Yoshiki Oodaira
email: oodaira@chs.ifs.tohoku.ac.jp

Wakana Tsuru

Satoshi Watanabe

Yuka Iga

S Y Mpo

O Tatin

A fundamental study of disappearance phenomenon of partial cavitation on the NACA 16012 hydrofoil

Keywords:

For the purpose of understanding of the basic characteristics of cavitation, a number of studies have been conducted using the hydrofoil. An experimental study using NACA16012 hydrofoil was conducted in order to investigate the relationship between cavitation and boundary layer. In the study, it was reported that partial cavitation disappeared in spite of decreasing cavitation number. The authors have reproduced the disappearance phenomenon using the same hydrofoil in the previous study at Tohoku University. Details of the disappearance phenomenon have not been described in the article, and research related to the phenomenon has never carried out and clarified by other researchers. Therefore, in the present study, reproduction experiment of the disappearance phenomenon is conducted with the cavitation tunnel at Kyushu University using NACA16012 hydrofoil and in order to clarify the mechanism of the disappearance phenomenon, lift and drag force and pressure distribution is measured in non-cavitation condition. As the experimental result, the disappearance phenomenon is observed and the boundary layer characteristics are different from other region the cavitation disappearing phenomenon occurs. In addition, by analyzing the pressure fluctuation at downstream of the hydrofoil, it was found that the disappearance phenomenon is observed in a certain frequency bands of cloud shedding.

INTRODUCTION

Cavitation is the unique phenomenon to a fluid machine with water as working fluid. Cavitation occurs when the local pressure of fluid is decreased roughly below the saturated vapor pressure. Cavitation is organized by the cavitation number σ defined as,

σ = p ∞ -p v 1 2 ρ ∞ U 2 ∞ , where p ∞ is mainstream static pressure [Pa], p v is saturated vapor pressure [Pa], ρ ∞ is density of liquid [kg/m 3 ] and U ∞ is mainstream velocity [m/s].
Cavitation has an undesirable effect on fluid machinery such as vibration and erosion. Especially, the cavitation on the hydrofoil has an important problem because mechanical properties such as lift and drag force are influenced by the form of the cavitation. Therefore, a number of studies have been conducted using the hydrofoil for understanding of the cavitation on the hydrofoil. An experimental study has been conducted in order to investigate the relationship between cavitation patterns and boundary layer [START_REF] Franc | Attached cavitation and the boundary layer: experimental investigation and numerical treatment[END_REF]. Figure 1 shows the cavitation patterns on the hydrofoil in deaerated water organized by the cavitation number σ and the angle of attack α conducted at Grenoble University. The line part in figure 1 represents non-cavitation region. Unexpected cavita-tion behavior is seen around α = 4. As cavitation number decreasing, cavitation patterns change as non-cavitation, partial cavitation, non-cavitation and finally super cavitation.

In that paper, it is described as follows that at this angle of attack, laminar separation occurs close to the leading edge immediately followed by transition, so cavitation can only develop for a short time and the unexpected behavior is seen. However, this hypothesis explain merely that the cavitation inception is delayed due to the specific boundary layer and it does not explain that why partial cavitation disappear in spite of cavitation number decrease.

As shown in figure 2, the authors have reproduced the disappearance phenomenon on NACA16012 hydrofoil in the previous study conducted at Tohoku University [START_REF] Iga | Experimental study of appearance of thermodynamic effect on cavitation in hot water[END_REF]. However, cavitation tunnel used in that study has large blockage effects, so it is necessary to consider whether the cavitation disappearance phenomenon occurs due to the influence of the tunnel or the phenomenon peculiar to the hydrofoil.

In the present study, in order to confirm that the cavitation disappearance phenomenon is not due to the blockage effect of cavitation tunnel, reproduction experiment is conducted using cavitation tunnel different from the tunnel used in previous study. Also, lift and drag force and pressure distribution on the hydrofoil are measured in non-cavitation condition in order to clarify the mechanism of the disappear-ance phenomenon. In addition, by analyzing the the pressure fluctuation at downstream of the hydrofoil measured in the previous study [START_REF] Iga | Experimental study of appearance of thermodynamic effect on cavitation in hot water[END_REF], relationship between unsteady characteristics of cavitation and cavitation disappearance phenomenon.

Angle of attack α (deg.)

Cavitation number σ (-) holes with a diameter of 0.5 [mm] are provided on the hydrofoil surface and connected to pressure transducers for measuring the pressure distribution on the suction surface of hydrofoil. Lift and drag force is measured by 2 gauges method using a cantilever beam with 4 strain gauges attached to each sides of a rectangular cross section. Lift and drag force and the pressure distribution is measured for 50 [s] at sampling frequency 10 [kHz] and time averaged. In this experiment, the flow velocity is kept fixed at 5.4 [m/s] (Re = 4.8 × 10 5 ) and 9.2 [m/s] (Re = 1.0 × 10 6 ). The static pressure is decreased gradually from non-cavitation condition to bubble cavitation condition. The change of the cavitation patterns is observed and cavitation number is recorded at the same time.

In order to investigate the unsteady characteristics of cavitation, FFT analysis of the pressure fluctuation at downstream of hydrofoil was conducted using measured in a vious study conducted with the cavitation tunnel at Tohoku University. It must be noted that it is a result of a different from the cavitation tunnel the hydrofoil characteristics measured.

RESULTS AND DISCUSSION

Reproduction experiment

Figure 4 shows the aspect of cavitation on NACA16012 hydrofoil at Re = 1.0×10 6 , α = 4. The inception of cavitation was observed at σ = 1.0. Cavitation get longer as the cavitation number decreasing (σ = 0.9), and unsteady sheet cloud surizing experiment is more repeatability than the incipient cavitation during the decompression experiment [START_REF] Billet | Scale effects on various types of limited cavitation[END_REF]. However in this study, the depressurizing experiment was conducted because the cavitation disappearance phenomenon did not occur in the pressurizing experiment due to lots of the remaining bubble created by cavitation.

Even though the angle of a ack of the disappearance phenomenon occurs is different from previous studies, this phenomenon is seen at three different cavitation tunnels.

erefore it is considered that the cavitation disappearance phenomenon is not caused by the influence of the tunnel but peculiar to NACA16012 hydrofoil.

The hydrofoil characteristics

We can obtain the knowledge about the boundary layer around the hydrofoil which is closely related to cavitation pa erns is obtained by measuring the li drag and the hydrofoil surface pressure distribution in the non-cavitation condition.

e li coefficient C L , drag coefficient C D and pressure coefficient C p are defined as,

C L = L 1 2 ρ ∞ U 2 ∞ A , C D = D 1 2 ρ ∞ U 2 ∞ A , C p = p ∞ -p 1 2 ρ ∞ U 2 ∞ ,
where A is the hydrofoil area, L is the li force, D is the drag force, and p is the measured pressure on the hydrofoil. Figure 6 shows the results of time-averaged li and drag coefficient in non-cavitation condition. It can be seen that the gradient of the li coefficient changes at the angle of attack α = 3 to 5 where cavitation disappearance phenomenon was observed at Re = 1.0×10 6 . Moreover, it can be seen the similar changes around the same angle of a ack, even at Re = 4.8×10 5 where cavitation disappearance phenomenon was not observed.

  

                    
Figure 7 shows the results of timeaveraged pressure distribution on the suction surface of the hydrofoil with noncavitation condition. In figure 7 (a), local pressure increase is seen around x/C = 0.55 only under the condition of Re = 1.0×10 6 and α = 3.5 where the cavitation disappearance phenomenon was observed. Since it is a time averaged result, the increase of local pressure is considered to be due to the stationary existence of separation bubble. As shown figure 7 (b), local pressure increase is not seen at Re =4.8×10 5 . Therefore, cavitation disappearance phenomenon occurs at the angle of attack where the separation bubble stationary exist on the midspan of the hydrofoil. Consideration on the boundary layer will be examined in detail by using oil film method.

x/C (-) Frequency (Hz)

=5 =3.5 =2 Pressure coefficient -C p (-) (a) Re = 1.0×10 6
Cavitation number σ (-)

α =5 α =6 α =7 α =8 α =9
} Cavitation disappearance occurs } Cavitation disappearance does not occur to cavitation number at each angle of attack is shown in figure 8. At α = 8, 9 cavitation disappearance phenomenon does not occur, the dominant frequency tends to decrease as cavitation number decreases. This is the characteristic tendency of general sheet cloud cavitation [START_REF]Spectral characteristics of sheet/cloud cavitation[END_REF]. However, at α = 6, 7 at which cavitation disappearance phenomenon occurs, the frequency takes a low value even in the high cavitation number. In addition, as the cavitation number decreases, the frequency after cavitation reappears is higher than the frequency before cavitation disappears. At the angle of attack at which cavitation disappearance phenomenon occurs, it can be seen that the frequency characteristic of cavitation changes. And focus on the frequency before and after cavitation disappearance, it can be seen that cavitation disappears in a certain range of frequency band (from 58 to 62 [Hz]) in spite of different cavitation number. Although the cavitation tunnel is different, it can be inferred that the stationary existence of separation bubble and the dominant frequency of cavitation may contribute to the cavitation disappearance phenomenon string together with the consideration of the boundary layer.

CONCLUSION

This study is aimed to clarify the mechanism of cavitation disappearance phenomenon. Reproduction experiment of this is conducted using cavitation tunnel established at Kyushu University. And lift and drag force and pressure distribution is measured in non-cavitation condition in order to investigate the basic characteristics of NACA16012 hydrofoil. In addition, unsteady characteristics of cavitation is investigated by analyzing the pressure fluctuation of the downstream of hydrofoil. 1. Compared with previous studies, even in the cavitation tunnel with smaller blockage effect at Kyushu Uni-versity, cavitation disappearance phenomenon is observed although the condition of the phenomenon. So, this phenomenon is unique to NACA 16012 hydrofoil. 2. It was found that the gradient of lift coefficient changes and there is a stationary existence of separation bubble at the angle of attack where cavitation disappearance phenomenon occurs. 3. The angle of attack at which cavitation disappearance phenomenon occurs, it was found that the unsteady characteristic of cavitation make a big chage and cavitation disappears at a certain range of frequency band.

Further study will be conducted for hydrofoil topics.

1. Measure the pressure distribution of the suction surface of the hydrofoil in the region cavitation disappears and compare with the pressure distribution in non-cavitation condition. 2. Conduct experiments to investigate the details of the boundary layer by oil film method.

