Toru Shigemitsu

Takumi Matsubara

Masahiro Sakaguchi

Junichiro Fukutomi

Influence of Back Shroud Shape on Performance and Internal Flow of Fluid Food Pump

Keywords: Centrifugal pump, Back shroud, Performance, Internal flow

Fluid machineries for fluid food have been used in wide variety of field i.e. transportation, filling, and improvement of quality of fluid food. Although, flow conditions of these are quite complicated because fluid food is different from water. Therefore, a design method based on the internal flow conditions is not conducted. In this research, a centrifugal pump having small number of blade was used to decrease shear loss and keep wide flow passage. Furthermore, a semi-open and two types of open impellers were prepared to investigate the influence of back shroud shapes on the performance and internal flow. The head of the pump using the open impeller was larger than that using the semi-open impeller because of the difference of the back shroud shape. In this paper, the internal flow conditions near the hub of the centrifugal pump having semi-open and open impellers were investigated by the numerical analysis results.

INTRODUCTION

It is said that world population will increase to 10 billion in several decades, so large quantities and stable supply of food are needed. Furthermore, a concern of food safety is also increasing nowadays. Fluid machines for fluid food have been used in wide variety of fields i.e. transportation, filling, and improvement of quality of fluid food. Flow conditions of those fluid machines are very complex and unclear because subject fluid of these fluid machines is fluid different from air and water. Ota et al. clarified influences of a tip clearance, a blade outlet angle and a blade number on the performance of a centrifugal pump using oil [START_REF] Ohta | Study on the Centrifugal Pump for High Viscosity Liquids (Effects of Clearance Ratio)[END_REF]- [START_REF] Ohta | Study on Centrifugal Pump for High-Viscosity Liquids (Effect of Impeller Blade Number on the Pump Performance)[END_REF] . On the other hand, internal flow condition of the pump using fluid food is not known, however, there are a few report related to the flow condition in a pipe, where fluid food flows [START_REF] Sandeep | Secondary Flow and Residence Time Distribution in Food Processing Holding Tubes with Bends[END_REF], [START_REF] Dutta | Velocity Distributions of Food Particle Suspensions in Holding Tube Flow: Experimental and Modeling Studies on Average Particle Velocities[END_REF] . Therefore, the design based on internal flow conditions is not conducted at present. The important issues requested for the fluid food pump are transportation performance, suction ability, flow rate control, quality maintenance, sanitary condition and so on. There are many kinds of the fluid food transportation pumps depending on the characteristics of fluid, i.e. a sanitary pump made from stainless steel, a screw pump and so on. A centrifugal pump was adopted as a fluid food pump in this research because the stable continuous transportation of fluid food was possible and the structure was simple compared to a positive displacement pump. The purposes of this research are to establish pump design method based on the internal flow condition of each fluid food and to make sure of the boundary line of fluid food, the centrifugal pump can apply to.

In this research, a centrifugal pump having small number of blade was used to decrease shear loss and keep a wide flow passage. Moreover, a semi-open impeller and two types of open impellers were adopted for this centrifugal pump to invetigate the back shroud shape influence on the performance and internal flow. Furthermore, the blade inlet and outlet angles were set as β1=β2=90° to facilitate the maintenance and cleaning in the pump. We investigated the effect of a tip clearance and viscosity on the performance and internal flow condition of the fluid food pump for low viscous fluid food in the previous research [START_REF] Kubo | Study on Performance and Internal Flow of Fluid Food Pump[END_REF], [START_REF] Kubo | The Influence of Tip clearance on Performance and Internal Flow Condition of Fluid Food Pump Using Low Viscous Fluid[END_REF] . Then, large vortex and flow separation were observed on a suction surface of a blade and strong unsteady flow could be generated in the pump [START_REF] Shigemitsu | Unsteady Flow Condition of Centrifugal Pump for Low Viscous Fluid Food[END_REF] . The flow conditions near the hub need to be investigated because the open impellers were adopted for some of the fluid food pump. Furthermore, it is important to clarify the influence of the back shroud shape on the performance and flow condition to improve the performance of the fluid food pump. Then, the flow condition near the hub of the centrifugal pump having the semi-open and open impellers were investigated by the numerical analysis results.

In this paper, the performances of the centrifugal pump having the semi-open and open impellers were clarifed by the experimental results. In addition to that, the flow conditions near the hub were shown based on the numerical analysis results.

