Tingyun Yin
email: yintingyunwt@163.com

Shouqi Yuan

Yin Luo

Ji Pei

Wenjie Wang

CAVITATION OPTIMIZATION FOR RESIDUAL HEAT RE-MOVAL PUMP USING ORTHOGONAL EXPERIMENTAL METHOD BASED ON PUMPLINX

Keywords:

To improve the performance of the centrifugal pump with a vaned diffuser, the influence of impeller geometric parameters on cavitation of the pump was investigated by Orthogonal Experimental Method (OEM) based on PUMPLINX. Impeller inlet diameter D1, inlet incidence angle Δβ, blade wrap angle φ were selected as the main impeller geometric parameters and the orthogonal experiment of L9(3*3) was done in this study. Three-dimensional steady simulations were conducted to predict the cavitation performance of pump using constant gas mass fraction model with 2 nd order upwind. The experimental results were justified by the variance analysis method. The inner flow of the pump was also analyzed in order to obtain the flow behaviors that can affect the pump performance. The results show that the impeller inlet diameter D1 has the greatest influence on the cavitation performance. The final optimized impeller accomplished better cavitation performance, which can meet the design requirements and the velocity distribution in the optimized impeller is more regular.

INTRODUCTION

Cavitation is a difficult problem to the development of hydraulic machinery, causing vibration, and noise, and affecting operation reliability of the pump. Residual heat removal pump is one of the key equipment in the millionkilowatt nuclear power plant, and it plays a vital role in the safety of the entire plant. Therefore, as an important characteristic of residual heat removal pump, the cavitation performance is usually studied.

Improving the cavitation performance by optimizing the impeller has been successfully achieved in a lot of studies. Xie [START_REF] Xie | Optimization of centrifugal pump cavitation performance based on CFD[END_REF] applied the numerical method to optimize slots on impeller blade near inlet in six groups of hydraulic model. The stimulation results showed that slotted impeller could improve the cavitation performance. Luo [START_REF] Luo | Effect of impeller inlet geometry on centrifugal pump cavitation performance[END_REF] analyzed the effects of impeller inlet geometry on centrifugal pump cavitation performance by using CFD. The pump cavitation performance can be significantly improved with new impellers designed by properly extending the blade inlet toward the pump inlet or enlarging the blade inlet angle. Wang [START_REF] Wang | Effects of Vane Wrap Angle on Cavitation Performance for Centrifugal Pumps[END_REF] redesigned the impeller with different vane wrap angles to study the effect of vane wrap angle on cavitation performance. It turned out that there would be an optimal value for vane wrap angle to obtain the best cavitation performance. Ran [START_REF] Tao | Optimization for Cavitation Inception Performance of Pump-Turbine in Pump Mode Based on Genetic Algorithm[END_REF] et al applied the genetic algorithm to modify the leading edge ellipse ratio and blade thickness on the front 20% meanline. By using CFD simulation, optimization was completed with obvious improvements on the cavitation inception performance.

The orthogonal design experiment is one of the most popular and efficient optimization methods to achieve the design objective which has many factors. Yuan [START_REF] Yuan | Orthogonal experimental study effect of main geometry factors of splitter blades on pump performance[END_REF] et al applied the orthogonal experimental to investigate the main parameters of the splitter blades on the pump performance and found that the splitter blades could make the curve of Q-H more flat and the pump with splitter blades could operate under overload conditions. Yan [START_REF] Hao | Numerical simulation on cavitation characteristics of large-scale axial-flow pumps based on orthogonal experiment[END_REF] optimized the cavitation performance of a large-scale axial-flow pump with the orthogonal experimental method and the numerical simulation. The results showed when the flow line of suction surface φ=0.7, the cavitation persistence length can be rapidly decreased and the cavitation volume fraction can be sharply cut. Shen [START_REF] Yanning | Orthogonal test design method based on numerical simulation for non-overload centrifugal pump with complex impeller[END_REF] et al applied the orthogonal experimental to optimize the parameters of the complex impeller. Through the test of specimen of the best design scheme, it demonstrated that the excellent selection by using numerical simulation of whole flow field was nonoverload and high efficiency.

In this study, computational fluid dynamics (CFD) technology was used to test three factors (impeller inlet diameter D1、inlet incidence angle Δ β、blade wrap angle φ) affecting cavitation performance of the residual heat removal pump by Orthogonal Experimental Method (OEM) to study the cavitation characteristics of pump, as well as internal flow instability induced, and to analyze the effects of three factors on the pump hydraulic performance. Besides, the inner flow characteristics of the final optimization were analyzed compared with the original one. And the mechanism of cavitation was further discussed to obtain the optimal parameters combination for pump design. Schematic map of optimization process as shown in Figure 1.

GEMOETRY AND EXPERIMENT 1.1 Gemoetry model

The 2D model of the radial diffuser pump is shown in Figure 1. The hydraulic components of the pump consist of the impeller, diffuser and volute. To be specific, the impeller is shrouded with 5 twist and backswept blades with the specific speed, ns=3.65nQ 0.5 /H 0.75 =111.7. The diffuser is shrouded with 7 radial vanes. The annular volute is designed to keep the pump operating safety. The design parameters and the main parameters of the diffuser and the volute will be shown in Table 1. The optimal parameters of the impeller will be presented in next section.

NUMERICAL SIMULATION

Mesh generation cavitation model

A grid generation method based on binary tree algorithm is used to generate the adaptive Cartesian grid in PUMPLINX.

The coarse grids are densified based on the dichotomy method in the high-required geometric precision region by using this technique. The grid will be cut in accordance with the surface of the fluid domain boundary to precisely fit the fluid domain on the boundary. The whole computational domain made up of the inlet, impeller, diffuser and volute consists of 1060291 grids. The details of the mesh are showed in Figure 3. In this paper, three-dimensional steady simulations were conducted to predict the cavitation performance of pump using constant gas mass fraction model with 2 nd order upwind.

BOUNDARY CONDITIONS

Boundary conditions of the centrifugal pump can be set parametricly, which the inlet pressure, out flow rate, and rotor are needed in PUMPLINX. Table 2 shows main boundary conditions required for PUMPLINX, and the working fluid parameters are showed in Table 3.

Table 2. Main Boundary Conditions

COMPARISON OF PERFORMANCE

By using the numerical simulations under 5 different flow rates, the predicted pump performance curve are obtained. To verify the accuracy of the simulation, the original pump, the size of which was scaled to 0.7, was tested in the open test rig. The rotating speed of the pump did not change. The performance of the model pump was obtained. The comparison of the performance between the experimental and the CFD is shown in Figure 5. The numerical results have a good agreement with the experimental results. At the design point, the head and net positive suction head required calculated by the PUMPLINX are 37.82m and 2.95m, respectively. While the experimental head and net positive suction head required are 37.9m and 2.93m. From the 80% to the 120% of the design conditions, the differences in the head are very small and the tendency of head from the numerical simulation is consistent with that of experimental data. It is clear that the numerical simulation can predict the performance characteristics of the residual heat removal pump accurately and the optimization can be achieved by using CFD. Comprised with the design parameters, the head can meet the required, but the net positive suction head required is high for the practical engineering application. In this paper, the parameter optimization of the impeller is analyzed.

ORTHOGONAL ARRAY EXPERIMENT

To achieve the best performance characteristics of the radial diffuser pump, the main parameters should be selected carefully and exactly. According to the empirical design of the impeller and studies on cavitation performance ever done, three main impeller geometrical parameters are selected, namely impeller inlet diameter D1, inlet incidence angle Δ β, blade wrap angle φ . The geometry of the diffuser and volute remain the same. Table 4 shows the different parameters of the impeller. The 9 impellers were designed by the software CFturbo 9.0, whose function is fast to export the 3D computational domain of the impeller. Then, the mesh of the 9 impellers is generated with the approximate number of grids with the original pump, and calculated by using the PUMPLINX with the same settings of the original pump. Table 5 shows the predicted head, efficiency and net positive suction head required of the 9 schemes under the design condition. Range analysis method was applied to identify the influence of the main parameters on the objective. The sum values of each level for each factors are defined as Ki, and the average values ki are defined as, the Ki and ki are calculated by the Eq.( 1) and (2).

   i 1 , i N i j i y K (1) i i i K N 1 k  (2)
Where i is number of level, j is number of factor, yi,j is the value of net positive suction head required, which is corresponding to the factor j in level i. N is the total number of levels. The analysis results for pump net positive suction head required under condition are shown in Table 6. To obtain the low net positive suction head required according to the Δ max-min values, the main parameters ' effect on the net positive suction head required is A>B>C, namely impeller inlet diameter is the most important factor of net positive suction head required, followed by inlet incidence angle, blade wrap angle. Thus, in this paper, mainly considering the lower net positive suction head required, the final optimal combination of the main parameter for low net positive suction head required is A3, B2 and C3, namely D1=280mm, Δβ=5.5 0 , φ=130 0 .

Table 5. Orthogonal Experimental Scheme

Table6. Range Analysis of Net Positive Suction Head Required

Δ max-min=difference between maximum and the minimum values of the NPSHR

FINAL OPTIMIZATION

According to the best combination of the main parameters (D1=280mm, Δ β =5.5 0 , φ =130 0 ), the optimized impeller is designed and simulated by using the same method. And the predicted performance is compared with the original one, as illustrated in Figure 6. Under the design condition, the net positive suction head required and head of the optimized pump is 2.32m and 72.39%. The net positive suction head required decreases 0.61m, which significantly improves the cavitation performance. Meanwhile, compared with the simulation results of the original impeller, the efficiency of the optimized impeller increases 0.6%, which meets the requirements.