Figure 1 .

 1 Figure 1. NACA16012: Cavitation patterns at Grenoble Univ. (Re = 1.0 × 10 6 )

Figure 2 .Figure 3 .

 23 Figure 2. NACA16012: Cavitation patterns at Tohoku Univ. (Re = 9.7 × 10 5 )

  (a) σ=1.0 (b) σ=0.9 (c) σ=0.8 (d) σ=0.7 (e) σ =0.65 (f) σ =0.35

Figure 4 .Figure 5 .

 45 Figure 4. Aspect of cavitation (Re = 1.0×10 6 , α =4)

Figure 6 .

 6 Figure 6. Li and drag coefficient in non-avitation condition

Figure 7 .

 7 Figure 7. Pressure distribution on the suction surface in non-cavitation condition

Figure 8 .

 8 Figure 8. Relationship between cavitation number and frequency at Tohoku Univ. (Re = 9.7 × 10 5 )

ACKNOWLEDGMENTS

This study was supported by IFS Graduate Student Overseas Presentation Award.


Qingfeng Ji

Weili Liao

Honggang Fan

Haixia Yang

Structural characteristic analysis of the guide vanes of a pump turbine which working at the slight opening region

Keywords:

In some pumped storage power station, the vibration and abnormal sound occur at the distributor of the pump turbine during the startup and shutdown process when the guide vane opening is very small. In order to find out the root cause of this unstable phenomenon, this paper conducted a deep research on the unstable phenomenon occurred at the small opening region during the startup process of a pump turbine in pump mode. In order to explain the unstable incentives of the pump turbine during the startup process, the static stress on the guide vanes during the startup process was analyzed. The results indicated that the maximum static stress of the guide vane occurred at junction of the end face of the guide vane and the shaft. Moreover, the deflecting of the mainstream can lead to the sudden change of the stress of the guide vane, caused the instability of the units.

INTRODUCTION

Reversible pump turbine (RPT) is designed for pumping water from a lower reservoir to a higher reservoir by using the surplus energy in the power grids. Furthermore, this water have to go down to generate electrical energy by RPT at peak hours or in case of emergency. RPT have to change the working condition between pump mode and turbine mode [START_REF] Eduard | Condition monitoring of pump-turbines New challenges[END_REF]. In order to meet the requirement of power grid, RPT usually works at low load off-design working condition. Due to RPT's special way of operating, it's hard to guarantee the stability of the units. Few years ago, both in TianHuangping pump storage power station [START_REF] Kong | Analysis of abnormal sounds in working condition change-over for high-head pumpturbine[END_REF] and YiXing pump storage power station [START_REF] Cai | The Research of the Abnormal Water Hammer Phenomenon based on the Unit 3 over Speed Test of Jiangsu Yixing Pumped Storage Power Station[END_REF], the abnormal sound and vibration occurs at the distributor of the RPT which in the pump mode when the guide vane opening is very small. In order to resolve this problem, B Nennemann [START_REF] Nennemann | YiXing pump turbine guide vane vibrations: problem resolution with advanced CFD analysis[END_REF] has conducted a detailed research about the abnormal phenomena which happened in YiXing pump storage power station by using 2-D periodical CFD simulation. Eventually, the author acclaimed that the unexpected bi-stable flow conditions and a self-excited torsion mode flutter vibration of the guide vane caused this problem, moreover, the vibration problem can be eliminated by modifying the shape of the guide vane successfully. Addressing the issue of TianHuangping, H G Fan [START_REF] Fan | Hydraulic torque on the guide vane within the slight opening of pump turbine in turbine operating mode[END_REF] studied the HT of the guide vane of the RPT during the startup process and shutdown process in turbine mode by 2-D periodical CFD simulation. His work indicated that repeating reversal of fluid occurs when the guide vane at slight opening during the shutdown process in turbine mode. This phenomena finally results in the dramatically increasing of the Hydraulic Torque (HT) which makes the vibration happens. Our previous work [START_REF] Fan | Flow Analysis of the Guide Vanes Region of Pump Turbine at the Slight Opening in the Pumping Startup Process[END_REF] has performed the 3-D CFD simulation of the startup process of the pump turbine of Tian Huangping pump storage power station which in the pump mode by implementing the dynamic meshing technique. The results indicates that the main flow between the guide vanes has experienced a deflection when the guide vane opening is very small, and this deflection leads to the sharply changing of the hydraulic torque on the guide vanes, if the transmission of the distributer cannot adapt this changing immediately, the vibration and abnormal sound will occur in the unit. With the same flow field analysis method of our previous work [START_REF] Fan | Flow Analysis of the Guide Vanes Region of Pump Turbine at the Slight Opening in the Pumping Startup Process[END_REF] of the dynamic process, this paper presents a structural characteristic analysis of the guide vanes when the pump turbine works in pump mode at slight opening region. The one-way FSI method was carried out to calculate the structural characteristic of guide vanes during the main flow deflection process. And the vibration characteristic of the guide vanes were analyzed. The result shows that the sharply changing of the stress is relevant to the defection of the mainstream.

METHEOD

In this paper, the simulation was performed by using the commercial CFD solver-ANSYS workbench. As same as our previous work [START_REF] Fan | Flow Analysis of the Guide Vanes Region of Pump Turbine at the Slight Opening in the Pumping Startup Process[END_REF], the geometry of prototype pump turbine of Tian Huangping pump-storage power station was implemented. The geometry model is shown in Figure .1 and the characteristics of the parameters are listed in table 1. Head (m) 500 610

In addition, tetrahedral mesh was used for the FEM model of the guide vanes, it has 5276840 elements and 8175137 nodes totally. Figure 2 and Table 2 shows the FEM model and material parameters of the guide vanes.

PARAMETER NONDIMENSIONALIZATION

In this paper, the Reynolds number and the dimensionless methods of the results were defined as follow:

2 2 Re nRD    (1) 2 V Kv gH  (2) () ref pp Cp gH    (3) m 2 0.5 V A L ref ref ref M C   (4)
Where, Re is Reynolds number, Kv is velocity coefficient, Cp is pressure coefficient, Cm is moment coefficient (the positive moment direction was defined as the closing direction of the guide vane). Rated

STATIC STRESS ANALYSIS AND RESULTS

Results of CFD simulation

A transient CFD simulation of the startup process of a RPT was carried out with the dynamic meshing technique. Figure 3 shows that the serious flow separation which at the tailing edge of the GV occurred, when the GVO (Guide vane opening) less than 4 degree, during the startup process of the RPT. This separation forced the mainstream stay in type Ⅱ,and the mainstream adhere to the surface of the right side GV (figure 4 shows the illustration of the flow pattren). Thus, the flow field had an intense change near the GV when the deflection happend. The euqation of the moment varitaion ratio is as follow:

m ave D A t  (5)
Where m is the moment varitaion ratio, Aave is the average moment amplitude and tD is the deflecting time.

Working condition

Figure 6 illustrates the Cm-GVO plot of GV 26# during startup process, the points on the curve are the working condition points in this paper. And it's easy to find out that the moment of the GV has a sharply increasing when the mainstream started deflecting. All of the working condition were classified based on the flow pattern and were listed in table 3.

Results of the static stress

Figure 7 divides the Cm -GVO plot of 26# GV into different flow pattern regions during the startup process. Figure 8 shows the maximum stress on the 26# GV at various GVO.

During the startup process, the maximum stress in GVs is negatively correlated to GVO. The greater GVO, the lower the maximum internal stress. But the hydraulic torque and maximum stress on GVs is far greater than any other flow pattern when the mainstream stays in Type Ⅱ.

In the latter part of Type Ⅱ flow pattern region, maximum stress decreases as GVO increases.

As the GVs open up, the incident angles on GVs decreases, the flow has less impact on GVs. According to the flow analysis, the hydraulic pressure on GVs decreases as GVs open up. This explains the stress decrease in the latter part of Type Ⅱ flow pattern region.

The hydraulic torque on GV slowly decreases during the deflection process. Torque perturbations are due to fluctuation of the mainstream that changes the flow field. Similar to the hydraulic torque, the maximum stress on GV fluctuates slightly yet remain stable in a larger scope.

CONCLUSION

The structural characteristic analysis of the guide vane during the startup process of the pump turbine in pump mode indicated that:

1. The GV internal stress decreases with greater GVO. And the maximum stress position varies with different GVO because of varied fluid excitement to GVs due to different mainstream flow pattern as GVO increases. The maximum stress occurs near the shaft -GV connecting positions at each GVO.

2. The mainstream deflection between each pair of GVs contributes most to the significant stress change when GVO is below 3°. And the stress changes less when GVO is above 3°. It meant GV internal stress changes significantly when the mainstream between each pair of GVs deflects from Type Ⅱ, while remains relatively stable in other flow pattern regions. and the significant GV internal stress change is due to the flow pattern change from Type Ⅱ to Type Ⅰ . These sharply changes are potentially damage to the structure of the units.

Figure 1 .

 1 Figure 1. CFD computational domain of the pump turbine

2 .

 2 (a) geometry of the guide vanes (b) mesh of single guide vane Figure FEM model of guide vanes

  rotated speed n=500 rpm, outlet radius of the runner R=1.0225 m, outlet diameter of the runner D2= 2.045 m, water density (20 ℃ ) ρ=998.2 kg/m 3 , water viscosity (20 ℃ ) =1.003x10 -3 , head of the pump turbine H=610 m, reference pressure Pref=0, reference velocity Vref=1 m/s, reference area Aref=1 m 2 , reference length Lref=1 m, and in this paper Re=6.55x10 6 .

Figure. 3

 3 Velocity contours of 26# guide vane (right one) during the startup process

Figure. 4

 4 Figure. 4 flow pattren of the main flow

Figure. 5

 5 Figure. 5 Average hydraulic moment variation rate of the different guide vane during the startup process

Figure 6

 6 Figure 6 Cm -GVO plot of 26# guide vane during startup process

Figure 7 Figure 8

 78 Figure 7 The Cm-GVO plot of 26# guide vane during the startup process

  distribution of 26# guide vane under different working condition during the startup process Considering that maximum stress occurs at different positions as GVO changes, local stress at fixed points on GVs where maximum stress once occurred are selected to compare the local stress variations throughout the startup process, as is shown in Figure 10. Curve "point 1" indicates the local stress plot at the maximum stress position in condition point 1, the rest and so on. Local stress at each selected positions decreases as GVO increases. Local stress at point 3, 4 and 5 vary significantly when GVO is below 3°, yet remain stable when GVO is above 3°. About the flow pattern regions, local stress at point 3, 4 and 5 vary most significantly in Type Ⅱ flow region and deflecting region, where flow pattern started deflecting. The mainstream deflects differently in different flow regions, produces different pressure fields around the GVs and therefore results in different internal stress distributions that were calculated based on the GV surface pressure field. When GVO is above 3°, the local stresses change less. The conclusion is that the GV stress changes most significantly when the flow pattern varies from Type Ⅱ to the others during the startup process. And it can lead to the damage of the guide vane dueing the operation.