EXPERIMENTAL APPARATUS AND METHOD

A specification of the pump, i.e. the design head, flow rate and specific speed was set based on the commercial fluid food pump in this research. Then, an original rotor was designed using a conventional design method of a centrifugal pump and a blade outlet angle was set as β2=22.5° which was recommended by the design method. After that, a radial blade(β2=90°) was adopted for the base model to pump fluid, of which viscosity was higher than that of water. Furthermore, the blade inlet angle was set as β1=90° to facilitate the maintenance and cleaning in the pump. Moreover, a semi-open and two types of open impellers were used to investigate the influence of the back shroud shape and small number of blade was set as z=4 in this centrifugal pump in order to suppress an increase of a disc frictional loss. The relations between the performance and a rotor having the special geometry, i.e. β1=β2=90°, semi-open and open impellers and small blade number, weren't known and it was difficult to estimate the performance of the pump with this special blade geometry. Therefore, design values of this pump were based on the original rotor having β2=22.5° and it wasn't the specific design values for the base model having β1=β2=90°. The design flow rate, head and rotational speed were Qd=65.3 l/min, Hd=2.5 m, and Nd=1650 min -1 based on the above assumptions. A specific speed was Ns=212 min -1 , m 3 /min, m and type number was K=0.52. Figure 1 and Table 1 show the geometry of the impellers and the primary dimensions of the test impeller. A two dimensional blade was used for the test impeller to facilitate the cleaning of the impeller and a tip clearance was c=0.5 mm. Diameters of the impellers were 100mm and four blades with single arc blade profile. The impellers were designed to keep the wide flow passage of solids, fibres and so on, and to reduce the friction and shear loss. Three type of impelllers with the different back shroud shape were prepared for this research under the same blade geomerty. TypeA was the semi-open impeller as shown in Fig. 1. TypeB and TypeC were the open impellers with the different back shroud shape. The back shroud areas of TypeB and TypeC were a half of the semi-open impeller TypeA. The back shroud was set near the pressure surface for TypeB as shown in Fig. 1 because the blade-to-blade main flow was concentraed on the pressure surface of the blade for TypeA [8]. On the other hand, the back shroud was set both pressure and suction surfaces of the blade for TypeC as shwon in Fig. 1. The pump used for this study had a single suction and single volute casing. A suction and discharge diameters were Din=50mm and Dout=40mm respectively and a cross-sectional shape of a volute was rectangle and clearance at the tongue of the volute casing was 3.7mm.

Figure 2 shows the schematic diagram of the experimental apparatus. For the pressure performance evaluation, the static head differences on the wall between 2Din upstream and 2Dout downstream of the rotor were measured by pressure transducers. Then, the total pump head were evaluated by adding the dynamic head difference of the sectional averaged axial velocity to the corresponding measured static head difference. The rotor was driven by a motor. The flow rate Q was obtained by an electromagnetic flow meter installed far downstream of the pump and torque was measured by a torque meter. Then the shaft power was calculated by the torque and the rotational speed measured by a rotational speed sensor.

The shaft power was evaluated by the torque eliminating the torque using a disc without the impeller in this performance test. Then, the hydraulic efficiency of the pump η was calculated as the ratio of the water power to the shaft power. Glycerol solution(40% volume density) was used in the experiment. The glycerol solution used in this research corresponded to a comparatively low viscous fluid food such as beer, sake, saccharose solutions, and soy sauce on the point of view of the viscosity range.

NUMERICAL ANALYSIS CONDITION

A numerical flow analysis was conducted to investigate the internal flow condition in detail. In the numerical analysis, the commercial software ANSYS-CFX was used and the numerical analysis was conducted with a numerical model which was the same with the test section of the experiment under the three dimensional unsteady condition. Fluid was assumed to be incompressible and isothermal water and the equation of the mass flow conservation and Reynolds Averaged Navier-Stokes equations were solved by the finite volume method. The standard wall function was utilized near the wall and SST model was used as the turbulence model. The numerical analysis was conducted at a design flow rate Qd and kinematic viscosity and density of the fluid were changed from water to simulate the glycerol solution used for the experiment. Figure 3 show the numerical grids used for the numerical analysis. An inlet boundary was 5Din upstream of the test section and an outlet was 5Dout downstream of it. A tip clearance was set c=0.5mm as the same with the experimental apparatus. The constant velocity and constant pressure were given as the boundary conditions at the inlet and outlet respectively. Mesh numbers of the numerical domain was about 6.0×10 6 grid points. The coupling between the rotor and casing was accomplished by the Transient Rotor Stator. The time step number per one rotor rotation was 180 and the time step was t=2.02×10 -4 s. The data of one rotor rotation were obtained after 6 rotor rotations in unsteady numerical analysis.