COMPARISON WITH THE INNER FLOW

Turbulent kinetic energy is used to describe the turbulent fluctuation. Its value and spatial non-uniformity identify the intensity of fluctuation diffusion and viscous dissipation. The Figure 7 illustrates the distributions of the turbulent kinetic energy. The biggest turbulence kinetic energy occurs at the outlet of the impeller and the inlet of the diffuser, the area of the high values is smaller in the optimal impeller and the optimization has improved the inner flow characteristics.

(a) the Net positive suction head required is a very important parameter of the pump performance. There is on accurate calculation method for NPSHR at present except to cavitation experiments. In this paper, numerical simulation of the cavitation experiment had been done by changing the inlet total pressure, and the vapor volume fraction distributions are compared between the optimal impeller and original impeller. Figure 8 shows when the inlet pressure of pump dropped to 32.5kp, the vapor volume fraction of optimal impeller was better than the original impeller, in which serious cavitation occurred, and it is more intuitive to use total volume fraction to do comparison, as shown in Figure 9. From the above numerical simulation results, it can be significantly seen that cavitation is distributed in the back of the blade inlet and the optimal impeller better the cavitation performance.

CONCLUSIONS

An L9 (3*3) orthogonal array was designed with the above three factors by using Orthogonal Experimental Method (OEM). The performances of the residual heat removal pump were predicted with PUMPLINX, and the best program for pump cavitation was obtained. In the above numerical simulations, cavitation is distributed in the back of the blade inlet, which meets the actual phenomenon. Under the design condition, the net positive suction head required decreases 0.61m, which significantly improves the cavitation performance. Meanwhile, compared with the simulation results of the original impeller, the efficiency of the optimized impeller increases 0.6%, which meets the requirements. The flow distribution in the optimized impeller is more regular. The area of the high turbulence kinetic energy is smaller in the optimal impeller and the optimization has improved the inner flow characteristics.

Figure 1 .

 1 Figure 1. Schematic Map of Optimization Process

Figure 2 .

 2 Figure 2. the 2D Model of The Radial Diffuser PumpTable 1. the Main Geometrical and Design Parameters of the Radial Centrifugal Pump

Figure 3 .

 3 Figure 3. the Open Test RIG

Figure 4 .

 4 Grids of Computational Domians

  (a) the Head Curve (b) The Net Positive Suction Head Required Curve

Figure 5 .

 5 Figure 5. Comparison of the Pump Performance between the Experiment and CFD

Figure 6 .

 6 Figure 6. Comparison of the Pump Cavitation Performance between the Optimization and Original

Figure 7 .

 7 Figure 7. Comparison of turbulence kinetic energy

  (a) the Vapor Volume Fraction Distribution in the Original Impeller (b) the Vapor Volume Fraction Distribution in the Optimal Impeller

Figure 8 .Figure 9 .

 89 Figure 8. Comparison of Vapor Volume Fraction Distribution in the Impeller

Table 3 .

 3 Working Fluid Parameters

Table 4 .

 4 Parameters of the Impeller

	Trail No.	Impeller inlet diameter D1(mm)	Inlet incidence angle Δβ( 0 )	Blade wrap angle φ (0 )
	1 2 3 4 5 6 7 8 9	260 260 260 270 270 270 280 280 280	3 5.5 8 3 5.5 8 3 5.5 8	110 120 130 120 130 110 130 110 120

4. RESULTS AND DISCUSSION

4.1 ORTHogonal experiment results based on the CFD

  Turbulence Kinetic Energy Distribution in the Original Impeller

			3 Δmax-min Rank	2.451 0.245 1	2.630 0.318 2	2.523 0.048 3
	(b) the Turbulence Kinetic Energy Distribution in the Original Impeller
	Trail No. 1 2 3 4 5 6 7 8 9	A 260 260 5.5 120 38.24 B C Head NPSHR Efficiency (m) (m) (%) 3 110 38.42 2.73 72.02 2.68 71.37 260 8 130 37.67 2.67 72.46 270 3 120 38.03 2.42 71.77 270 5.5 130 37.61 2.48 72.47 270 8 110 38.01 2.61 71.15 280 3 130 37.67 2.42 72.46 280 5.5 110 37.69 2.32 70.62 280 8 120 37.63 2.61 71.38
		Levels 1 2	Net positive suction head required (A) (B) (C) D1(mm) Δβ( 0 ) φ( 0 ) 2.696 2.572 2.554 2.503 2.492 2.571

ACKNOWLEDGMENTS

This study is supported by National Science & Technology Pillar Program (Grant NO. 2011BAF14B04), State Key Program of National Natural Science of China (Grant NO. 51239005) and National Natural Science Foundation of China (Grant NO. 51409123). Many thanks are given to the 4Cpump research group.


Xuelin Tang
email: xl-tang@mail.tsinghua.edu.cn

Hui Gao

Wei Yang

Zhuqing Liu

Zhifeng Yao

Yulin Wu

Numerical investigations and PIV measurement of the internal cavitation flows inside a centrifugal pump with vaned diffusers

Keywords:

A transparent acrylic centrifugal pump model with vaned diffusers which is suitable for PIV (Particle Image Velocimetry) measurement has been built. The internal flow field inside the pump under different flow-rate conditions for different NPSH (Net Positive Suction Head) has been measured using the PIV technique with fluorescent particles. The RNG k- turbulence model has been employed to calculate the three-dimensional unsteady turbulent flows in the pump and the predicted results are examined and certified by the PIV measurement data. The predicted results of cavitation performance and internal flow velocity fields agree with the PIV data. It can be concluded that the cavitation greatly affects the flow in the impeller channel and even jammed the impeller passage with successive decrease in cavitation allowance. And simultaneously, reverse flows happen at the outlet of the impeller and inside the diffuse blades, and even the lowvelocity retarding area emerges at the outlet of volute passage near the wall. The results obtained in this paper provide some supports for further investigation of cavitation influence on internal flow in hydraulic machinery.

INTRODUCTION

The cavitation phenomenon often occurs in the operation of pump. When cavitation appears , the internal flow of the hydraulic machinery is subject to interference and destruction, leading to the change of operation characteristics. So the analysis of internal cavitation flow characteristics has become an important research topic in the field of hydraulic machinery [START_REF] Jun | Numerical prediction of the hydrodynamic performance of a centrifugal pump in cavitating flows[END_REF].

With the increasing development of science and technology, the introduction of numerical simulation to traditional pump design method can effectively investigate the properties and rules of cavitation flow, predict cavitation characteristics, optimize the hydraulic machinery using inverse design method and save the test cost. And the introduction of test to traditional pump design method can directly show the actual cavitation characteristics and make up for the ill-considered factors in the numerical simulation such as: the mechanical loss and volumetric loss which cannot be calculated [START_REF] Dupont | Cavitating flow calculations in Industry[END_REF].

PIV test technology is so far the most comprehensive, applicability strongest and have the minimum destroy to the flow field in all test methods. Several PIV studies have been focused on the internal flow in centrifugal pump impellers [START_REF] Yang | Experimental study on the flow field in the blade to blade of a centrifugal pump by PIV[END_REF]. Foeth [START_REF] Foeth | Time resolved PIV and flow visualization of 3D sheet cavitation[END_REF] used the PIV testing technology to test the flow field around 3D hydrofoil and the cavitation occurrence; development and shedding along with the process of velocity distribution were accurately observed. The wake between the model pump impeller and the fixed guide vane was studied by Akin and Rockwell [START_REF] Akin | Flow structure in a radial flow pumping system using high-image-density Particle Image Velocimetry[END_REF] using PIV testing technology and the characteristics of flow separation and flow attached again were verified employing the instantaneous flow and velocity contours. Shao Jie [START_REF] Luo | The effect of impeller inlet geometric parameters on cavitation performance of centrifugal pump[END_REF] applied the PIV testing technology to a small centrifugal pump model to study the internal unsteady flow and the results of velocity and streamline distribution were compared to the results of numerical calculation in order to verify the feasibility of DES model based on SST k -ε model. Some studies were concerned with the numerical simulation in the cavitation flow. Medvitz [START_REF] Medvitz | Performance Analysis of Cavitating Flow in Centrifugal Pumps Using Multiphase CFD[END_REF] adopted multiphase CFD model to simulate the centrifugal pump cavitation flow and the results showed that the head coefficient drops rapidly when the cavitation number is lower than the critical cavitation number. Gan [START_REF] Gan Jiaye | The numerical calculation of cavitation flow in a mixed pump impeller[END_REF] used completely cavitation model to simulate the cavitation flow in a mixed flow impeller and accurately predicted the area where the impeller occurred and the development of cavitation.

In this paper, the RNG k- turbulent model with full cavitation model has been employed to simulate the threedimensional unsteady turbulent cavitation flows in the pump. And the PIV experiment has also been carried out to certify the accuracy of the method. The numerical results were compared with the test results.

EXPERIMETNAL APPARATUS 1.1 Centrifugal Impeller

The centrifugal pump is made by transparent plexiglass and all surfaces inside the flow field of the pump are polished. The physical diagram of the pump under investigation is shown in Figure 1. The design parameters of centrifugal pump are as follows:

3 =14.5 m / h Q , 3 =14.5 m / h Q , =1500 / min nr.
The main dimensions of the test impeller are summarized in Table 1.

The PIV testing area of the pump

The third velocity component can be thought of as zero in the measuring plane because 2d PIV testing technology is used in this experiment. So the measure plane shown in Figure 2 is selected to make sure the third velocity component in the area is small enough.