Figure. 10

 10 Figure.10 Stress plot of different maximum stress point during the startup process

Table 1

 1 Parameters of the pump-turbine

	Parameters Runner outlet diameter D2 (mm) Guide vane height b0 (mm) Runner blade number Guide vane number Stay vane number Rated speed (rpm)	Value 2045 262 9 26 26

Table 2

 2 Material parameters of the guide vanes

	Parameters Guide vane height b0 (mm) Guide vane number Material density ρ (kg/m 3 ) Young's modulus E (GPa) Poisson's Ratio γ	Value 262 26 7850 206 0.28

Table 3

 3 Classification of the flow pattern zone under the different working condition during the startup process Flow pattern

	region	Working condition number	GVO (degree)
		1	2.03
	type Ⅱflow pattern region	2 3 4	2.11 2.30 2.50
		5	2.69

ACKNOWLEDGMENTS

The authors would like to thank National Key R&D Program of China (2016YFC0401905), National Natural Science Foundation of China (No. 51279084, No. 51679196) and State Key Laboratory of Hydro Science & Hydraulic Engineering (2014-KY-05) and for their financial support.


Jian Wang

Yong Wang

Matevž Dular

Experimental study on cavitating flow in an axisymmetric system

Keywords: Cavitating flow, Cavitation erosion, Visualization, Collapse, Orifice plate

We experimentally studied the cavitating flow in an axisymmetric system, aiming at finding reliable approaches to analyze cavitation. Three image post-processed methods were applied to analyze the test data. The first image post-processing method was applied to obtain the mean value and standard deviation distribution, which indicate the erosion area may cover almost all the cavitation developing route and the most vulnerable erosion area locates near the cavitation collapse site. It is coincides with the erosion tests analyzed through the pit-count algorithm method. The cavitation circulation frequency was invested via PSD analysis method.

It shows that the frequency linearly decreasing with decreasing cavitation number. Additionally, the cavitation intensity effect on cavitation erosion was quantitatively studied based. It is found that the damages are strongly enhanced when increasing the flow velocity. Moreover, the growth rate of eroded pits number is actually stepwise instead of linear (similar to our previous work in a venturi tube), which supports the idea that the cloud cavitation collapse is the primary reason for erosion.

The present approaches applied here shows good potential ability of investigating cavitating flows and can be utilized for other apparatus.

Introduction

Cavitation is defined as bubbles generation. It happens where the local pressure is below the liquid saturation pressure, and then the bubbles collapse when they are *Corresponding Author: Tel:+86 13775530081 E-mail address: arieskin@126.com convected downstream, because of pressure recovery. If the collapse occurs in the vicinity of solid wall, the shock wave or micro jet generated during this process would erode the surface of the rigid boundary. As the energy impacting on the surface exceeds the material's fatigue strength, then erosion happens. Such damage pattern is found almost in every hydrodynamic machines, like vessels, propellers, pumps and so on, resulting in enormous economic lose [START_REF] Wang | An improved turbulence model for predicting unsteady cavitating flows in centrifugal pump[END_REF][START_REF] Liu | Partially-averaged navier-stokes model for predicting cavitating flow in centrifugal pump[END_REF][START_REF] Wu | Numerical analysis of flow in a Francis turbine on an equal critical cavitation coefficient line[END_REF][START_REF] Luo | Comparison of cavitation prediction for a centrifugal pump with or without volute casing[END_REF][START_REF] Ji | Partially-Averaged Navier-Stokes method with modified k-ε model for cavitating flow around a marine propeller in a non-uniform wake[END_REF][START_REF] Ji | Numerical investigation of three-dimensional cavitation evolution and excited pressure fluctuations around a twisted hydrofoil[END_REF].

Remarkable attempts have been made to clarify the physical mechanism of cavitation erosion, since last century in 1917 Rayleigh discussed about the issue of cavitation erosion on ship propellers [START_REF] Rayleigh | On the pressure developed in a liquid during the collapse of a spherical cavity[END_REF]. The intensity of the shock wave emitted by bubbles collapse is one of the most important factors resulting in cavitation erosion, which could reach several GPa [START_REF] Wang | Shock wave development in the collapse of a cloud of bubbles[END_REF][START_REF] Field | Cavitation in impacted drops and jets and the effect on erosion damage thresholds[END_REF]. Various measurement approaches are applied to obtain the impact load on solid surface. But it seems like that different approaches might get different results. Franc et al. [START_REF] Franc | Impact Load Measurements in an Erosive Cavitating Flow[END_REF] tested the cavitation generated by two parallel circular plates with a straight nozzle on one of them in the center, and the other one was placed 2.5mm away as the eroded sample. A maximum impact load of 500N was measured via a conventional pressure transducer as the cavity velocity ranges from 45m/s to 90m/s. However, Carnelli et al. [START_REF] Carnelli | Application of spherical nanoindentation to determine the pressure of cavitation impacts from pitting tests[END_REF] only got an order of 20N impact load on a same test system by using the spherical nanoindentation measurement. Also, Hattori et al. [START_REF] Hattori | Prediction method for cavitation erosion based on measurement of bubble collapse impact loads[END_REF] obtained similar results (a maximum impact load of 20N) at flow velocity up to 184m/s by following the standard ASTM G124-95. Furthermore, Soyama et al. [START_REF] Soyama | A new calibration method for dynamically loaded transducers and its application to cavitation impact measurement[END_REF] employed the poly vinylidene fluoride (PVDF) to measure the impact load in a cavitating jet device. It turned out to be up to 200N when the jet velocity at the nozzle outlet ranges from 126m/s to 155m/s. The discrepancy between these tests was significant. One of the reasons lies in that the size of the pressure transducers adopted in these works are still larger than the tiny bubbles, although they are "micro". The measured impact load may be caused by several bubbles collapse. And the lower sampling frequency of the pressure transducer, compared with the fast progress of collapse, could also influence the test result.

Another prevalent measurement is the pitting test, first proposed by Knapp in the 1950s [START_REF] Knapp | Recent Investigations of the Mechanics of Cavitation and Cavitation Damage[END_REF][START_REF] Knapp | Accelerated field tests of cavitation intensity[END_REF]. By exposing the solid material to a cavitating flow, it is used as a transducer during the erosion incubation period, which means the material only suffers plastic deformation without mass loss.

In this way, one can obtain the distribution of pits and even the number and area, but such erosion experiments still require a long exposure time [START_REF] Bachert | Experimental Investigations Concerning Erosive Aggressiveness of Cavitation in A Radial Test Pump with The Aid of Adhesive Copper Films[END_REF][START_REF] Hutli | Experimental study on the influence of geometrical parameters on the cavitation erosion characteristics of high speed submerged jets[END_REF][START_REF] Laguna-Camacho | A study of cavitation erosion on engineering materials[END_REF][START_REF] Franc | Incubation time and cavitation erosion rate of work-hardening materials[END_REF][START_REF] Rijsbergen | High-Speed Video Observations and Acoustic-Impact Measurements on a NACA 0015 foil[END_REF]. Hutli et al. [START_REF] Franc | Incubation time and cavitation erosion rate of work-hardening materials[END_REF] tested the cavitation erosion process by exposing the copper samples to a high speed submerged cavitating jet for 30min.

Rijsbergen et al. [START_REF] Rijsbergen | High-Speed Video Observations and Acoustic-Impact Measurements on a NACA 0015 foil[END_REF]

Experimental setup

All the experiments were conducted in a cavitation loop test platform in the Laboratory for Water and Turbine Machines, University of Ljubljana. As seen in Fig. 1, the entire closed test system was driven by a loop pump, a upstream tank which can be used for water heating, a downstream tank which is connected with a compressor and a vacuum pump for controlling the system pressure, a test section composed of two parallel plates, an electromagnetic flow meter, ABB WaterMaster V with a measurement uncertainty of 2% and two pressure transducers, ABB 266AST with ±0.04% measurement uncertainty. Hence, the measurement uncertainties due to instruments was estimated less than 3%. The detailed test rig facilities can be referred in [START_REF] Petkovšek | Simultaneous observation of cavitation structures and cavitation erosion[END_REF]. In the attempt of obtaining the full cavitation erosion data from the very beginning of the cavitation being generated, two valves are mounted upstream and downstream from the test section, far enough for avoiding disturbance. To assure the test repeatability, the dissolved gases were measured by the Van-Slyke method [START_REF] Brand | Ein physikalisches verfahren zur bestimmung von geloesten und ungeloesten gasen in wasser[END_REF] after running the test rig for 30 min at a low pressure, since the dissolved gases have significant influence on cavitation. In the presented work, the gas content of 15 mg/L in water was maintained.

Results and discussions

Fig. 4 shows the instantaneous cavitation structures under various cavitation numbers. These successive images were recorded by high speed camera at a shooting frequency of 2773fps for 2min.

As seen, the cavitation structure is annular, and the size increases with the decreasing cavitation number. For σ=3.36, the length reaches 26.9mm (from the cavitation inception point to collapse point), while it is only 6.1mm at σ=5. [START_REF] Knapp | Accelerated field tests of cavitation intensity[END_REF]. Also, because the distance between two plates is only 3mm, we can note that the generated cloud cavitation is composed of numerous tiny bubble clusters.

For predicting the eroded areas, the successive images of the cavitation structure were post-processed by calculating the standard deviation valve. Each image is treated as matrix, within which every each element wave or jet flow. Whereas the cavitation evolution is a little irregular as the cavitation number is higher. Besides, one may find that the distribution is not strict symmetrical. We reckon that it is probably caused by the uneven screw intension.

