cavitation flow in the gap,we set the upstream pressure to 392.40 kPa,490.50 kPa and 588.60 kPa (relative pressure) and change the downstream pressure. As most cases are similar, we only present the case ofpu=490.50 kPa in this section. This case represents the typical working hydraulic heads in the navigation locks. The results of all cases will be discussed in the section of the choked cavitation effect.

2.2.1 Wall pressure distribution

The time-averaged pressure distribution in the gap shown in Fig. 7 indicates that it is difficult to identify the inception of the cavitation by means of numerical simulations because a considerable number of repeated calculations of different pressure combinations are involved. The low cavitation number increases the negative pressure zone (Fig. 7). Figure 8(a) shows the time-averaged pressure distribution along the valve panel under different operating conditions when the upstream pressure is constant.When the downstream pressure is below 147.15 kPa,the throat and diffusion sections are in the state of the saturated vapor pressure (Fig. 8(a)). When the downstream pressure is increased, the pressure at the throat and diffusion sections is also increased accordingly because the throat and diffusion sections are in the state of the cavitation blockage when the downstream pressure is lower than a certain value, in the state of the saturated vapor pressure of the water.

Fig. 6 (Color online) Instantaneous images of four different cavitation models of the gap flow in the valve lintel. (a) Incipient cavitation (clear water occupies the entire flow tract and some discrete bubbles occur in the aerator with counterclockwise vortices). (b) Developing cavitation (cavitation bubbles extend into the diffusion part and form a white cavitation cloud.A clear water region is found on the left side). (c) Intensive cavitation (white attached cavitation clouds are observed on the two wall boundaries, while there is still a narrow and unstable clear water region in the middle of the two cavitation clouds). (d) Choked cavitation (white cavitation clouds occupy the entire flow tract and the clear water region is no longer there)

Fig. 7 (Color online) Relative pressure distributions within the valve lintel gap for p u =490.50 kPa . (a) p d =0 kPa , (b)pd=49.05 kPa , (c) pd =98.10 kPa , (d) pd =147.15kPa , (e) pd =196.20 kPa , (f) pd =245.25 kPa , (g) pd=294.30 kPa, (h) pd =343.35kPa, (i) pd=392.40 kPa

Fig. 8 (Color online) Time-averaged pressure distribution along the valve panel and velocity along the throat at pu=490.50 kPa

Therefore, when the pressure at the throat reaches the saturation pressure of the liquid at the working liquid (water) temperature, which is the ultimate negative pressure in the valve lintel gap, the pressure at the throat and diffusion section will not change with the continuous decrease of the downstream pressure.

2.2.2 Flow velocity profiles along the throat

Figure 9 shows the flow velocity distributions.The sequence of the velocity evolution is accompanied with different downstream pressures. Figure 8(b) indicates that the velocity distribution along the throat varies with different downstream pressures when the downstream pressure is lower than the critical value. On the other hand, the volumetric flow rate will not vary with the change of the downstream pressure when the flow is choked.

2.2.3 Volume fraction distribution of the vapor phase

Figure 10 shows the time-averaged volume fraction distributions of the vapor phase. The following process is demonstrated: The four cases from (i) to (f) are all under the non-cavitation conditions because the gap pressure is high and no cavitation occurs. From (f) to (d), the clear water zone becomes increasingly narrow and completely disappears as the left and right areas of the gap cavity cloud converge along with the further development of the cavity in the valve lintel and the valve panel.This process is the beginning of choked cavitation conditions. From (d) to (a), the gap is congested by the cavitation pocket, and the pressure inside the gap should correspond to the vapor pressure at the operating temperature[27], and those four cases are all under the choked cavitation conditions. Besides, the numerical results of the vapor phase distribution are consistent with the experimental ones as discussed in section 2.

Fig. 9 (Color online) Contours of velocity in the valve lintel gap under the same operating conditions as described above

Fig. 10 (Color online) Contours of the volume fraction (vapor) within the valve lintel gap under the same operating conditions described above

2.3 Choked cavitation effect

To study the cavitation flow and the choked cavitation effect, the downstream cavitation number is introduced in this paper, which was used in some similar studies of the cavitation Venturi tubes and orifice plates and was defined as[6,7,28]

wherevthis the mean velocity at the throat(calculated by the volumetric flow rate divided by the throat area).downstream cavitation number (Fig. 11), it is seen that the volumetric flow rate per unit width remains constant when the pressure ratioprreaches a critical value of approximately 0.53 (Fig. 11(a)) or the cavitation number is equal to 0.63 (Fig. 11(b)) and even when the pressure ratio or the cavitation number is further decreased. Similar cavitation Venturi tubes studied by Ghassemi et al.[4,16,23]show that the critical pressure ratios are 0.70, 0.80 and 0.72. In other words,when the downstream pressure reaches a certain threshold, the further development of the cavitation leads to the so-called choked cavitation. Then the volumetric flow rate reaches a peak, remains constant,and does not respond to the continuous decrease of the pressure ratio or the cavitation number.