The test loop

The configuration of the whole test loop is shown in Figure 3. The test loop includes a centrifugal pump, a water-sealing gate valve used to adjust the flow at the inlet, a transparent cavitation tank for storing test fluid and controlling cavitation test, a vacuum pump extracting vacuum and pipeline to connect each component. The water flow rate is measured by an electromagnetic flowmeter installed in the outlet pipe and the water head is measured by two pressure transmitters located at the inlet and the outlet line respectively. A torque sensor is installed between the pump and the frequency modulation motor to measure the speed of the impeller and the shaft torque. The angular speed is fixed to 1500 rpm for this study.

The PIV testing system includes the laser (New Wave of Gemini 120) and the CCD camera (PCO Sensicam 1.3K×1K). The operating frequency of the laser is 15Hz and the wavelength of the laser is 532 nm. The pulse energy of a single laser is 120 mJ.

THE EXPERIMETNAL STUDY AND NUMERICAL SIMULATIONS OF THE CAVITATION FLOW 2.1The experimental study of the pump

The cavitation test

The combination of the vacuum pump and the control valve is used to control the inlet pressure of the pump so as to achieve the cavitation condition in this study. The internal flow field is measured when the pump cavitation occurs.

In the process of the test, the inlet pressure gauge of the pump is the vacuum degree, so the calculating formula for effective cavitation allowance is as follows:

2 1 1 2 a v a p v p p NPSH g g g g        (1)
where a p is the barometric pressure, 1 p is the relative pressure of the pump inlet, v p is the vaporization pressure under the test temperature. The cavitation performance curve of three typical flow conditions of the pump in the test is shown in Figure 4.

From Figure 4 as well as the criterion of critical cavitation allowance, we can come to a conclusion: the critical cavitation allowance increases with the increase of flow rate and the critical cavitation allowance of the small flow condition is 2.84 m, 4.09 m of the rated flow condition and 5.08 m of the large flow condition.

The internal flow test results

The internal flows inside the pump are measured under three working conditions: design flow rate ( The test data of three typical flow conditions demonstrate that there exists a low velocity stagnant zone at the outlet of the diffuse blade near the wall and the flow regularity is similar to each other. With the reduction of cavitation allowance, the flow pattern at the diffuse blades changes obviously. The streamline distortion occurs at the outlet of the diffuse blade and even there is a backflow phenomenon when the cavitation allowance is lower than the critical cavitation allowance. Overall, the internal flow condition under the influence of cavitation in the diffuse blade area is smaller than that in the impeller area.

The numerical simulation

In this study, the RNG k- turbulent model is adopted in the calculation of numerical simulation. The second-order central difference scheme is used for the pressure term and the velocity item, the second-order difference scheme is adopted for the turbulent kinetic energy and the turbulent kinetic energy dissipation rate item. The SIMPLEC algorithm is applied to solve the discrete equation.

The inlet of the pump is set as velocity inlet boundary condition, the outlet is set as outflow boundary condition and the wall is considered as no slip wall. After the examination of the grid independence, the total grid number in the study is 1150440 and the number of nodes is 199737.

The critical cavitation allowance curve

The comparison of the steady and unsteady numerical simulation results under three typical flow conditions with the experimental data are shown in Figure 8.The Figure 9 shows the critical cavitation allowance curve.

From Figures 8, the numerical results of both the steady and unsteady calculation is in consistent with those of the test. The cavitation performance curve has no significantly change with the decrease of the cavitation allowance under the same flow rate at first. But there is a sharp decline of the head when the cavitation allowance is lower than the critical cavitation allowance. The NPSH of 3% head drop is predicted as 4.09 meters which is close with the test result of 3.92 meters under rated flow condition. The overall error of the test data with the unsteady calculation results is within 10% and within 5% for the unsteady calculation results. From Figure 9, it can be seen that the critical cavitation allowance increases with the increase of flow rate. The reason why the calculated critical cavitation allowance is smaller than the testing one under the same flow condition is that the numerical calculation only considers the hydraulic loss while the fluid flow through the channel will be affected by surface roughness and complex boundary surface in the actual testing process.

The internal flow characteristics

To further verify the accuracy and reliability of numerical calculation, the comparison of the predicted internal cavitating flows inside the pump with the PIV test is also needed. In this paper, the unsteady numerical simulations of the internal flows under cavitating conditions are carried out. For the rated flow condition, the cavitating flows are obtained under three different conditions according to the degree of cavitation with no cavitation (NPSH=5. The static pressure distribution in axial cross section for rated flow at completely cavitation condition

Figure13

The volume fraction distribution in axial cross section for rated flow at completely cavitation condition As you can see from the figures above, the flow patterns inside the impeller are consistent with the PIV data. So the unsteady numerical simulation can not only provide predicted results close to testing data of the cavitation performance but also reproduce the internal cavitating flows inside the pump. However, the value of the relative velocity inside the impeller passage or the absolute velocity in diffuse blade region is slightly higher than the experimental results.

The static pressure and gas volume fraction distribution at the cross section of the impeller area are provided by selecting five images at the fixed moments in a period as shown in Figure 12 and Figure 13.

It can be seen from the figures above that the lowest pressure obviously exists the suction side of impeller blade surface near the inlet. The distribution of static pressure in the impeller region decreases at first and then increases and the static pressure at the impeller outlet reaches its maximum. Cavitation bubble come into being when the static pressure of the impeller inlet is lower than the vaporization pressure .And with the reduction of NPSH, the number of cavitation bubbles increases gradually destroying the internal flow and make the distribution of static pressure in the impeller disorder. The cavitation bubble is mainly within the impeller passage, but the diffuser and volute area will also appear a small amount of bubbles under serious cavitation condition. The distribution of the gas volume fraction is corresponding to the distribution of static pressure. Under the rated flow conditions, the suction surface of four impeller blades happen cavitation, but the distribution is not uniform. While the gas volume distribution in each passage is symmetrical under fully cavitation condition, which is conformed to the cavitation flow characteristics. The gas volume fraction distribution of small flow and large flow conditions are the same with that of the rated flow condition.

Conclusions

(1)The external characteristic performance and internal cavitation flow of PIV test under different operating conditions agree well with computational results by RNG k- turbulence model.

(2)The critical cavitation allowance increases with the increase of flow rate. And the calculated critical cavitation allowance is smaller than the testing one under the same flow condition.

(3) When the cavitation allowance is lower than the critical cavitation allowance, vortexes comes first at the back of the blade in the impeller area and the vortex will continue to become bigger and move toward the outlet as the cavitation allowance continues to decrease. For the guide vane area, only the streamline distribution changes as the cavitation allowance changes and no vortex forms. The calculated relative velocity inside the impeller passage or calculated absolute velocity of diffuse blade is higher slightly than the experimental results.

(4) The gas volume fraction distribution of small flow and large flow conditions are the same with that of the rated flow condition. The cavitation bubble is mainly within the impeller passage, but the diffuser and volute area will also appear a small amount of bubbles under serious cavitation condition.

Figure 1 .

 1 Figure 1. Physical map of the pumpTable 1. Impeller geometry Geometry Symbol Value/Unit Impeller Inlet diameter D1 60 /mm Outlet diameter D2 155/ mm Blade number Z1 4 Blade width b1 10/ mm Diffuse blade Inlet diameter D3 157 /mm Outlet diameter D4 197 /mm Blade number Z2 9 Blade width b2 13/ mm 1.2 The Experimental Set-Up

Figure 2 .

 2 Figure 2. The measuring plane

Figure 3 .

 3 Figure 3. Physical models of the test circuitsThe water flow rate is measured by an electromagnetic flowmeter installed in the outlet pipe and the water head is measured by two pressure transmitters located at the inlet and the outlet line respectively. A torque sensor is installed between the pump and the frequency modulation motor to measure the speed of the impeller and the shaft torque. The angular speed is fixed to 1500 rpm for this study.The PIV testing system includes the laser (New Wave of Gemini 120) and the CCD camera (PCO Sensicam 1.3K×1K). The operating frequency of the laser is 15Hz and the wavelength of the laser is 532 nm. The pulse energy of a single laser is 120 mJ.2. THE EXPERIMETNAL STUDY AND NUMERICAL SIMULATIONS OF THE CAVITATION FLOW 2.1The experimental study of the pump

Figure 4 .

 4 Figure 4. The experimental cavitation performance curve of the pump

3 =14Figure 5 .

 35 the rated flow condition, the internal flows are obtained for different NPSH according to the degree of cavitation with no cavitation (NPSH=5.35 m), partial cavitation (NPSH=4.09 m), completely cavitation (NPSH= 3.57 m), respectively; similarly, for the small flow (NPSH=4.71m, NPSH=2.84m and NPSH=2.02 m) and for the large flow condition (NPSH=6.10 m, NPSH=5.08 m and NPSH =4.54 m). The relative velocity and streamline distribution in the impeller area are shown in Figure 5 ~ Figure 7. Considering the laser irradiation is covered by the guide vane and the machining accuracy of the guide vane wall is not high, the flow passage between the four guide vanes is selected to analyze the absolute velocity and streamline distribution in the guide vane area. The shaped lines of two guide vanes are eliminated in the INSIGHT -3G software. And the results can be seen in Figure 5 ~ Figure 7. The relative velocity and streamline distribution in impeller and the absolute velocity and streamline distribution in diffuse blade of the small flow conditionIt can be seen from the figures that the relative velocity distribution in impeller becomes relatively disorder with the decrease of cavitation allowance especially under small flow and rated flow conditions. When the cavitation allowance is lower than the critical cavitation allowance, vortexes come to occur at the back of the blade. At this time a large number of cavitation bubbles block the impeller passage and cause serious damage to the internal flow of impeller passage.