Fig. 6 shows the damaged aluminum foil after being exposed in cavitation for 5min, at σ=3.41. A ring-type eroded area can be seen directly by naked eyes. The diameter is approximately 40mm, identical with the predicted standard deviation distribution in Fig. 5. The cavitation erosion progress on the aluminum foil was recorded by high speed camera. We employed pit-count algorithm to detect the erosion pits [START_REF] Petkovšek | Simultaneous observation of cavitation structures and cavitation erosion[END_REF][START_REF] Keil | Cloud cavitation and cavitation erosion in convergent divergent nozzle[END_REF]. Fig. 7 shows the recorded instantaneous raw images of aluminum foil in pairs and the post-processed images by pit-count algorithm, where the brighter area implies overlapping pits.

The chosen operation condition is σ=3.41 and the time interval between each pair is 900∆t, here ∆t=43.5ms.

The brighter area indicates the damaged pits, compared with the original image at t=0. As seen from the experimental images, the pits are growing gradually in the region where cavitation collapses. Much clearer pits can be observed in the post-processed images, benefited from the pit-count algorithm, which eliminates the disturbances from illumination reflections or system vibration, resulting in more sharp contrast. One can see great agreement between experimental images and measured images. At the beginning of the exposure, the pits scatter nearby the collapse sites without many overlaps. However, since the cavitation tends to erode the place where has already been damaged or has surface defect [START_REF] Dular | Pit clustering in cavitation erosion[END_REF], some of the erosion pits overlap each other consequently, giving rise to higher brightness after long exposure.

To investigate the cavitation intensity effect on erosion, two more tests were carried out by keeping the cavitation number constant as 3.41. Under this circumstance, the cavitation size remains unchanged, but the nozzle throat velocities are different. Based on the case in Fig. 8, where v=12.9m/s, the higher velocity condition was accomplished by increasing both the flow rate and upstream pressure, while the lower velocity was achieved in an opposite approach. The test results are given in Fig. 14.

It is implied that the cavitation intensity has profound impact on cavitation induced damage. We can see

  tested cavitating flow around a 2D NACA hydrofoil, aiming at studying the sheet cavitation collapse behavior and related erosion mechanism. The detailed cavitation evolution structures were captured from side and top views by two high speed cameras. And the cavitation erosion was measured by oil painting method for 1 hour. Such long testing time makes it difficult to investigate the relationship between cavitation structures and erosion. A common way to solve this issue is the utilization of softer material as the transducer [21-22]. Recently, our previous work applied a thin aluminum foil attached on the solid surface as an erosion sensor. The tests only lasted few seconds to minutes, which enabled us to record the cavitation erosion progress and the cavitation evolution synchronously by high speed camera [23-25]. The results presented a strong demonstration that the cloud cavity separation and collapse are responsible for the damage. In the meantime, great effort has been made on the prediction of cavitation erosion, based on various mechanisms [26-31]. And most of them were conducted by CFD method. A recent study performed by Gavaises et al [32] predicted the cavitation area in an axisymmetric geometry (similar to the current work) via both visualization and CFD approaches, in order to correlate the eroded location with the location of cavitation development. The simulation results had great agreement with the visualizations. In the present study, we performed jet flow cavitation tests through two adjacent parallel plates with a nozzle tube-orifice plate. Aiming at providing insight into the details of the correlation between cloud cavitation collapse and cavitation-induced erosion, and establishing fully experimental methods for investigating different cavitating devices, three image post-processing approaches were employed to analyze the cavitation characteristics. The prediction results were validated by cavitation erosion tests.

Fig. 1 Fig. 2

 12 Fig. 1 Test rig ①Loop pump ②Downstream tank ③Orifice plates ④Upstream tank ⑤⑥Valves ⑦Electromagnetic flow meter ⑧Heater ⑨Pressure transducer ⑩Compressor ⑪Vacuum pump The test section was manufactured by Plexiglas for visualization. As mentioned above, it consists of two square parallel plates with a dimension of 150mm×150mm. The flow comes through a nozzle tube at the center of the upstream plate, where the cavitation generates at the throat. And then it moves downstream through four symmetrical outlet holes on the periphery. The downstream plate is used

Fig. 3 Fig. 4

 34 Fig. 3 Aluminum foil attached downstream plate

Fig. 5

 5 Fig. 5 Mean value (upper) and standard deviation distribution (below) under various cavitation numbersindicates one pixel. More details concerning to the applied post-processing approach can be found in ref.[START_REF] Dular | Relationship between cavitation structures and cavitation damage[END_REF].

Fig. 5

 5 Fig. 5 presents the mean value and standard deviation distribution under various cavitation numbers. From which, we can identify the cavitation covering area via the mean value distribution, and predict the potential eroded area through the standard deviation distribution. One can notice that the size of the mean value and standard deviation distributions are almost identical with each other. However, the contours are not that analogous when the cavitation number drops below 3.41. The standard deviation indicates that the most potential erosion region locates on the periphery with highest deviation value, while the most covered cavitation region given by the mean value lie in the middle of the contour (The scales of the standard deviation and mean value are used for better image contrast to identify the cavity covered region and potential erosion region). When the cavitation number goes above 3.57, the distributions are nearly the same. The reason is that as the cavitation number decreases, the cavitation evolution process becomes more distinct and regular. As a result, the likely eroded area concentrates at the sites where collapse occurs, making it prone to be damaged by emitted shock

Fig. 6

 6 Fig. 6 Cavitation damaged aluminum foil at σ=3.41

Fig. 7 Fig. 8

 78 Fig. 7 Raw images of recorded aluminum foil (upper), post-processed images (middle) and predicted results of the eroded area at this instant (down) at σ=3.41

Fig. 9

 9 Fig. 9 Erosion pits number as a function a time (a) 1min (b) 5sFrom the above analysis, one can find it is not a straightforward work to determine the cavitation evolution

Fig. 10

 10 Fig. 10 plots the results of PSD analysis under various cavitation numbers. In each graph, the highest amplitude value corresponds to the exact cavitation circulation frequency. It can be noticed that as the cavitation number declines, the circulation frequency becomes much easier to distinguish from the PSD curve -the maximum amplitude is much clearly when σ bellows to 3.41. For higher cavitation numbers, the discrepancy of the amplitude against frequency turns less obvious. It is because the cavitation evolution, as mentioned above, is not that regular when σ is higher, contributing to sort of irregular circulation. Even though, the maximum amplitude can still be picked out easily. In general, one can find the cavitation circulation frequency decreases with the decreasing cavitation number, which means the cavitation circulation becomes longer and longer, as shown in Fig. 8.

Fig. 10 PSD

 10 Fig.10 PSD analysis on cavitation circulation frequency under various cavitation number


Bo Hu

Dieter Brillert

Hans Josef Dohmen

Friedrich-Karl Benra

Investigation on thrust and moment coefficients of a centrifugal turbomachine

Keywords:

In radial pumps and turbines, the centrifugal through-flow is quite common, which has strong impacts on the core swirl ratio, pressure distribution, axial thrust and frictional torque. The impact of centrifugal through-flow on above parameters are still not sufficiently investigated with different circumferential Reynolds numbers and dimensionless axial gap widths.

A test rig is designed at the University of Duisburg-Essen and descirbed in this paper. Based on the experimental results, correlations are determined to predict the impact of the centrifugal through-flow on the core swirl ratio, the thrust coefficient and the moment coefficient with good accuracy. Part of the 3D Daily&Nece diagram from a former study of the authors is extended with centrifugal through-flow. The results will provide a data base for calculation of axial thrust and moment coefficient in order to design radial pumps and turbines with smooth impellers.

INTRODUCTION

Rotor-stator cavities are common devices in radial pumps and turbines. The typical geometry of a rotor-stator cavity is shown in Figure 1.

Figure 1. Geometry of a rotor-stator cavity

The through-flow in such cavities can be either radial inward or radial outward and it impacts the radial pressure distribution acting on the turbomachine rotor in a certain manner. The study of the flow in a rotor-stator cavity has significant relevance to many problems encountered in turbomachinery. The thrust coefficient and the moment coefficient are two major concerns in radial pumps and turbines. The investigation of the flow in rotor-stator cavities can provide more confidence for calculating the axial thrust (direction see Figure 1) and the frictional torque M in radial pumps and turbines.

Since evaluating and is quite important for the design of turbomachinery, a lot of researches are accomplished on these topics. Von Kármán [START_REF] Kármán | Über laminare und turbulente Reibung[END_REF] and Cochran [START_REF] Cochran | The flow due to a rotating disk[END_REF] gave a solution of the ordinary differential equation for the steady, axisymmetric, incompressible flow. Daily and Nece [START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF] examined the flow of an enclosed rotating disk both analytically and experimentally. Kurokawa et al. [4~6] studied and in a rotor-stator cavity with both centrifugal and centripetal through-flow. Poncet et al. [START_REF] Poncet | Turbulent rotating disk flow with inward throughflow[END_REF] studied the centrifugal through-flow in a rotor-stator cavity and obtained two equations of the core swirl ratio K for both the Batchelor type flow and the Stewartson type flow based on the local flow rate coefficient (positive for centrifugal through-flow). Schlichting and Gersten [8] organized an implicit relation based on the results of Goldstein [START_REF] Goldstein | On the resistance to the rotation of a disc immersed in a fluid[END_REF] for under turbulent flow conditions. Debuchy et al. [START_REF] Debuchy | On the flow behavior in rotor-stator system with superimposed flow[END_REF] determined an explicit equation of K for the Batchelor type flow which is valid over a wide range of the local flow rate coefficient: 0.5 (negative for centripetal through-flow). Launder et al. [START_REF] Launder | Transitional, and Turbulent Flows in Rotor-Stator cavities[END_REF] provided a review of the current understanding of instability pattern that are created in rotor-stator cavities leading to transition and eventually turbulence. Will et al. [12~14] investigated the flow in the side chamber of a radial pump. Recent experimental investigations for large global Reynolds number with or without through-flow have been conducted by Coren et al. [START_REF] Coren | Windage sources in smooth-walled rotating disc systems[END_REF], Long et al. [START_REF] Long | Windage Measurements in a Rotor Stator Cavity with Rotor Mounted Protrusions and Bolts[END_REF] and Barabas et al. [START_REF] Barabas | Experimental and numerical determination of pressure and velocity distribution inside a rotor-stator cavity at very high circumferential Reynolds numbers[END_REF]. Based on the experimental results, Bo Hu et al. [START_REF] Hu | Investigation on the flow in a rotor-stator cavity with centripetal through-flow[END_REF] This study is focused on the impact of centrifugal through-flow on and , so that the influence of both the centripetal (Bo Hu et al. [START_REF] Hu | Investigation on the flow in a rotor-stator cavity with centripetal through-flow[END_REF]) and the centrifugal through-flow can be better understood. The definitions of the significant dimensionless parameters in this study are given in Eq. (1.1~1.10).