3. Discussions

Fig. 11 (Color online) Volumetric flow rate per unit width versus (relative pressure)

When the aeration valves are closed, the valve lintel gap flow has four modes of operation: the incipient, the developing, the intensive, and the choked cavitation modes. In the choked cavitation mode, the cavitation clouds are expended to the diffuser section and then fully congest the valve lintel gap. Under this condition, the flow process in the convergent area can be regarded as a 1-D isentropic flow. The density can also be regarded constant and equal to the liquid density at the operating temperature[9]. The velocity along the throat part is calculated by assuming that the static pressure of the fluid is decreased to the vapor pressure before the fluid is vaporized in the throat, thereby the volumetric flow rate becomes constant and independent of the downstream pressure.

The relative volumetric flow rate in the valve lintel gap could be expressed in dimensionless form, whereqw,chois the choked volumetric flow rate at the corresponding upstream pressure.

For three different upstream pressures, the normalized volumetric flow rates are plotted versus the pressure ratio in Fig. 12. One can see that only slight differences could be observed among the different upstream pressures due to the wide range of variation of the relative volumetric flow rates against the pressure ratio. A similar trend was reported by Ashrafizadeh and Ghassemi[14], Cioncolini et al.[7]and Mishra and Peles[29]too.

Fig. 12 (Color online) Normalized volumetric flow rate versus the pressure ratio

For a steady and choked cavitation flow, the static pressure in the throat can be assumed to be equal to the saturation vapor pressure of the water.The following Bernoulli’s energy equation[14]is applied between the inlet and the throat part

whereρldenotes the density of the water,puis the pressure at the inlet,α1,α2are the kinetic energy correction factors andα1=α2= 1. The flow process in the convergent section can be regarded as a 1-D isentropic flow[16].cζis the local head loss coefficient of the convergent part and its value depends on geometrical features and typically has no correlation with the Reynold number because the flow is in the region of quadratic resistance law[29].

Given thatvu≈ 0 and the position head difference is ignored, the following equation is used when the choked cavitation occurs:

Substituting Eq. (12) into Eqs. (9) and (10), we have

wherepvis the saturation vapor pressure at the inlet temperature.

Hence, when the choked flow occurs, the pressure ratio and the cavitation number are linearly related, with a slope of (1+ζc) . At choking,pthis assumed to be equal to the saturated pressure. Hence,Eq. (12) further indicates that the choked volumetric flow rate is proportional to the square root of the difference between the upstream pressure (absolute pressure) and the saturated water pressure. This result is highly consistent with the conclusions of Long et al.[6]and Cioncolini et al.[7], wherein the choked mass flow rate is proportional to the square root of the upstream pressure through the micro-orifice.

Figure 13 shows that the three curves of the dimensionless cavitation number (σ) versus the pressure ratio (Pr) are almost overlapped. For a given gap in this study, the local head loss coefficient of the convergent part is approximately 0.19.Particularly, when the choked cavitation occurs, by applying the least square fitting procedure to a linear function, the fitting line under the choked cavitation conditions indicates a clear linear relationshipσ=(1+0.19)Pr(the red magnified zone in Fig. 13,R2=0.99). As Eq. (13) reveals, that is the reason whyσis linearly correlated withPrand its slope is (1+ζc). This result is consistent with the general trends reported in literature[6,30].

Fig. 13 (Color online) Cavitation number as a function of the pressure ratio and the magnified image of choked cavitation conditions

4. Conclusions

In this study, experimental tests and CFD analyses are performed to study the cavitation flow in the gap of the navigation lock filling and the discharging valve’s lintel. The main results are summarized as follows:

(1) In view of the different cavitation evolution processes, the flow pattern of the gap flow in the valve lintel can be divided into four modes, namely, the incipient (a), the developing (b), the intensive (c), and the choked (d) cavitation modes. In particular, the choked cavitation conditions are crucial to the gap flow in the valve lintel. The characteristic feature of this type of cavitation is that the volumetric flow rate remains unchanged once the pressure ratio or the cavitation number is lower than a certain threshold. In this case, the pressure in the gap decreases into the saturated vapor pressure of the water at the operating temperature, and the white cavitation clouds occupy the entire flow tract.

(2) For a given geometric shape of the valve lintel gap, the choked cavitation conditions are reached at the same critical pressure ratio and the cavitation number (0.53, 0.63, respectively) regardless of different upstream pressures. When the cavitation flow is choked, the volumetric flow rate remains almost constant and is proportional to the square root of the difference between the upstream pressure(absolute pressure) and the saturated vapor pressure of the water.

(3) In the choked cavitation flow, the cavitation number and the pressure ratio are linearly related. The slope of the fitting curve is (1+ζc) and equal to the ratio of the critical choked cavitation number to the critical choked pressure ratio (0.63/0.53 in this study).