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. The relative velocity and streamline distribution in impeller and the absolute velocity and streamline distribution in diffuse blade of the rated flow condition

Figure 9 .

 9 Figure 9. The comparison of critical cavitation allowance curve

  22 m), partial cavitation (NPSH=3.92 m), completely cavitation (NPSH=1.51 m); similarly, three cases for the small flow condition according to the degree of cavitation with no cavitation (NPSH=4.47 m), partial cavitation (NPSH=2.57 m), completely cavitation (NPSH=1.22 m) and three cases for the large flow condition according to the degree of cavitation with no cavitation (NPSH=5.82 m), partial cavitation (NPSH =4.96 m), completely cavitation (NPSH = 2.94 m). But only the rated flow results under completely cavitation condition are analyzed herein. The calculated relative velocity and streamline distribution in impeller region and absolute velocity and streamline distribution in diffuse blade region are shown in Figure 10 and Figure 11.

Figure10Figure 11

 11 Figure10The calculated relative velocity and streamline distribution in impeller of the rated flow condition Unit(m/s)

Figure12

  Figure12The static pressure distribution in axial cross section for rated flow at completely cavitation condition

Table 1 .

 1 Impeller

	1	geometry Symbol Value/Unit Impeller D1 60 /mm Outlet diameter Geometry Inlet diameter D2 155/ mm Blade number Z1 4 Blade width b1 10/ mm Diffuse blade Inlet diameter D3 157 /mm Outlet diameter D4 197 /mm Blade number Z2 9 Blade width b2 13/ mm

.2 The Experimental Set-Up

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51179192, 51479196, 51139007), the Program for New Century Excellent Talents in University (NCET) (Grant No. NETC-10-0784), the National Hi-Tech Research and Development Program of China ("863" Project) (Grant No. 2011AA100505) and the Chinese Universities Scientific Fund (Grant No. 2015QC090).


Mingxu Qi

Chaochen Meijie Zhang

Ma Isromac

Influences of Dis-tuned Tip Clearance on the Discrete Aerodynamic Noise in Centrifugal Compressor

Keywords:

With the purpose of decreasing the discrete aerodynamic noise in turbocharger compressor, the influences of dis-tuned impeller tip clearances on the aerodynamic noise are numerically and experimentally investigated. The dis-tuned tip clearance configuration is realized by ensuring either a bigger tip clearance size for main blades while a smaller one for splitter blades, or a smaller tip clearance size for main blades while a bigger one for splitter blades. The influences of dis-tuned tip clearance on centrifugal compressor performances are validated and results indicate that the dis-tuned tip clearance method used in current study has minor influences on compressor performances. Based on full unsteady CFD results, near field discrete aerodynamic noise analysis is performed on the tip clearance dis-tuned compressors, while far field aerodynamic noise is experimentally investigated. Both numerical and experimental results show that the dis-tuned tip clearance configurations have significant influences on compressor discrete aerodynamic noise, with maximum noise SPL decreased by 8 dB under the compressor operation points investigated in current research..

INTRODUCTION

With the emission restriction standard being more and more strict, turbocharging technology has been widely used in boosted engines for passenger and commercial vehicles and has the trend to be a standard technology to match the even more strict emission restrictions. However, due to the high rotating speed and highly strong unsteady flow in compressor and turbine, the turbocharger acts as one of the major noise source in the engine system and introduces discrete noises and turbulence noises. The aerodynamic noise generated by turbocharger compressor consists of broadband noise and discrete noise. As a major noise source, the discrete noise level is related with impeller rotating speed and blade number. Liu investigated the aerodynamic noise in centrifugal compressor with numerical methods and concludes that the maximum sound power is generated by the impeller [1] . Wen studied the aerodynamic noise of centrifugal compressor by experimental method and results show that there is narrow band tip clearance noise at 1/2 blade passing frequency, and its amplitude increases with rotating speed increasing [2] . Reference [START_REF] Raitor | Sound generation in centrifugal compressors[END_REF] carried out experimental investigation on centrifugal compressor noise, and results show that dominant noise generation mechanisms depend on the tip speed Mach number at the compressor inlet. Meanwhile, rotor-stator interaction is also considered to be a major noise source in turbomachinery. Dipole and monopole tonal noise in the frequency domain is predicted in [START_REF] Khelladi | Predicting tonal noise from a high rotational speed centrifugal fan[J][END_REF] by an aero-acoustic model based on the FW-H equation, and the results show that the interaction between the impeller and the diffuser plays an important role in centrifugal fan noise and is deemed to be the origin of the tonal noise. As another similar evidence, interaction between impeller and volute tongue is also experimentally proved to be a strong noise source in low speed centrifugal fan [5] . In order to reduce or control the level of aerodynamic noise, a lot of treatments or approaches are investigated. An optimization model of the blade circumferential angle was developed to minimize the A-weighted sound level and good results were achieved in reference [START_REF] Wu Xianjun | An optimization model by using unequal spacing blades for turbine noise reduction[END_REF]; Active control on the tip clearance flow with air injection was used to reduce the noise level and improve the performances on an axial flow compressor [7] , which was proved to be successful; Wake generator was used in reference [START_REF] Polacsek | Fan interaction noise reduction using a wake generator: experiments and computational aeroacoustics[END_REF] to adjust the phase of the impeller blade passing frequency in a fan and the sound pressure level was reduced by 8dB. Such approaches are effective on reducing compressor noise level, but the corresponding structures are normally applied to large sized axial compressors or fans and are not practical for small size centrifugal compressors with high rotating speeds. Considering the fact that the discrete noise in centrifugal compressor is with frequency of BPF and with the purpose of dis-tuning the tip clearance flow, dis-tuned tip clearance method is utilized in current research to investigate its influences and potential effects on centrifugal compressor noise reduction. The dis-tuned tip clearance is realized by un-equivalent clearance size setting for the main blades and splitter blades, and both numerical end experimental investigations are thus carried out.

CASE DESCRIPTION

The tip clearance dis-tuned methods is utilized on a turbocharger centrifugal compressor in current study. The centrifugal compressor op erates from rotating speed of 40000 r/min to 90000 r/min, with impeller diameter being 90mm. The impeller consists of 7 main blades and 7 splitter blades, with designed tip clearance size being 0.5mm. Two tip clearance dis-tuned models are developed by adjusting the actual tip clearance of main blades and splitter blades. Investigated models are summrized as follows: MODEL 1, the tip clearance is set to be 0.4mm for main blades and 0.6mm for splitter blades; MODEL 2, the prototype of the compressor, with tip clearance being 0.5mm both for main blades and splitter blades; MODEL 3, the tip clearance is set to 0.6mm for main blades and 0.4mm for splitters. Numerical siumulations are carreid out on the compressor stage (including impeller, vaneless diffuser and volute) for all the different settings. For experimental study, the impeller with these 3 kinds of tip clearance settings are machined with 5-axial milling method, and during the test, only the impeller is replaced while other components of the compressor are retained. Meridional dimensions of the investigated models are illustrated in figure 1 and described in table 1. The solver is based on a structured multi-block, multi-grid approach, and solves the time-dependent Reynolds averaged Navier-Stokes equations using finite volume method, with either the algebraic turbulence model of Baldwin-Lomax, one equation model of Spalart-Allmaras or two-equation models of k- models used for closure. For the present calculations, the Spalart-Allmaras turbulence model is selected and the second-order central scheme with addition of second and fourth order artificial dissipation is used. V-cycle multi-grid technology, coupled with local time stepping and implicit residual smoothing method is utilized for time marching acceleration. Computational domain consists of impeller, vaneless diffuser and volute. Mixing plane procedure is used at rotor-stator interface for compressor performances prediction and experimental validation. Non-Linear Harmonic (NLH) methods is used for unsteady flow simulations, which solves steady state N-S equations with derterministic stress prediction in time domain, and represents the flow into unsteady flow in frequency domain. In all the NLH simulations, third order harmonics and second order purtabations are modeled. Multi-block structured mesh is used for the computational domain. Totally about 2.2 million grid cells are modelled for the compressor stage, with non-dimensional first layer cell size (y + ) being about 1-5. Corresponding solid surface grid of the computational domain is shown in figure 2. The near field aerodynamic of the compressor is calculated at the upstream of impeller inlet and Formulation 1A raised by Farassat [START_REF] Farassat | Derivation of Formulations 1 and 1A of Farassat[END_REF] for FW-H equation is used for the calculation. To enable the caluclation of the near field noise at this planar surface, the pressure fluctuations are resolved by the unsteady flow simulations.

Experiment Setup

Both the compressor performance tests and the far field noise measurement are conducted on the compressor performance test rig developed in Beijing Institute of Technology. Total pressure and temperature are measured at the inlet and outlet of the centrifugal compressor for efficiency and pressure ratio testing. The air mass flow rate is measured by a double folium curve flow meter. The far field aerodynamic noise is directly measured by using the SPL meter. During the tests, the background noise level is always much lower than that of the compressor, which fits the requirement of noise measuring standard. Two SPL meters are placed on the same planar of the compressor shaft. One of the SPL meter locates at upstream and 1 meter away from compressor inlet, another SPL meter locates at perpendicular direction and 1 meter away to the compressor shaft. Figure 3 shows the measuring positions of the measuring positions of compressor far field noise.