(1.1) A plenty of researches, such as those by Kurokawa et al. [START_REF] Kurokawa | Roughness Effects on the Flow along an Enclosed Rotating Disc[END_REF] and Poncet et al. [START_REF] Poncet | Turbulent rotating disk flow with inward throughflow[END_REF], show that the pressure distribution along the radius of the disk can be estimated with the core swirl ratio K with Eq. (3.1) both with and without through-flow. Will et al. [12~14] determined Eq. (3.2) to evaluate the pressure distribution along the radius of the disk for the incompressible, steady flow. It is obtained directly from the radial momentum equation when the turbulent shear stress is neglected. In a rotor-stator cavity, the cross sectional area changes in the radial direction. Consequently, the pressure must also change since the mean velocity changes in the radial direction according to the continuity equation. Based on Eq. (3.2), the pressure along the radius can be calculated with Eq. ( 4) based on the values of K. K is a variable along the radius of the disk. A simplification is made as follows: K is a fixed value every 1 mm in the radial direction. Then, the approximate pressure distribution along the disk can be calculated with Eq. ( 4).

(1.2) (1.3) ̇ (1.4) (1.5) (1.6) (1.7) | | (1.8) ( ) ( ) , (1.9) ∫ ( ) (1.10)

THEORETICAL ANALYSIS

represents the pressure at x=1. Due to the construction of the geometry, there is no pressure tube at x=1. The closest pressure tube in the front cavity is at x=0.955. The value of is calculated combining the measured pressure at x=0.955 with Eq. ( 4) based on the core swirl ratio along the radius.

( ) ∫ ( ) (4
) Where:

∫ ∑ ( ) , ( )
The difference of the force on both sides of the disk is the main source for the axial thrust , calculated with Eq. ( 5).

(calculated with Eq. ( 6)) and respectively represent the force and the thrust coefficient on the front surface of the disk (in the front chamber, shown in Figure 1), while (calculated with Eq. ( 7)) and are those on the back surface of the disk (in the back chamber). represents the radius of the hub (see Figure 1). The back chamber (G=0.072), shown in Figure 1, is viewed as an enclosed cavity. The values of are obtained when =0 and the axial gaps of the both cavities have the same size for different Re (under that condition = ). After obtaining those values, the values of with different values of can be calculated with Eq. ( 8).

(

) (6) ( ) ( ) (7) ( ) 5

NUMERICAL SIMULATION

To predict the cavity flow, numerical simulations are carried out using the ANSYS CFX 14.0 code. Considering the axial symmetry of the problem, a segment (15 degree) of the whole domain is modeled and a rotational periodic boundary condition is applied. Structured meshes are generated with ICEM 14.0. The domain for numerical simulation when G=0.072 is depicted with yellow color in Figure 4. The mesh on the cross section at the position "I" and position "II" (see Figure 4) are depicted in Figure 5. The simulation type is set as steady state. Barabas et al. (2015) found that the simulation results from the SST turbulence model in combination with the scalable wall functions are in good agreement with the measured pressure in a rotor-stator cavity with air. The deviations of the pressure measurements are less than 1%. Hence, in this study, the same turbulence model and wall functions are used. The turbulent numeric is set as second order upwind. The non-slip wall condition is set for all the walls. The boundary conditions at the inlet and the outlet are pressure inlet and mass flow outlet, respectively. The values of the pressure at inlet are set according to the pressure sensor at the pump outlet. The convergence criteria are set as in maximum type. The maximum value of in all the simulation model is 13.4.

TEST RIG DESIGN AND EXPERIMENTAL SET-UP

The test rig is supplied with water by a pump system, shown in Figure 6. The shaft is driven by an electric motor. A frequency converter is used to adjust the speed of rotation (0~2500/min) with the absolute uncertainty of 7.5/min. In this study, only the axial gap of the front chamber is changed by installing six sleeves with different length. Other parameters of the experiments in this study are given in Table 1.The cross section of the test rig is shown in Figure 7. measured results are estimated with the root sum squared method. The measured range of the torque meter is 0~10

(V) (I) (VI) (II) (VII) (III) (VIII) (IV) (IX) (X) (

I). Sleeves (to change the axial gap), (II). Guide vane (24 channels), (III). Front chamber, (IV). Disk, (V). Back cover, (VI). Linear bearing, (VII). Tension compression sensor, (VIII). Thrust plate, (IX). Nut, (X). Shaft

The measured range of the pressure transducer is 0~2.5 bar (absolute pressure). The measured range of the thrust transducers is -100~100 N. The input voltage signals are the following ranges: 0~10 V for the pressure transducers and the torque transducer, -10 V~10 V for the axial thrust transducers. The absolute accuracy of the data acquisition system (with NI USB-6008) is 4.28 mV in this study. The random noise and zero order uncertainty are neglected because they are very small. The uncertainties of the measured results, noted as , are calculated in a former study of the authors (Bo Hu et al. [START_REF] Hu | Investigation on the flow in a rotor-stator cavity with centripetal through-flow[END_REF]), given in Table 2.

RESULTS AND DISCUSSION

Velocity distribution

All the velocities are made dimensionless by dividing .The velocity profiles at three radial positions for Re=1.9 and G=0.072 (wide gap) are shown in Figure 8. The dimensionless radial velocities are not exactly zero in the central cores ( ), shown in Figure 8 (a~c). From the distribution of tangential velocity , there are central cores at all the investigated radial positions where the values of are almost constant along , shown in Figure 8 (d~f). The values of the tangential velocity are smaller at when increases, depicted in Figure 8 (d,e). The trend of are in good agreement with the measured in the literature (such as from Poncet et al. [START_REF] Poncet | Turbulent rotating disk flow with inward throughflow[END_REF] and Debuchy et al. [START_REF] Debuchy | On the flow behavior in rotor-stator system with superimposed flow[END_REF]). The values of | | become smaller towards the shaft. The velocities for are in the reference [START_REF] Hu | Investigation on the flow in a rotor-stator cavity with centripetal through-flow[END_REF].

x=0.955 x=0.79 x=0.57 (a) (b) (c) (d) (e) (f) (g) (h) (i) =1262 =3787 =5050

Main K curves

To evaluate the pressure distribution, the values of K should be estimated. Although some correlations are determined to predict the values of K with centrifugal through-flow, such as Eq. (2.1), Eq. (2.2) and Eq. (2.3), there is still an uncertainty on the impact of G on K. The geometry of the cavity,especially at the inlet and the outlet, will also have large influence on K. Based on Eq. ( 4), the pressure difference between the two pressure tubes number e and number e+1 can be calculated with Eq. ( 9). ̅ represents the average value of K between the two adjacent pressure tubes. There are 12 pressure tubes in the front chamber from r=0.05 m (x=0.455) to r=0.105 m (x=0.954). Since the radial distances between the adjacent pressure tubes are small, the application of the average values of K between the tubes may not result in large errors. The values of for the experimental results are calculated with .

The values of K ( ̅ ) therefore can be verified based on the pressure measurement with Eq. [START_REF] Debuchy | On the flow behavior in rotor-stator system with superimposed flow[END_REF].

( ) ( ) ̅ ( ) ( ) (9)
√ ( ) ( ) ( ) ( ) (10
) Based on the results from both numerical simulations and pressure measurements, Eq. ( 11) is determined to describe the impact of G on K. The experimental results based on pressure measurements are compared with those from simulation and those calculated by Eq. ( 11) in Figure 10.

The results from Eq. ( 11) are in good agreement with those from numerical simulations and experiments.

Relatively large errors only occur when 0.01, which can be attributed to the application of the average values of K in and around the transition zone of the two flow types, where K decreases dramatically. In the future, more pressure taps will be manufactured at the low radius to eliminate the error. The influence of G on K is based on the results in Figure 10. Poncet et al [START_REF] Poncet | Turbulent rotating disk flow with inward throughflow[END_REF] and Debuchy et al [START_REF] Debuchy | On the flow behavior in rotor-stator system with superimposed flow[END_REF] ignored the impact of G on K based on the results from LDV measurements. In most of the radial pumps and turbines, G is a variable along the radius. simplified correlation is required with good accuracy over the whole range of G. Based on the measurements, Eq. ( 12) is correlated to predict the values of K when G ranges from 0.018 to 0.072. The results are compared in Figure 11. The results from Eq. ( 12) are in good accordance with both the simulation results and the experimental results. In this paper, Eq. ( 12) is applied during the calculation of the pressure instead of Eq. [START_REF] Launder | Transitional, and Turbulent Flows in Rotor-Stator cavities[END_REF]. The correctness of Eq. ( 12) will be further verified based on the LDV measurements in the future.

Batchelor/Couette type flow ( ):

( )

Stewartson type flow ( ):

( ) There are some results of K, which, however, do not fit the resuts from Eq. ( 12), especially at x=0.955 for wider gaps. Some of the results are shown in Figure 12 (a). Near the outlet, there is a area change from the front cavity to the channel in the guide vane for G=0.036, 0.054 and 0.072. (see Figure 1 and Figure 4).

The measured pressure at x=0.955 is strongly influenced by the geometry at the outlet of the testrig (see Figure 12

Axial thrust

Based on the measurements, Bo Hu et al. [START_REF] Hu | Investigation on the flow in a rotor-stator cavity with centripetal through-flow[END_REF] determined an empirical equation for the thrust coefficient in a rotor-stator cavity with centripetal through-flow. It is organized based on the experimental results for centripetal through-flow. When compared with the experimental results in this paper, it is modified for centrifugal through-flow. In this study, is positive for centrifugal through-flow. It is written as:

( ) ( ) ( ) (13)

Part of 3D Daily&Nece diagram

The moment coefficients can be predicted according to the flow regimes. The typical tangential velocity profiles for regime III and regime IV are shown in Figure 15. Near the distinguishing surface, there is a mixing zone, where regime III and regime IV coexist in the cavity (ignored in this study). The distinguishing surface for centripetal through-flow (Bo Hu et al. [START_REF] Hu | Investigation on the flow in a rotor-stator cavity with centripetal through-flow[END_REF]) is also plotted to make it a complete diagram.