Experimental Validation on Numerical Methods

The predicted compressor total pressure ratio and totalto-total efficiency are validated with experimental data. The comparisons between numerical and experimental data of the three dis-tuned models under rotating speed of 60000 r/min are presented in figure 4. As can see from the results, with the numerical methods described in above sections, the predicted pressure ratio and efficiency for MODEL 1, 2 and 3 match the experimental result reasonablly. The location of the peak efficiency point on the speed line and the trends of the curves (pressure ratio against massflow rate and efficiency against massflow rate) are correctly captured. The maximum error of efficiency between the predicted and tested data is within 4% and that of the pressure is less than 2%, which can be considered as acceptable. The calculated aerodynamic noise at passage passing frequency under peak efficiency point of 60000 r/min for the three models are compared with experimental results, as shown in figure 5. For all the models investigated, the numerical methods under predict the SPL of PPF noise by 13dB to 18dB. which However, the influences of the dis-tuned tip clearance on aerodynamic noise are proved to be in the same trend between numerical simulation and experiment. MODEL 1 is proved to decrease the compressor aerodynamic noise level while MODEL 3 tends to increase the noise level.

Considering the facts that the measurement is conducted in a non-anechoic space and the similar SPL vairation trends predicted by numerical and experimental approaches, the results can be considered to be reasonable.

Influences of Dis-tuned Tip Clearance On Compressor Performances

Even though the dis-tuned models can decrease the aerodynamic noise level in centrifugal compressor, the performances are still the most important things should be considered, i.e., reduction of noise should not lead penalties on compressor performances. Thus, the performances of the centrifugal compressor with 3 different dis-tuned tip clearance model under 4 different speed lines are experimentally tested and the pressure ratio as well as efficiency are compared in figure 6. The tested compressor rotating speeds increases from 50000 r/min to 80000 r/min. As can be seen from the results, for all the operating speed lines being tested, MODEL 1 has almost the same performances with MODEL 2 (prototype) both on efficiency and pressure ratio. While slightly decreased pressure ratio and efficiency are observed for MODEL 3. The difference between MODEL 3 and prototype increases as compressor rotating speed increases.

Anyway, the maximum decrease on pressure ratio and efficiency is still within 1%. Since the tip clearance size of main blades in MODEL 1 is decreased while in MODEL 3 is increased, it can be inferred that the size increase of main blade tip clearance has a greater impact on compressor performances than the increase of splitter blades tip clearance. Hence, to consider the influences of dis-tuned tip clearance on compressor performances, one can say that dis-tuned tip clearance method used in current research has minor influences on compressor performances, which can even be kept unchanged by decreasing main blades tip clearance while increasing that of splitter blades. With this conclusion, potential reduction of compressor discrete noise with the dis-tuned model mentioned above can thus be deemed as practical in engineering viewpoint.

Near Field Aerodynamic Noise Analysis

The near field discrete aerodynamic noise is numerically calculated at the monitoring surface upstream of the impeller inlet, and the distribution of the monitoring points on the surface is a 7X4 matix, which has 7 stations along circumferential direction and 4 points along spanwise direction on each station, as shown in figure 7. The peak efficiency and near choke operation points under 60000 r/min are investigated in this part. For each operation point, the data collection time is set to 0.001 second and resolution frequency is iset to 1000Hz. Meanwhile, the SPL at PPF of 7000Hz tends to be much higher than that at BPF of 14000Hz. Comparisons between the dis-tuned tip clearance models and prototype indicate that for MODEL 1, the SPL at 7000Hz is decresed by 3.2dB at peak efficiency point and 2dB at near choke point, while for MODEL 2, the SPL at 7000Hz is increased by 2.5dB and 3dB respectively.

The SPL contour of near field discrete aerodynamic noise of 3 different models at the monitoring surface are shown in figure 9-a and 9-b, with results for peak efficiency and near choke operation points included. As a comparison between different operation points for the same model, it can be seen that for all the models investigated, the SPL level on the monitoring surface at near choke point is higher than that at peak efficiency point. Meanwhile, the SPL tends to increase along spanwise direction and gets to its maximum level in the tip region.

When comparing the SPL among different dis-tuned models under same operation point, one can also find that the dis-tuned tip clearance has significant influence on the near field discrete aerodynamic noise. Under peak efficiency operation point, though similar contour distribution can be found for all the three models, the maximum value of SPL differs a lot. By taking MODEL 2 as reference, the averaged SPL value on the monitoring surface for MODEL 1 is obviously lower than MODEL 2 by difference being about 3 dB, while the SPL of MODEL 3 is higher than MODEL 2, especially in the region near the tip. Similar results can be found for the 3 models under near choke operation point: SPL of MODEL 1 is about 2 to 3 dB lower than MODEL 2 in a global view on the monitoring surface, while SPL of MODEL 3 is about 3-4 higher than MODEL 2.

Far field aerodynamic noise analysis

The far field aerodynamic noise of the centrifugal compressor with three dis-tuned tip clearance settings are measured at peak efficiency point of rotating speed of 50000 r/min, 60000 r/min and 70000 r/min. Two SPL meters locate at 1 meter away from the compressor are installed for noise measuring, as shown in figure 3. During the tests, the compressor is driven by a radial turbine, and the far field aerodynamic noise measured also includes the noise of the turbine. Since the blade number of the turbine impeller equals 10 and is identical to the blade number of the compressor, the noise source can thus be identified by frequency. Under rotating speed of 5000 r/min, 60000 r/min and 70000 r/min, the compressor passage passing frequency (PPF) equals to 5833Hz, 7000Hz and 8166Hz accordingly, while the turbine blade passing frequency (BPF) equals to 8333Hz, 10000Hz and 11666Hz. The PPF sound pressure amplitudes of the three models at 1# monitoring point is listed in table 2, in which, Δp represents the relative value upon MODEL 2. A positive value means the sound pressure amplitude of the model is higher than that of the prototype, while a negative value means the sound pressure amplitude of the model is lower than that of the prototype. The sound pressure spectrum at 1# monitoring point for the 3 models are presented in figure 10. From the spectrum characteristics, both broadband noise and discrete noise can be observed. The sound pressure amplitude of broadband noise for all the models at any rotating speed is much lower than that of discrete noise, and with the increase of rotating speed, the broadband noise also increases.

The discrete noise consists of compressor passage passing frequency (PPF) noise and turbine blade passing frequency (BPF) noise. Comparisons among the three models indicate that the PPF amplitude of MODEL 1 is significantly decreased at rotating speed of 60000r/min and slighly decreased at rotating speed of 50000r/min and 70000r/min, as compared to the prototype (MODEL 2). While for MODEL 3, the PPF ammplitude is always higher than the prototype. These results again confirm that the dis-tuned tip clearnance model for MODEL 1 is able reduce the discrete aerodynamic noise of the investigated compressor, while the one for MODEL 3 worsen the discrete noise level.

The measured SPL under peak efficiency point of speed line 50000r/min, 60000r/min and 70000r/min are shown in figure 11 for easy comparisons on discrete noise under PPF. With the chart, the influences of the dis-tuned tip clearance model on the discrete noise can be clearly pointed out. With the dis-tuned model of decreasing the tip clearance of main blades while increasing that of splitter blades, MODEL 1 can decrease the discrete noise, and the maximum reduction on SPL is up to 7dB at 60000 r/min. While with the dis-tuned model of increasing the tip clearance of main blades and decreaseing that of splitter blades, MODEL 3 leads to discrete noise enhancements, with the increased SPL being up to 8dB under 50000 r/min.

With above discussion, it can be concluded that the distuned tip clearance control has significant influences on compressor discrete noise, either in near field noise analysis or in far field noise measurements. Similar conclusions are given by both numerical simulations and experimental measurements. However, dis-tuned tip clearance method does not necessarily reduce the discrete noise. Its influences on the discrete noise reduction depend upon the detailed dis-tuning method. In current study, the prefered dis-tuned method is to decrease the tip clearance of the main blades while increase the tip clearance of the splitter blades.

CONCLUSION

In order to decrease discrete aerodynamic noise in smallsize high-speed centrifugal compressor, two dis-tuned tip clearance models are developed and their influences on the performances and discrete aerodynamic noise of compressor are investigated with both numerical and experimental methods. With above discussion, following conclusions can be given on the compressor investigated in current research:

1) The dis-tuned tip clearance method utilized in this paper has minor influences on the compressor performances. With the method of increasing the main blades tip clearance size while decreasing the splitters', the compressor pressure ratio and efficiency can be kept unchanged.

2) The dis-tuned tip clearance method has influences on the compressor discrete aerodynamic noise. Both the near field and far field discrete noise can be reduced by dis-tuning the tip clearance size of main blades and splitter blades. For the compressor investigated in current research, the proposed dis-tuning method for noise reduction is to decrease the tip clearance size of main blades while increase the tip clearance size of splitter blades.

Figure 1 .

 1 Figure 1. Meridional dimensions of the centrifugal compressor (left: MODEL 1, right: MODEL 3)

Figure 2 .

 2 Figure 2. Solid surface grid of the computational domain

Figure 3 .

 3 Figure 3. Far field noise measurement setup

Figure 4 .

 4 Experimental validation on compressor performances

Figure 5 .

 5 Figure 5. PPF SPL validation

Figure 6 .

 6 Figure 6. Influence of dis-tuned tip clearance on compressor performance

Figure 7 .

 7 Figure 7. Near field discrete noise monitoring pointsFigure8-a and 8-b illustrate the specturm characteristics of the discrete aerodynamic noise for three different models at one monitoring point locates at 75% span. One can see that for all the models investigated, the discrete aerodynamic noise mainly apears at 7000Hz (the compressor passage passing frequency, PPF) and 14000Hz (the compressor blade passing frequency, BPF). Meanwhile, the SPL at PPF of 7000Hz tends to be much higher than that at BPF of 14000Hz. Comparisons between the dis-tuned tip clearance models and prototype indicate that for MODEL 1, the SPL at 7000Hz is decresed by 3.2dB at peak efficiency point and 2dB at near choke point, while for MODEL 2, the SPL at 7000Hz is increased by 2.5dB and 3dB respectively.

Figure 8 .Figure 9 .

 89 Figure 8. Spectrum characteristic of discrete aerodynamic noise (Left: model 1 Middle: model 2 Right: model 3)

3 PPFFigure 10 .

 310 Figure 10. Noise sound pressure spectrum at 1# monitoring point (Left: MODEL 1, Middle: MODEL 2, Right: MODEL 3)

Figure 11 .

 11 Figure 11. PPF noise comparison

Table 1 .

 1 Tip clearance settings

	Model	Tip clearance/mm Main blades Splitter blades
	1	0.4	0.6
	2-Prototype	0.5	0.5
	3	0.6	0.4

2. NUMERICAL AND EXPERIMENTAL SETUP 2.1 Numerical Method Flow solver EURANUS, integrated in the NUMECA FINE/Turbo CFD package, is used in current research.

Table 2 .

 2 ComparisonWith the given data of sound pressure amalitude in the table, one can see that for MODEL 1, the dis-tuned tip clearance decreases the PPF sound pressure under rotating speed of 50000 r/min and 60000 r/min, comparing to that of the prototype (MODEL 2). While for MODEL 3, the dis-tuned tip clearance increases the sound pressure under all the rotating speeds.

			of PPF amplitudes of three models at 1#
				monitoring point
	n(r/min) Model PPF(Hz) Sound pressure(Pa)	ΔP(Pa)
		1	5833	0.22	-0.02
	50000	2	5833	0.24	0
		3	5833	0.69	0.45
		1	7000	0.34	-0.42
	60000	2	7000	0.76	0
		3	7000	1.15	0.39
		1	8166	0.60	0
	70000	2	8166	0.60	0
		3	8166	0.86	0.26

ACKNOWLEDGEMENT

The authors would like to thank for the from National


Xin Liu

Lingjiu Zhou

Xavier Escaler

Zhengwei Wang

Numerical investigation of the cavitation effect on natural frequencies of a hydrofoil

Keywords:

The added mass effects on a NACA0009 hydrofoil under cavitation conditions determined in a cavitation tunnel have been numerically simulated using finite element method (FEM).

Based on the validated model, the effects of averaged properties of the cavity considered as a two-phase mixture have been evaluated. the simulation results point out that the average properties of the cavity, considered as a mixture of vapor/water, have a significant effect on added mass coefficients and mode shapes. These effects appear to be more important for higher modes of vibration. For instance, the first torsion mode can evolve towards the second bending mode when reducing the cavity void ratio from 1 to 0.9.

INTRODUCTION

If large scale cavitation attaches to a solid surface, additional uncertainties will be added to the FSI (Fluid-Structure Interaction) phenomenon. Both cavitation flow and classic FSI problems have been studied for years separately, but it is still a challenge to estimate added mass effects in a cavitation flow. The mixture of water and vapor phases that forms the macroscopic hydrodynamic cavities can create averaged properties that are difficult to quantify and that vary from the expected effects of pure liquid water flow. Moreover, new boundary conditions at cavitation regions are not fully understood. In fact, many studies have noted the cavitation effects on FSI for various bodies, especially propeller blades and hydrofoils. Nevertheless, estimations of such effects are only seldom found.

The FEM-based acoustic fluid approach, that avoids the modeling effort and the time duration of the 2-way FSI simulation, can be easily applied to model complex structural and fluid geometries with high efficiency particularly at the design stage for safety assessment. In spite of that, this method has not yet been used to estimate the added mass effects on hydrofoils under cavitation conditions due to the difficulties of modelling the cavity properties.

The added mass effects under cavitation conditions on a NACA0009 hydrofoil determined in a cavitation tunnel will be numerically simulated using finite element method (FEM). A coupled acoustic fluid-structural system comprising the tunnel test section will be solved for different fluid conditions. The numerical results for a NACA0009 hydrofoil with partial cavitation have been compared with the experimental ones obtained by De La Torre, et al. [START_REF] La Torre | Experimental investigation of added mass effects on a hydrofoil under cavitation conditions[END_REF].

The hydrofoil modes of vibration will be determined for only air, only water and for various vapor-water conditions. In particular, several leading edge attached partial cavities with different dimensions and made of pure vapor will be simulated.

METHODS

In FSI problems, the structural dynamics equation must be considered along with the Navier-Stokes (NS) and the continuity equations. The discretized structural dynamics equation can be formulated based on the structural elements. The NS and continuity equations are simplified to get the acoustic wave equation assuming that the fluid is compressible, which means that its density changes due to pressure variations (i.e. acoustic medium), and that there is no fluid flow [START_REF]ANSYS Inc. Ansys theory reference[END_REF].

The governing finite element matrix equations of multiphase FSI are expressed in Equation ( 1):

              S T 0,
F, M 00 u ρR M 0 p p ρR 0 M u C0 0 p +0 C 0 0 0C p K -R -R +0 K                                                          S liquid F,liquid liquid F,gas F,gas gas u F p F 0= 00 F p K                      (1)
where MF, CF and KF are the fluid mass, damping and stiffness matrices, respectively. R is a coupling matrix that represents the effective surface area associated with each node on the fluid-structure interface. FF is the applied fluid pressure vector at the interface obtained by integrating the pressure over the area of the surface. P and U are the fluid pressure and the structure displacement vectors, respectively. MS, CS and KS are the structure mass, damping and stiffness matrices, respectively, and FS is the structural load vector.

Numerical Model

The simulated case corresponds to a NACA0009 hydrofoil, as shown in Figure 1, under various cavitation conditions experimentally tested at the EPFL High-Speed Cavitation Tunnel. On the left of Figure 3, a photograph of the typical attached partial cavitation as described in De La Torre, et al. [START_REF] La Torre | Experimental investigation of added mass effects on a hydrofoil under cavitation conditions[END_REF] is shown. The foil is made of aluminum alloy with a Young's modulus of 0.54 GPa, a Poisson's ratio of 0.33 and a density of 2770 kg/m 3 . The FSI domain, which comprises the whole test section of the cavitation tunnel, was meshed with hexahedral elements as shown in Figure 2. The hydrofoil was fixed with an incidence angle of 0º. One lateral section of the hydrofoil was constrained in the model. The rest of hydrofoil surfaces were in contact with the fluid. A FSI condition was set at these interfaces so that the displacement of the structure is identical to that of the fluid in normal direction. Considering the coupling characteristics of vibration in water, the fluid boundaries were modeled as fully reflective. The water properties were set as 1000 kg/m 3 for density and 1450 m/s for speed of sound.

Regarding the modelling of the attached cavity, it was considered to be of a constant length along the whole span as shown on Figure 3. Acoustic elements were used like for the water domain. At the interface of the vapor cavity with the surrounding water, the acoustic pressure and the normal component of the water/vapor velocity are continuous. The vapor properties were set as 0.174 kg/m 3 for density and 340 m/s for speed of sound. For simplification purposes, the simulated cavity represents a constant thickness and length domain which obviously differs from the actual cavity shape. Regarding the cavity thickness, that could not be measured during the experiments, the visual impression was that it was quite constant, as shown in Figure 4.

Results and Discussion

The effects on the added mass coefficients and on the mode shapes have been evaluated when the cavity is assumed to be a two-phase mixture. During the experiments in the cavitation tunnel, De La Torre et al. [START_REF] La Torre | Experimental investigation of added mass effects on a hydrofoil under cavitation conditions[END_REF] observed that the attached cavity morphology was not homogeneous. In most of the cases, two different regions, named as transparent and foamy, could be identified depending on the visual aspect. The transparent region was considered to be mainly composed by a single phase of vapor and the foamy one was considered to be mainly composed by a homogeneous mixture of vapor and water.

In our cavity model, it is possible to simulate a non-pure vapor cavity by assuming the averaged properties of a twophase flow based on the void fraction, α, which is defined as:

V V V   (2)
where VV is the vapor volume and V is the total volume of the cavity. Consequently, the cavity density, c, can now be calculated as:

  1 cV L       (3)
where V is a vapor density of and L is the water density, which have been taken as 1.7410 -2 (saturated vapor at 20 C) and as 1000 kg/m 3 , respectively.

The speed of sound in a bubbly air/water mixture at atmospheric pressure tested by Brennen [START_REF] Brennen | Homogeneous bubbly flows -in Cavitation and bubble dynamics[END_REF] as well as the cavity density are plotted in Figure 5 as a function of void fraction. These data have been taken as the reference values to carry out the numerical simulations with a cavity of fixed length (l/c = 0.318) detaching from the leading edge and fixed width of 1.8 mm. The significant differences in the f3 mode shapes shown in Figure 7 between the air and cavitation conditions obtained with the acoustic model are in close agreement with experimental measurements. Based on such data from Brennen, a series of cases have been simulated considering the void ratio in the range from 0.6 to 1 as shown on Figure 6. When the void ratio is below 0.9, the natural frequency of each mode does not change significantly. But when the void ratio is larger than 0.9, particularly above 0.99, the natural frequency of each mode changes sharply. The mode transition, as shown as in Figure 8 and Figure 9, exits but cannot distinguished easily. Moreover, the mode transition can happen more than once for the same order mode, such as f5, f6 and f7. The two modes of transition can intersect at a narrow frequency range. For example, as shown in Figure 7, the intersection void ratio is 0.965; before this value, the frequency and modal shape of f4 keep unchanged while the frequency and modal shape of f3 is approaching to f4 along with the void ratio increase. At α = 0.965, f3 and f4 have a very close frequency and almost the same shapes. After 0.965, it turns that the f3 is unchanged while f4 begins to change both frequency and modal shapes.