Moment coefficient

According to the experimental results from Han et al. [START_REF] Hongbiao | Exploring fluid resistance of disk rotor based on boundary layer theory[END_REF], the moment coefficient on the cylinder surface of the disk, noted as , can be estimated with Eq. ( 14) for smooth disks.

|

| 𝑡 (𝑙𝑔 𝜐 )

Comparing the torque measurements by the authors with the results from Daily and Nece [START_REF] Daily | Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks[END_REF] and Dorfman [START_REF] Dorfman | Resistance of a rotating rough disc[END_REF] [START_REF] Dorfman | Hydrodynamic resistance and the heat loss of rotating solids[END_REF], two correlations are determined to predict the moment coefficient (for a single surface of the disk), given in Eq. ( 15) and Eq. ( 16).

Regime III

( ) (15)
Regime IV

( ) (16)
The experimental results of are compared with those from Eq. ( 15) and Eq. ( 16), depicted in Figure 17. For G=0.018 and G=0.036, most of the flow regimes are regime III, shown in Figure 17 (a) and Figure 17 (b). The experimental results of are in good accordance with those from Eq. ( 15) in general. When G increase to 0.036, the flow regimes change from regime III to regime IV with the increase of Re for =0, 1262 and 2525, which can also be found based on the experimental results of . For G=0.054 and G=0.072, most of the flow regimes are regime IV. The results of from experiments are in good agreement with those from Eq. ( 16) in general. The regime III may occur at small Re and large . At the same values of , the intersection points of the curves from Eq. ( 15) and those from Eq. ( 16) are close to those in Figure 16 (a) between the curves for G and . The difference can be attributed to the existence of the mixing zone. The amounts of increase with the increase of . The values of drop faster with the increase of Re for smaller values of G. Compared with those for centripetal through-flow (Bo Hu et al. [START_REF] Hu | Investigation on the flow in a rotor-stator cavity with centripetal through-flow[END_REF]), the centrifugal through-flow will result in larger values of at the same values of | |, which is in accordance with the conclusion of Dibelius et al. [START_REF] Dibelius | Experiments on Friction, Velocity and Pressure Distribution of Rotating Discs[END_REF]. Eq. ( 16) Exp Eq. ( 15) Eq. ( 16) = Exp Eq. ( 15) Eq. ( 16) Exp Eq. ( 15) Eq. ( 16) = Exp Eq. ( 15) Eq. ( 16) On the distinguishing lines (see Figure 16), the results from Eq. ( 15) should be equal to those from Eq. ( 16). The results of / at the distinguishing lines are presented in Figure 18. The differences, attributed to the existence of the mixing zone, are very small in general and cover an amount less than 5%. Based on the results from Figure 18, Eq. ( 15) and Eq. ( 16) can be used to predict the transition from regime III to regime IV with good accuracy. All the results show that the moment coefficient can be predicted with Eq. ( 15) and Eq. ( 16) based on the 3D Daily&Nece diagram. ). The applications of the equations will become wider by introducing the impact of surface roughness of the disks in the next step. All the results of K will be verified based on the velocity measurements with a two-component LDV system. The distinguishing lines will be modified based on the measured velocity components in both tangential and radial directions in the future. The outlet geometry has a relatively large influence on the results of K, which deserves further study. The impacts of boundary condition (at both inlet and outlet) and internal flow structures on and should also be investigated in the future.

CONCLUSIONS

The influence of centrifugal through-flow on the velocity, radial pressure distribution, axial thrust and frictional torque in a rotor-stator cavity with different G is strong.

Based on the pressure measurements, an empirical correlation is determined to predict the impact of Re, on K when G ranges from 0.018 to 0.072. A correlation is determined, which enables to predict the influence of G, Re and on the thrust coefficient for a smooth disk (

). Part of the 3D Daily&Nece diagram is obtained by distinguishing the tangential velocity profiles for centrifugal through-flow. Four distinguishing lines and the approximate distinguishing surface are presented. Using the equations for the axial thrust coefficient and the moment coefficient, the influence of the centrifugal through-flow can be better predicted when designing radial pumps and turbines with smooth impellers. This makes the correlations of a huge worth for the designers.

Some more attention will be drawn in the future to the impact of the disk roughness. The 3D Daily&Nece diagram and Eq. ( 12) will also be modified based on the velocity measurements with a two-component LDV system.

  the values of in a rotor-stator cavity with centripetal through-flow. They also extended part of the 2D Daily&Nece diagram into 3D by distinguishing the tangential velocity profiles with a third axis of through-flow coefficient based on the simulation results. Based on the experimental results, two equations were determined to describe the impact of , Re and the dimensionless axial gap width G on for regime III (merged disk boundary layer and wall boundary layer, namely Couette type flow) and regime IV (separated disk boundary layer and wall boundary layer, namely Batchelor type flow).

Figure 2 .

 2 Figure 2. Velocity profiles for both Batchelor type flow and Stewartson type flow

Figure 3 (Figure 3 .

 33 Figure 3. Main K-curves

Figure 4 .

 4 Figure 4. Domain for numerical simulation at G=0.072

Investigation on thrust and moment coefficients of a centrifugal turbomachine - 4

 4

Figure 5 .

 5 Figure 5. Mesh on the cutting plane

Figure 6 . 1 .

 61 Figure 6. View of the test rig

Figure 7 .

 7 Figure 7. Cross section of the test rig

Figure 8 .Figure 9 .Figure 9 .

 899 Figure 8. Velocity profiles for Re= and G=0.072

BatchelorFigure 10 .

 10 Figure 10. Mean -curves

Figure 11 .

 11 Figure 11. Mean -curves

Figure 12 .

 12 (b)). Based on the simulation results, there are small vortices near the outer radius of the disk. Hence, part of the measured values at x=0.955 are not used during the calculation of K. K (a) Part of the results do not fit Eq. (12) (b) Surface streamlines near the outlet (from simulation) Large differences of K attributed to the geometry near the outlet 4.2 Pressure coefficient A reference pressure is taken at the dimensionless radial coordinate x=1. Due to the restriction of the geometry of the test rig, there is no pressure tube at x=1. The closest tube is at x=0.955. The pressure values at x=1 are from Eq. (4) based on the values of K from Eq. (12). The values of pressure coefficient are positive because the pressure drops towards the shaft. In Figure 13, the values of are plotted versus the non-dimensional radial coordinate x. The through-flow coefficient is used as a parameter. The experimental results show that decreases with the increasing , Re and G in general. The experimental results are in good agreement with those from equations. When Re=2.79 , the uncertainty of the is 1.3 , which is very small compared with the measured results. Hence, the error bars are neglected in Figure 13 (d~f).

Figure 13 .

 13 Figure 13. Distribution of along the radius

  Where 0.018 G 0.072, 5050, Re . The comparison of the results of for different G and are shown in Figure 14. Bp represents the calculated thrust coefficient based on the pressure calculation along the radius of the disk, which are calculated combining the measured pressure with Eq. (4) based on the values of K (calculated based on every 1 mm along the radius for Batchelor type flow or Stewartson type flow) from Eq. (12). In the transition zone, the equation of K for Batchelor type flow is used. The values of are smaller for wider axial gaps in general. The values of decrease with increasing . In a rotor-stator cavity with centripetal through-flow ( is negative) studied by Bo Hu et al. [18], however, the values of increase with increasing | |. The experimental results of are in good agreement with those based on the pressure calculation and Eq. (13).

Investigation on thrust and moment coefficients of a centrifugal turbomachine - 10 (Figure 14 .

 1014 Figure 14. Mean -curves

Figure 15 .

 15 Figure 15. Typical tangential velocity profiles for regime III and regime IV

Figure 16 .

 16 Figure 16. Part of the 3D DailyNece diagram

Figure 17 .

 17 Figure 17. Main -curves in dependence of G

Figure 18 .

 18 Figure 18. Results of / at the distinguishing lines

  Two correlations are determined to predict the influence of centrifugal through-flow on for the two zones with good accuracy for the smooth disk (). At the distinguishing lines, the results from the two equations are very close. The values of for centrifugal through-flow exceed those for centripetal through-flow at the same values of | |.

  front surface of the disk Fab Force on the back surface of the disk FS Full scale G Dimensionless axial gap K Core swirl ratio at =0.5 ̅ Average value of K between the two adjacent

Table 2 .

 2 Uncertainties of the measured results

	p (bar)	(N)	M (Nm)	Re
	4.04	2.43	3.00	9.01	4.1

ACKNOWLEDGMENTS

This study is funded by CSC (China Scholarship Council) and the chair of turbomachinery at University of Duisburg-Essen.


Botond Barabas
email: botond.barabas@uni-due.de

Dieter Brillert

Hans Josef Dohmen

Friedrich-Karl Benra

The Behavior of the Modal Properties of Weak and Strong Coupled Acoustic Modes at Different Pressure Levels in a Rotor-Stator Test Rig

Keywords:

The application of high pressure radial compressors is expected to have an above-average growth rate in future. When vibrations are matter of investigations, impellers surrounded by dense gases cannot be treated separately but as a coupled system comprising the impeller structure and the fluid. In this paper experimental results gathered in a rotor-stator test rig at different gas pressures are presented. The modal properties of a weakly coupled acoustic mode and a strongly coupled acoustic mode are determined. The results show no influence of the gas pressure on the natural frequency of the weakly coupled acoustic mode. The damping of it is approaching a minimum with increasing gas density. The natural frequencies of the strongly coupled modes diverge from each other when the gas pressure is increased.

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INTRODUCTION

High pressure radial compressors are expected to have an above-average growth rate in future. The main application fields of these machines are at oil exploitation sites as reinjection gas supplies or at Carbon-Capture-and-Storage power plants for carbon dioxide compression. The operational pressures and densities of the working fluids are intended to increase with these applications. Together with the increasing fluid density, aeroacoustics and aeroelasticity have a great impact when it comes to vibration issues and high cycle fatigue. Especially, the aeroacoustics in the side cavities of radial compressors play a major role in the operational reliability of radial compressors.

One of the first publications about a fundamental aeroacoustic excitation source in turbomachines has been published by Tyler and Sofrin [START_REF] Tyler | Axial Flow Compressor Noise Studies[END_REF]. They reveal excitations at discrete frequencies as an interaction between rotor and stator components, the so called Tyler-Sofrin modes. Ehrich [START_REF] Ehrich | Acoustic Resonances and Multiple Pure Tone Noise in Turbomachinery Inlets[END_REF] conducted further studies that he has focused on natural frequencies of acoustic modes in annular cavities influenced by axial flow and flow rotation. A problem orientated numerical investigation of acoustic modes in the side cavities of radial compressors has been conducted by König [3], who reveals a coupling of the acoustic modes in the two side cavities with each other. The paper compares 2D-analytical and 3D-FEM methods for acoustic mode calculations. Petry [START_REF] Petry | Experimentelle Untersuchung aeroakustischer und aeroelastischer Phänomene in Hochdruck-Radialverdichtern[END_REF] has been the first who showed the excitation of acoustic resonances in side cavities of radial compressors with Tyler-Sofrin modes in his experiments.