Conclusion

The added mass effects on a NACA0009 hydrofoil under cavitation conditions have been numerically calculated using acoustic FEM. The simulation results point out that the average properties of the cavity, considered as a mixture of vapor/water, have a significant effect on added mass coefficients and mode shapes. These effects appear to be more important for higher modes of vibration. For instance, the second bending mode can evolve towards a second bending mode when reducing the cavity void ratio from 1 to 0.9.

Figure 1 .

 1 Figure 1. Truncated NACA0009 hydrofoil.

Figure 2 .

 2 Figure 2. Discretized finite element mesh of the FSI domain with the surrounding water.

Figure 3 .

 3 Figure 3. Lateral view of the cavity numerical model.

Figure 4 .

 4 Figure 4. View of the cavitation from the top.

Figure 5 .

 5 Figure 5. Speed of sound in a bubbly air/water mixture at atmospheric pressure and cavity density as a function of void ratio.

Figure 6 .

 6 Figure 6. Natural frequency evolution as a function of cavity void ratio for first seven order modes.

Figure 7 .

 7 Figure 7. Measured and predicted f3 mode shapes.

Figure 8 .Figure 9 .

 89 Figure8. Example of the first kind of mode transitions for f3 and f4, and some modal shapes at particular void ratios.

ACKNOWLEDGMENTS

This work was supported by National Natural Science Foundation of China (Nos. 51279083 and 51409148).


Liu Demin
email: liudemin999@126.com

Liu Xiaobing

Numerical optimization on the "S" characteristics of model pump-turbine

Keywords: Pump turbine, S shape, guide vane outflow angle

The performance of a reversible pump turbine with S-shaped characteristics is of great harmful to the transient processes such as start-up and load rejection. When working point transition is required, then process in turbine mode must be fast, efficient and reliable. Especially during start-up, the machine must operate at a stable speed close to the no-load condition (i.e. close to the runaway characteristic) in a large operating head range while not involving a special stabilizing technique and all synchronized guide vanes are open slightly. When pump turbine operates in "S" zone, two direction flow phenomena such as turbine direction and pump direction coexist in the flow passage together and the interaction effect between turbine and pump flow is responsible for "S" zone. Many vortexes are formed because the interaction, furthermore, the vortex cause energy loss and discharge fluctuation. In order to improve the "S" characteristics of model pump-turbine and improve the safety margin of prototype pump-turbine, a criterion which can be used to justify whether pump turbine may operate in the "S" zone on model design is proposed. The criterion is the guide vane outflow angle is larger than impeller installed angle at no-load working point at different guide vane opening. When the guide vane outflow angle is larger than runner blade geometry installed angle at leading edge (high pressure side), the "S" zone will be removed from operation zone. How to explain this phenomenon? In fact, when the guide vane outflow angle increases, the flow discharge increase, so the main flow at turbine direction covers the main passage and the instability source such as pumping effect (the partial flow is from runner to guide vane which is like the pump mode, but at the same time the main flow is turbine mode) is suppressed, finally the S-shaped is improved.

INTRODUCTION

All over the world, the production of energy from renewable variable sources such as wind and solar is increasing massively. In order to integrate more and more wind and solar power in a given network, complementary power sources with high flexibility are necessary. Among them, Hydro Pumped Storage Plants (PSP) using reversible Pump-Turbine machines is an ideal complement, due to their fast response capability and their ability to store large amounts of potential energy when base load production exceeds the consumption on the grid and to release this stored energy when the power demand is higher.

Flexibility of Pumped Storage in operations is a requirement. Frequent starts and stops and fast mode changes are increasingly required, particularly from pumping mode to the generating mode. Reversible Pump-Turbine machines need to be designed for high availability, reliability and stability, especially during the start-up process in turbine mode when peak power production is required [1].

High head pump-turbine machines are more subject to the S-shaped characteristic because they are directly impacted by their own hydraulic design characteristics. Indeed, due to their low specific speed factor leading to a centrifugal design, the S-shaped phenomena in the Turbine quadrant becomes even more important and then more difficult to be mastered than for other lower head machines. Moreover, this issue will be even more difficult to master if a large head operating range is needed because the lowest head level will be shifted toward the most critical S-shaped region.

During start-up, there is no-load on the turbine shaft and the machine must be operated at stable speed close to the runaway characteristic. One necessary coupling condition is to have a rotation frequency level of the turbine in line with the grid frequency level. Just before the coupling sequence, the turbine rotation frequency must be stable without speed oscillations. If those conditions are not fulfilled, the start-up process can be slowed down or even impossible if no special stabilizing technique is used. So the S shaped curve of pumpturbine on n 11 -Q 11 must be out range of the operation zone. And the safety margin must be enough to ensure unite start-up.

Reversible machines must operate efficiently in both modes-Pump and Turbine modes. This is the reason why a design compromise must be found between the Pump and the Turbine modes, including not only the continuous normal operating area but also the offdesign operating range such as the start-up areas. Balancing of such hydraulic behavior is important for any kind of reversible Pump-Turbine. Because the Sshaped zone has so great damage to unite stability operation, many researchers and engineers pay more attention to S-shaped zone optimization.

Houdeline [2] gives the optimal result of significant improvement of the S-shaped characteristic of high head pump-turbine using all synchronized guide vanes based on CFD tools as well as model tests; however the optimization criterion is not given.

Yan [3] has compared with S-curve between the CFD calculation based on the commercial code ANSYS-CFX13 with the standard k-ε model and experiment result. Yan has proposed a numerical method which is effective to simulate S shaped curve and pressure fluctuation; however the criterion of S shaped curve is still not given.

Gentner [4] has proposed the criterion near no load at constant wicket gate opening in dimensional and dimensionless presentation for instability is dQ 11 /dn 11 >0. After many times numerical simulation or test, the curve n 11 -Q 11 is obtained and criterion can be estimated based on dQ 11 /dn 11 >0. Furthermore, Gentner [4] has published two vortices structure in runner channel such as fully developed vortices and vortex structure partially developed. With fully developed vortices, fluid enters the runner only at the shroud side, while vortices block the rest of the cross section. The inflow at the hub side enters the secondary vortex and these vortices are the main contribution to the S shaped curve. The main driver of the primary vortex is a strong cross flow on the pressure side from hub to shroud at the leading edge. If this cross flow can be reduced or avoided, the characteristic of the S-shape curve can be avoided or reduced along with the associated system fluctuations. Gentner's research achievement has given hydraulic engineer a direction of optimization to improve the Scurve.

Hasmatuchi [5,6] has compared the flow angle on runner inlet between best efficiency point and runaway point. When the pumped-turbine operates on S-curve zone, the flow angle is smaller. So Vlad's achievement has also given a direction to optimize the runner to improve the S-curve.

Hasmatuchi [7,8] has shown the experiment result of pump-turbine operating under off-design conditions in generating mode. The main flow under off-design conditions is dominated by one stall cell rotating with the impeller at sub-synchronous speed in the vane-less space between the impeller and guide vanes. It is likely the result of flow separations developed in several consecutive impeller channels, which lead to their blockage. Obviously, this rotating hydrodynamic instability is responsible for hydraulic unbalance and strong structural vibrations.

To sum up, many researchers pay attention to the S-shaped curve phenomena and principle, unfortunately, a reasonable criterion whether the designing turbine may run in S-shaped zone is still not obtained by present. Guide vane flow angle at no-load working points is an effective parameter to estimate whether the S-shaped zone could exist in the range of operation zone.

Consequently, this paper focuses on the flow analysis and criterion application in the S-shaped zone. The structure of the paper is arranged as follows: Detailed specifications of the pump-turbine are introduced in section 1. Section 2 describes the CFD model, mesh distribution, disposal of boundary condition and numerical procedures. Section 3 presents the results and discussions, such as the predicted results of discharge-speed characteristics curve, flow angle formulate and comparison. At last, conclusions are made.

Pump turbine specifications

The whole pump-turbine configuration including spiral casing, stay vanes, guide vanes, runner and draft tube is considered in the physical model, as shown in Fig. 1. The computing domain enables to overcome the influence of boundary conditions, periodic interfaces, and pitch ratio of rotor-stator interface, and it especially considers the non-axisymmetric inflow from the spiral casing.

Mathematical model and computational details 2.1 Mathematical model

The continuity equation and momentum equation are adopted in simulation.

Continuity equation

( ) 0 t ρ ρ ∂ + ∇⋅ = ∂ u 1 Momentum equation ( ) ( ) 2 D D eff eff p t ρρ μ μ λ =-∇ +∇ + + ∇ ∇ ⋅ u f uu 2
The turbulence stress is taken into account with turbulence viscosity coefficient in a standard SST model proposed by Menter [9] .

Grid generation

Rotating and stationary domains are matched using rotor stator interfaces. The global mesh encompassed 9 million nodes with smallest skewness angles of 20 degrees. As displayed in Fig. 2, special refinement is applied in the runner and in the guide vane domain. The mesh independent is validated before the scheme comparison. When the mesh number is 9,000,000, the comparison between simulation efficiency 0.930 and experimental value 0.928 has shown good agreement in

Boundary condition 1) Inlet condition.

The mass flow inlet at the spiral casing (Turbine mode) is used, whose profile is assumed to be uniform.