When vibrations are matter of investigations, impellers surrounded by dense gases cannot be treated separately but as a coupled system comprising the impeller structure and the fluid. Each nodal diameter natural mode is at least split into two modes with two corresponding natural frequencies. For a weak coupling the coupled natural frequencies approximately equal the uncoupled structural and acoustic natural frequencies, respectively. The stronger the coupling the merrier the coupled natural frequencies differ from the uncoupled ones.

The effect of dense surrounding fluids on the natural frequencies of pump impellers has been investigated numerically by Liang et al. [START_REF] Liang | Numerical simulation of fluid added mass effect on a francis turbine runner[END_REF] and experimentally by Rodriguez et al. [START_REF] Rodriguez | Experimental investigation of added mass effects on a Francis turbine runner in still water[END_REF]. They conducted their investigations by keeping the fluid pressure constant and changing the type of the fluid. Their findings are decreasing natural frequencies of the impeller and increasing damping ratios if the impeller is surrounded by water instead of air. Magara et al. [START_REF] Magara | Natural Frequency Shift in a Centrifugal Compressor Impeller for High-Density Gas Applications[END_REF] have investigated the natural frequencies of a radial compressor impeller for different surrounding gas pressures experimentally. For some natural frequencies of the structure their results show a decreasing trend for increased pressures; however, other natural frequencies increase at the same time. They conclude an influence of the stiffness of the surrounding gas beside the added mass effect onto the structure.

Sandberg and Göransson [START_REF] Sandberg | A Symmetric Finite Element Formulation for Acoustic Fluid-Structure Interaction Analysis[END_REF] have conducted a numerical finite element approach to investigate a coupled fluid-structure interaction of an aircraft fuselage. They have found striking differences between the uncoupled and coupled eigenfrequencies. In their numerical investigation of coupled eigenmodes of a nuclear vessel Sigrist et al. [START_REF] Sigrist | Dynamic analysis of a nuclear reactor with fluid-structure interaction Part I: Seismic loading, fluid added mass and added stiffness effects[END_REF] discover a change of mode shapes at increasing fluid densities. This effect is observed in the results when added mass and coupling effects are taken numerically into account.

It becomes apparent that aeroelasticity and aeroacoustics in side cavities of radial compressors is a complex research field. Investigations in the past have revealed some effects, which cannot be fully described, yet. In order to gather fundamental understanding of these effects a rotor-stator test rig has been built up at the University of Duisburg-Essen. The main advantage of its design is that contrary to real compressors the excitation frequency and strength can be set independently from gas parameters and disk rotational speeds. This decoupling of excitation source and the system, which should be excited, allows the identification of influence parameters. Only a few publications [START_REF] Liang | Numerical simulation of fluid added mass effect on a francis turbine runner[END_REF][START_REF] Rodriguez | Experimental investigation of added mass effects on a Francis turbine runner in still water[END_REF][START_REF] Beirow | Mistuning and Damping Analysis of a Radial Turbine Blisk in Varying Ambient Conditions[END_REF] have investigated the influence of fluid density variations on the aerodynamic damping of disks in radial turbo machines. Though, a more thorough knowledge of the aerodynamic damping of acoustic and of structural modes could improve the operational reliability of high pressure radial compressors. In this paper the influence of the surrounding gas pressure variation on the damping and natural frequencies of weakly and strongly coupled acoustic modes are presented. The experimental data is gathered with a non-rotating disk.

EXPERIMENTAL SETUP AND METHODOLOGY 1.1 Test rig

The measurements presented in this paper are conducted in the rotor-stator test rig that has been introduced in the framework of aerodynamic and acoustic measurements in Barabas et al. [START_REF] Barabas | Experimental and Numerical Determination of Pressure and Velocity Distribution Inside a Rotor-Stator Cavity at very High Circumferential Reynolds Numbers[END_REF][START_REF] Barabas | Identification of Coupled Natural Frequencies in a Rotor-Stator Test-Rig for Different Gas Properties[END_REF]. Geometrically, the test rig represents simplified side cavities of radial compressors. In Figure 1 one stage of a radial compressor is sketched. The side cavities are the spaces between the casing and the impeller's hub and shroud. They are marked green and blue in Figure 1, respectively. These cavities are simplified in the test rig (Figure 2). The impeller represented as a plain disk is embraced by a plain walled casing. To distinguish between the cavity where the loudspeakers excite the fluid and the most sensors are mounted this cavity is called the front cavity and the other rear cavity. The sensors' and loudspeaker's allocations are sketched in the figure, too. The axial gap widths (s) in both cavities can be adjusted independently from each other covering the range of gap widths of modern radial compressors. Within this experimental investigation the gap widths in both cavities are set equally. The disk thickness (b) has been chosen to enforce at least one pair of acoustic and structural modes, with the same number of nodal diameters, to have similar uncoupled natural frequencies at ambient conditions. This ensures a strong coupling between the acoustic and structure mode. Furthermore, the geometric dimensions of the cavities and the disk have been chosen in a way that other acoustic and structure natural frequencies with equal mode shapes differ from each other. This guarantees a weak coupling between those. A picture of the test rig, taken from the front cavity side, is presented in Figure 3.

Instrumentation

The acoustic specific instrumentation of the test rig with the uncertainty and repeatability analysis is described in detail in Barabas et al. [START_REF] Barabas | Identification of Coupled Natural Frequencies in a Rotor-Stator Test-Rig for Different Gas Properties[END_REF]. The test rig is equipped with loudspeakers as excitation sources to overcome the absence of blade-vane interaction. The applied acoustic specific pressure transducers and strain gages are summarized in Table 1. The transducer positions in meridional view and in normal view are presented in Figure 2 and Figure 4, respectively. However, even the identification of low order modes is an impossible task with a finite number of pressure transducers. As remedy, a novel rotatable pressure transducer has been developed and built in the front cavity casing of the test rig. It is based on an idea of rotatable microphones [START_REF] Cicon | Investigation of Continuously Traversing Microphone System for Mode Measurement[END_REF]. Together with a specific evaluation method the pressure pattern of an acoustic mode at a constant excitation frequency can be obtained in circumferential direction.

In Figure 4 the test rig is sketched in the normal view with the sensor positions. The pressure sensors are marked with a "P", the strain gages with a "G", and "LS" refers to the loudspeakers. The character inside the parentheses represents the location: The rear cavity is labeled with "RC" and the disk is labeled with "D". If the location character is omitted the sensor is mounted in the front cavity. The presented results within this paper have been recorded with the pressure sensor "P147" and the strain gage "G142 (D)", mounted on the disk. The fluid is excited with the loudspeaker "LS 2". For a better differentiation they are marked red in the figure. sensors are presented. Labels: "P": pressure sensor, "G": strain gage, "LS": loudspeaker, "RC": rear cavity, "D": disk

Methodology

Experimental Procedure and Boundary Conditions

The basis for the evaluation of the natural frequencies and the modal damping values are frequency sweep experiments over a specific interval. The maximum allowed sweep velocity S max,exc without falsifying the measured resonance height, frequency, and damping is set according to Ewins [START_REF] Ewins | Modal Testing: Theory and Practice[END_REF] to

𝑆 max,exc = 3.6 ⋅ 𝑓 𝑟 2 𝜁 2 (1)
with f r as the resonance frequency and ζ as the critical damping ratio.

The developed pressure patterns at certain frequencies are measured with the rotatable pressure sensor in circumferential direction. The measurement procedures are described in Barabas et al. [START_REF] Barabas | Identification of Coupled Natural Frequencies in a Rotor-Stator Test-Rig for Different Gas Properties[END_REF].

The experiments are conducted with air as fluid at different pressure levels. They are summarized in Table 2. The speed of sound changes due to the pressure increase less than 1% and is only considered for the weakly coupled acoustic mode. . The data blocks for the STFT have been chosen small enough to allow a good resolution in time. The overlap of 75% between two adjacent blocks avoids data loss. With the knowledge and application of the excitation frequency to time relation, a good frequency resolution can be achieved, too.

However, because the excitation force of the loudspeakers on the fluid cannot be determined directly it is assumed to be constant in the small frequency range around a resonance. Hence, the Frequency Response Function (FRF) is not calculated; instead, the analysis is based on the direct response of the transducers, not set in relation to the excitation force. Comparisons of the response signals' amplitude between different pressure levels should be treated with caution.

There are different methods to determine the modal parameters of a system. Each has its advantages and disadvantages for particular vibrating systems. In the framework of this paper the peak-amplitude method is applied, as presented in Ewins [START_REF] Ewins | Modal Testing: Theory and Practice[END_REF]. It can resolve the natural frequencies of the strongly coupled modes and the natural frequencies and the damping of the weakly coupled mode, because it is well separated from adjacent mode. Because the structural and mechanical damping is very small compared to the aerodynamic damping they are neglected within this paper.

Analytical approach

The experimental results of the natural frequencies are compared with a qualitative analytical approach presented by Magara et al. [START_REF] Magara | Natural Frequencies of Centrifugal Compressor Impellers for High Density Gas Applications[END_REF] to estimate the natural frequency shift of a similar coupled system with a disk enclosed in a gas cavity. They couple the equations of motions of the gas and the structure and assume an inviscid and compressible gas. They obtain

�𝜔 ac,r 2 -𝜔 2 ��𝜔 st,r 2 -𝜔 2 � -2𝜔 2 𝜌 f ⋅ 𝑐 2 𝜌 st ⋅ 𝑏𝑏 = 0 (2)
with 𝜔 ac,r and 𝜔 st,r as the uncoupled natural frequencies with the same mode shape of the fluid and disk, respectively. 𝜌 f and 𝜌 st correspond to the density of the fluid and the structure. The fluid's speed of sound is denoted with c and the geometry is represented by b as the disk thickness and s as the axial gap width of the cavity (Figure 2). The minuend of Equation ( 2) can be graphically represented as a parabola, the subtrahend as a line. Two solutions exist at the crosssections. These are the coupled natural frequencies of the system. Magara et al. [START_REF] Magara | Natural Frequencies of Centrifugal Compressor Impellers for High Density Gas Applications[END_REF] have introduced the terms "disc origin" and "fluid origin mode" for the respective branches of the parabola, depending on the uncoupled natural frequencies. The terms are renamed to "structure dominant" and "acoustic dominant" within this paper. The vibration can be characterized with the phase relation between the axial disk motion and the pressure pattern's resulting axial force onto the disk, too. In the model of Magara the lower frequency branch is called "in-phase mode" and the higher frequency branch "out-of-phase mode".