2) Outlet conditions. Pressure outlet is used at the draft tube (Turbine mode), static pressure (p=0 Pa) is specified for all cases.

3) Other conditions. No-slip condition is assumed on all the solid walls, and standard wall function is used to calculate the turbulence kinetic energy and turbulence dissipation frequency near the wall. The interaction between rotation of runner and stagnant zones are dealt with by means of rotator and stator method.

Solution strategy

The software CFX13 is used to carry out the numerical simulation. In the calculation, the second order upwind scheme is used for discretization of convective term and the second order central scheme for discretization of diffusion term. The coupled solver is used and numerical convergence is set to a maximum of 10 -4 .

Calculation result and analysis 3.1 Flow in pump-turbine

In this paper, the total 4 guide vane openings and 28 working points are calculated in Tab.3. The calculated points include optimal working points, runaway points and turbine braking working points. The vortex effect which is named pumping effect (the partial flow is from runner to guide vane which is like the pump mode, but at the same time the main flow is turbine mode) could block the flow passage and result in energy loss. According to V.hasmatuchi [8] results, the Sshape is likely the result of rotating flow separation developed in several consecutive runner channels such as guide vane opening 10 o in Fig. 4. In other words, when the slope becomes positive close to the runaway point leading to the first inflection of the S-shaped characteristic in the turbine quadrant(i.e. the net head increases while the discharge decreases), then a separation flow leading to a pumping effect occurs at the runner channels entrance while the flow globally remains oriented in the turbine direction. This pumping effect is reverse direction of main flow coming from the guide vane. The interaction between turbine flow and pumping effects is the root cause of the unsteady and complex flow. The S-shaped characteristic is responsible for the speed oscillations of the machine close to the no-load point. According to the CFD calculation results, the velocity flow angle the included angle between inflow direction and tangential direction can be obtained in Fig. 5. From Fig. 5, the variety of the flow angle is periodic with guide vane numbers Z 0 , so 20 peak-topeak values appear in a period. The variety of flow angle is regular at guide vane inlet section in Fig. 5 (a) because of the flow is smooth, however the variety of flow angle is irregular at guide vane out section in Fig. 5 (b) which is caused by the interaction between guide vane wake, pumping effect (the partial flow is from runner to guide vane which is like the pump mode, but at the same time the main flow is turbine mode) and runner interaction appearing at the guide vane out [10] in Fig. 5(c).

Figure. 1

 1 Figure. 1 Computation domain of the pump-turbine model. Specifications of the machine are listed in Tab.1:Table 1. Main specifications of the tested pump turbine runner diameter at inlet(mm) 560 runner diameter at outlet(mm) 300 runner blade number Z 1 9 guide vane number Z 0 20 Maximum guide vane opening( o ) a 32 rotational speed(rpm) 1200 specific speed (m. kW) 135.7

Fig. 2

 2 Fig.2 (a) and the efficiency value has shown the independence to mesh number. The mesh detail is shown in the Tab.2. The mesh number can satisfy the calculation requirement and confirm the ability of the model to simulate flow characteristic of model pumpturbine. The y+ on guide vane and runner blade is shown in Fig. 2(b) and (c).

  (a) Mesh independent (b) y+ on guide vane (c) y+ on runner blade Figure.2 Mesh independent and y + distribution of the reduced pump-turbine.

Fig. 3

 3 Fig.3 shows the n 11 -Q 11 curve distribution on hill chart. The four group curves represent four guide vane openings such as 7 o , 8 o , 9 o and 10 o . At guide vane 7 o , 8 o and 9 o , the curves are still negative slope (The interaction point between equal guide vane opening line and runaway line slope is dQ 11 /dn 11 <0) and S-shaped (The interaction point between equal guide vane opening line and runaway line slope is dQ 11 /dn 11 >0) characteristic appears at guide vane 10 o . The positive slope always exists at guide vane 10 o near the runaway point j, which shows the pumped-turbine operates in the S-zone starting from runaway working point to the braking working points.

Figure. 3

 3 Figure.3 n 11 -Q 11 at different guide vane opening Fig.4 presents the flow structure in flow passage at no-load working points (on runaway line) and different guide vane opening 7 o , 8 o , 9 o and 10 o . One can notice that no vortex takes place in the flow passage at guide vane 7 o and 8 o , however, vortexes is developed in two flow passages and several flow passages at guide vane opening 9 o and 10 o , respectively. From different guide vane opening curve in Fig.3, more vortexes take place

  Flow structure in flow passage at different guide vane opening at no-load points

  Velocity flow angle and pumping effect3.2 S-shaped criterionOn guide vane opening 9 o , the working points a, b, c, d, e, f and g in Fig.3are chosen to compare the guide vane out flow angle distribution. To some extent, there is a correlation between vortex in runner channel and S-shaped curve, which is proved by Gentner's research achievement. Nevertheless, the quantity relation is more important to improve the Sshaped curve. Compared the guide vane out flow angle and S shaped curve, the guide vane out flow angle has some effect on S shaped curve.The guide vane outflow angle β from the CFD result is usually negative value and flow angle α is the complementary angle of guide vane outflow angle β . In order to comparisons at different working points and obtain the correlation between flow angle and S shaped taking place. The flow angleα is written in Eq. (3). α is different at different working points as shown in Tab.4 and Fig.6, and even if guide vane opening is same. From the Tab.4, the guide vane outflow angle α is 22.0 o on working point e; vortex is developed in two passages in Fig.6 (c), while the 11 -Q 11 curve is negative slope in Fig.3. No vortex takes place in the flow passage in Fig.6 (a) and (b). The guide vane outflow angle α is 5.6 o on working point f and vortexes appear in four passages in Fig.6 (d), nevertheless the n 11 -Q 11 curve is positive slope in Fig.3. And guide vane outflow angle α is -12.4 o on working point g, the positive-slope becomes obvious on n 11 -Q 11 curve in Fig.3, many runner channels are filled with vortexes in Fig.6 (e).

Table. 4 Figure. 6

 46 Figure.6 Streamline structure at different working points of same guide vane opening 9 o

Table. 5

 5 Flow angle at different working points 7

Figure. 7

 7 Figure.7 Guide vane outflow angle at guide vane opening 7 o , 8 o , 9 o and 10 o

Table 1 .

 1 Main specifications of the tested pump

	turbine
	runner diameter at inlet(mm)	560
	runner diameter at outlet(mm)	300
	runner blade number Z 1	9
	guide vane number Z 0 Maximum guide vane opening( o ) a	20 32
	rotational speed(rpm)	1200
	specific speed (m. kW)	135.7

Table 2 .

 2 Mesh detail of the tested pump-turbine

		Mesh number(*10 6 )	y +	Minimum Skew angle( o )
	Spiral case	0.5	150	22
	Stay vane	0.6	50	32
	Guide vane	2.4	20	54
	runner	5.2	15	52
	Draft tube	0.3	60	35

Table 3 .

 3 working points on the different guide

		opening
		7degree	8degree
		n11(r/min) Q11(L/s) n11(r/min) Q11(L/s)
	Optimal	46.99	46.27	47.56	46.83
	working	46.96	61.52	47.58	62.33
	points	46.75	91.88	47.44	93.23
	Runaway point	46.57	104.00	47.30	108.00

To validate the conclusions robust, an optimization result considering the guide vane out flow angle and runner inlet geometry installed angle is shown in Fig. 8 and the n 11 -Q 11 curve of three schemes X 1 , X 2 and X 3 at same wicket gate opening 8 o are shown. Three curves have the similar characteristic negative slope, in other words, the S-shaped is eliminated on the runaway points on three schemes.

The flow angles at three runner schemes X 1 , X 2 and X 3 are shown in Fig. 9. Compared the guide vane outflow angle value, the value is almost same. However, the difference among three schemes X 1 , X 2 and X 3 still exists on positive slope critical point labeled by red circle in Fig. 8 and Fig. 9. The guide vane outflow angle value is 21.58 o , 21.92 o and 22.02 o from scheme X 1 , X 2 and X 3 in Fig. 10. The value is biggest at scheme X 3 , so the safety margin of S curve is biggest.

For three schemes, the S-shaped curve doesn't appear on n 11 -Q 11 curve at no-load point. The stable head limit for a grid coupling operation is shifted outside the operating range (Fig. 8). In such a way, the grid coupling is possible for any head conditions and without requirement of special stabilizing technique.

Compared guide vane outflow angle and the Sshaped curve, the 20 o see above is the geometry installed angle at runner leading edge (high pressure side). So the criterion is the guide vane outflow angle at some guide vane opening is smaller than the installed angle at runner leading edge and the n 11 -Q 11 curve may appear positive-slope at no-load points.

Conclusions

This paper mainly focuses on the guide vane outflow angle and the "S" shaped curve. By choosing SST turbulence model, mesh independent validate, "S" shaped curve are calculated by the numerical method. A criterion of justy the S shaped curve for optimization is provided. The following conclusions can be made:

(1) The flow field is shown on the runaway point. When the hydraulic-turbine operates on no-load point, the runner channels are blocked by the vortex and pumping effect could suppress flow in turbine direction in runner channel.

(2)A criterion which judges whether the S shaped appear on n 11 -Q 11 curve at no-load points is provided. When the guide vane outflow angle is smaller runner blade geometry installed angle at leading edge, the S shaped appears on n 11 -Q 11 curve at no-load points.

(3)Furthermore, if the guide vane outflow angle is bigger, the conclusion is the S shaped safety margin is better. To be honesty, the robust of conclusion is still validated by many projects.

Acknowledgement

The authors gratefully acknowledge the support from the National Natural Science Foundation of China (No.