The uncoupled natural frequencies 𝜔 ac,r and 𝜔 st,r for Equation ( 2) cannot be determined directly from the measurements, because the modes with natural frequencies at f* = 0.93 and at f* = 1 are already strongly coupled. Therefore, the uncoupled natural frequencies are determined via solving Equation ( 2) with the experimental data from the 1 bar case. The resulting values for the uncoupled natural frequencies 𝜔 ac,r and 𝜔 st,r are used to calculate the natural frequencies at higher gas densities.

RESULTS AND DISCUSSION

In this part the results of the weakly and strongly coupled modes are presented. This paragraph starts with the results for the weakly coupled mode and thereafter the results for the strongly coupled acoustic mode are presented.

Weakly coupled mode

The mode that has been chosen for the investigation of the weakly coupled mode has a natural frequency of f* = 0.97 at ambient conditions. The response of the pressure sensor at ambient conditions is presented in Figure 5. A peak is visible at the non-dimensional frequency f* = 0.97. On the contrary, the response of the strain gage mounted on the disk (Figure 6) does not show a pronounced response at the non-dimensional frequency f* = 0.97. Thus, the coupling between the fluid and structure is very weak; furthermore, the excited mode is an acoustic mode. The pressure pattern in circumferential direction of the acoustic dominant mode is presented in Figure 7. The data is recorded with the rotatable pressure sensor as introduced in Barabas et al. [START_REF] Barabas | Identification of Coupled Natural Frequencies in a Rotor-Stator Test-Rig for Different Gas Properties[END_REF] and shows the envelope of the pressure at different circumferential positions for two opposite chronological phases (0 and π). The pressure pattern is a superposition of an acoustic mode with zero and five nodal diameters. Due to this superposition only two nodes can be identified where the envelopes of both phases cross each other. This is at a circumferential position of 90° and 120°. If the superpositioned mode with zero nodal diameters was not present, the pressure envelopes would additionally cross each other at 10°, 40°, and 170° circumferential position. The five nodes between 0° and 180° circumferential position can be determined from the time-sequence animation. The influence of the surrounding gas density on the natural frequency of the weakly coupled mode is presented in Figure 8. Only a marginal influence of the increasing density on the natural frequency can be read from the diagram. It has to be pointed out, that the influence of the speed of sound cannot be neglected in this case. Therefore the natural frequency plotted in the diagram is compensated and related to the speed of sound at ambient conditions. However, the surrounding gas density does not influence the natural frequency of the acoustic dominant mode. In Figure 9 the STFT results of the acoustic dominant mode in dependence of the surrounding gas pressure are plotted over the non-dimensional frequency. The curves are compensated with the speed of sound. A difference between the case at 1 bar and the other gas pressure cases is obvious. The response amplitude of the 1 bar case is significantly lower and the peak is flatter than the respective properties of the other cases. Compensated with the speed of sound.

The damping is analyzed with the peak-amplitude method and the results are plotted in Figure 10. The apparent behavior from the qualitative STFT curves is confirmed with the peakamplitude method. The damping decreases with increasing densities and approaches a minimum at a density around 11 kg/m 3 . A further increase of the gas density does not influence the damping of the weakly coupled acoustic mode. It is assumed, that the damping behavior of this particular mode is influenced by two effects that superpose each other. The first one is a generally decreasing damping ratio of weakly coupled acoustic mode with increasing gas density. The second one is the influence of an adjacent mode. This adjacent mode could be the already mentioned strongly coupled acoustic mode at a non-dimensional frequency of f* = 1. The difference of the natural frequencies of both modes is the smallest for the 1 bar case. It increases, because the natural frequencies of the strongly coupled modes diverge from each other when the gas pressure is increased. The influence of the adjacent mode is hardly visible for a gas density higher than approximately 11 kg/m 3 where the damping nearly remains at a constant minimum level.

Strongly coupled mode

The strongly coupled modes that have been chosen for this investigation have a non-dimensional natural frequency f* = 0.93 and f* = 1. Their response peaks can be well recognized as peaks in the pressure sensor data in Figure 5, though the results of the strain gage in Figure 6 give a clearer result. Both modes can be observed in the data of both different transducer types, which means a strong coupling with each other. Based on the strain gage data the excited mode at f* = 0.93 has a five times higher response amplitude as the excited mode at f* = 1. Therefore, the mode with the lower natural frequency is the structure dominant mode and the mode with the higher natural frequency the acoustic dominant. The next step is the identification of the excited modes before the natural frequencies of the strongly coupled modes can be investigated in dependence of the gas density. The envelopes of the pressure pattern in dependence of the circumferential position are presented for the structure dominant mode (f* = 0.93, Figure 11) and the acoustic dominant mode (f* = 1, Figure 12). The pressure envelopes are based on the data from the rotatable pressure sensor. Four nodes over 180° in circumferential direction can be observed, respectively. Hence, the number of nodal diameters is m = 4. The pressure signals of the transducers in the front and rear cavity are in phase opposition to each other, this means the number of axial nodes is l = 1 and a net pressure force takes effect on the disk. The number of nodal circles can be determined with n = 0, due to the inphase signal of the outer and inner pressure transducer. The variation of the surrounding gas pressure shifts both natural frequencies, the one of the acoustic dominant as well as the one of the structure dominant mode. This phenomenon has been shown by Magara et al. [START_REF] Magara | Natural Frequencies of Centrifugal Compressor Impellers for High Density Gas Applications[END_REF] and repeated in former experiments in the test rig at lower gas pressures (Barabas et al. [START_REF] Barabas | Identification of Coupled Natural Frequencies in a Rotor-Stator Test-Rig for Different Gas Properties[END_REF]). In Figure 13 the measured and the analytically determined natural frequencies (Magara [START_REF] Magara | Natural Frequencies of Centrifugal Compressor Impellers for High Density Gas Applications[END_REF]) are plotted over the gas density. They show qualitatively the same diverging behavior for the acoustic and structure dominant mode when the surrounding gas pressure is increased. Towards higher densities the measured acoustic dominant natural frequencies show a deviation from the calculated ones. On the first sight, the analytical and the measured natural frequencies of the structure dominant mode appear to fit well.

CONCLUSIONS AND OUTLOOK

In this paper the modal properties of a weakly coupled acoustic mode and a strongly coupled acoustic and structure mode are presented. The vicinity of the strongly coupled acoustic mode at low gas pressures is most likely the reason for the higher damping of the weakly coupled acoustic mode . For this particular weakly damped acoustic mode it may be concluded that the damping is influenced by a close mode but not by the increasing density of the fluid, whereas the natural frequency of it is neither influenced by the vicinity of a mode nor by the increasing density of the fluid

The increase of the surrounding gas pressure lets the natural frequencies of the strongly coupled acoustic and structure modes diverge from each other.

In the future, the detected effects within this paper should be analyzed more intensively. The findings that have been made for a particular mode are going to be analyzed for the general case. The knowledge about adjacent modes that influence each other's damping needs a more thorough investigation. Furthermore, the influence of disk rotation on the behavior of the modal properties will be investigated in future experimental campaigns. Those results could be applied in the design of radial compressors and if the aerodynamic damping behavior is understood better, the operational reliability of radial compressors could be improved.

Figure 1 .Figure 2 .

 12 Figure 1. A meridional sketch of a radial compressor stage with the side cavities.

Figure 3 .

 3 Figure 3. A picture of the test rig.

Figure 4 .

 4 Figure 4. Normal view of the rotor-stator test rig with the sensors' allocations. Within this paper results of the red marked sensors are presented. Labels: "P": pressure sensor, "G": strain gage, "LS": loudspeaker, "RC": rear cavity, "D": disk

Figure 5 .

 5 Figure 5. STFT result of the frequency sweep at ambient conditions (pressure sensor data, P147).

Figure 6 .

 6 Figure 6. STFT result of the frequency sweep at ambient conditions (strain gage data, G142).

Figure 7 .

 7 Figure 7. Results of the rotatable pressure sensor at f* = 0.97, the weakly coupled mode.

Figure 8 .

 8 Figure 8. The natural frequency of the weakly coupled mode in dependence of the surrounding gas density. Compensated with the speed of sound.

Figure 9 .

 9 Figure 9. The STFT of the weakly coupled mode recorded with a pressure sensor in dependence of the surrounding gas pressure.Compensated with the speed of sound.

Figure 10 .

 10 Figure 10. The damping of the weakly coupled mode in dependence of the surrounding gas density.

Figure 11 .

 11 Figure 11. Results of the rotatable pressure sensor at f* = 0.93, the structure dominant mode.

Figure 12 .

 12 Figure 12. Results of the rotatable pressure sensor at f* = 1, the acoustic dominant mode.

Figure 13 .

 13 Figure 13. Frequency shift due to pressure variation

  the Modal Properties of Weak and Strong Coupled Acoustic Modes -7 until ρ = 11 kg/m 3

Table 1 .

 1 Summary of the measurement instrumentation.

	Installed	Type of transducer	Quantity
	location
	Impeller	Strain gage	2
	Impeller	Piezoresistive pressure transducer	2
	Rear cavity	Piezoresistive pressure transducer	2
	Front cavity	Piezoelectric pressure transducer	2
	Front cavity (Rotatable transducer)	Piezoelectric pressure transducer	1

Table 2 .

 2 : Summary of the experimental boundary conditions.

	Pressure level [bar]	Density [kg/m 3 ]	Speed of sound [m/s]
	1	1.17	342.6
	5	5.92	343.2
	9	10.77	343.4
	13 17	15.68 20.47	344.2 344.8
	Evaluation Procedure
	The measured data of the sweep experiments are evaluated with a Short Time Fourier Transformation (STFT) in Matlab ®

ACKNOWLEDGMENTS

This project is funded by BMWi (Bundesministerium für Wirtschaft und Energie). The authors are grateful for the financial support under contract number 03ET2011E.