RESULTS AND DISCUSSIONS

Performance curves

Figure 4 shows the pump performances of TypeA, B and C using the 40vol% glycerol solution obtained by the experiment. The experiments were conducted in wide flow rate range from the shut-off to large flow rates. The horizontal axis shows a flow rate. Figures 4(a),(b) and (c) show the head, shaft power and efficiency respectively. Focused on the head curve of TypeA, the head at the design flow rate was 3.55m, which was higher than the design head Hd=2.5m. The head curve had flat shape from 0.5Qd -1.0Qd and a negative slope of head curve was observed in large flow rate Booster pump •

Q/Q d [%] Q/Q d [%]
H [m] Q/Q d [%] 0 1 2 3 4 5 H [m] Q/Q d [%]
LTypeB [m] ■ LTypeC [m] ▲ ηTypeA [m] • ηTypeB [m] ■ ηTypeC [m]
region. The head of TypeB and C were larger than that of TypeA because the circumferential velocity increased by the back shroud edge. The shaft power increased with the increase of flow rate and showed the minimum value at shut-off flow rate for all types. On the other hand, the shaft powers of TypeB and C were larger than that of TypeA in all flow rates. The maximum efficiency ηmax=56.5% was obtained at 1.4Qd for TypeA and the maximum efficiency of TypeB and TypeC were ηmax=49.2 and 49.7% respectively at 1.8Qd. It was considered that the efficiency of open impellers were lower than that of semi-open impeller because of the flow from the clearance of the back shroud. The performances of TypeB and C with the different back shroud shape were almost the same and the influence of the back shroud shape on the performance was small. The pump performance data obtained by the numerical analysis using fluid simulating the 40vol% glycerol solution were compared with the experimental results and its results are shown in Fig. 5. The head, shaft power and efficiency for TypeA obtained by the numerical analysis accorded well with the experimental results at the design flow rate. On the other hand, the head for TypeB and C obtained by the numerical analysis were larger than that obtained by the experimental results and the shaft power of the numerical analysis for TypeB and C were almost the same with the experimental results. Then, the efficiency of the numerical analysis for TypeB and C were larger than that of the experimental results. However, the difference between the experimental and numerical results was not so large, therefore the internal flow condition was investigated by the numerical analysis results. The length of the velocity vectors is constant, so it shows the flow direction of the relative velocity and the contours show the relative velocity magnitude. θr is a rotation angle of a leading edge of the blade from the axial horizontal line. The rotational direction of the rotor is a counter crock-wise direction. The rotation angles of Figs.6 were θr=0°. The b and B2 show an axial length from the hub and an outlet flow passage width respectively and its definitions are illustrated in Fig. 7. It was found in Figs.6 that the fluid flowed out mainly from the pressure surface region of the blades, although separation regions were observed on the suction surface of the blade. There were vortexes in the separation region near the suction surface of the blade. The fluid near the trailing edge of the blade flowed out along the suction surface of the blade. These flow conditions near the hub were similar to each impeller TypeA, B and C, however the main flow condition of open impeller TypeB and C showed the slight difference from that of TypeA. Therefore, the influence of the back shroud shape difference on the axial cross sectional flow condition was small.

Internal flow conditions

Figures 8(a)-(c) show relative velocity vectors on the cylindrical plane at the radial position r=45mm. The flow rate and rotation angle were the same with those of Figs. 6. The rotational direction of the impeller was right direction of the paper. It could be found in Figs. 8(a)-(c) that the flow conditions near the suction surface(SS) of the blade were complex

H [m] Q/Q d [%] L[kW]
η exp. [%] H cal. [m] L cal. [kW] η cal. [%] H exp. [m] L exp. [kW] η exp. [%] H cal. [m] L cal. [kW] η cal. [%] H exp. [m] L exp. [kW] η exp. [%] H cal. [m] L cal. [kW] η cal. [%]
because of the separation near the SS of the blade. On the other hand, the flow conditions near the pressure surface(PS) of the blade for TypeA was more uniform than that for TypeB and C. The interaction between the main flow and flow behind the back shroud was generated for TypeB and C. The shear loss is an important factor for the fluid food pump to evaluate the quality of the fluid food. Then, the shear rate distributions of the open impeller TypeB and C were investigated by the numerical analysis results. Figures 9(a 1. The head of TypeB and C were larger than that of TypeA because the circumferential velocity increased by the back shroud edge. The shaft power increased with the increase of flow rate and showed the minimum value at shut-off flow rate for all types. On the other hand, the shaft powers of TypeB and C were larger than that of TypeA in all flow rates.

2. The performances of TypeB and C with the different back shroud shape were almost the same and the influence of the back shroud shape on the performance was small.

Figure. 1

 1 Figure. 1 Test pump rotors

Figure. 3 Figure. 2

 32 Figure. 3 Numerical grids

Figure. 4

 4 Figure.4 Performance curves obtained by experiment

Figures 6 (

 6 Figures 6(a)-(c) show axial cross sectional relative velocity vectors over the contours of the relative velocity in the impeller near the hub(b/B=0.18). The flow rate was design flow rate Qd.The length of the velocity vectors is constant, so it shows the flow direction of the relative velocity and the contours show the relative velocity magnitude. θr is a rotation angle of a leading edge of the blade from the axial horizontal line. The rotational direction of the rotor is a counter crock-wise direction. The rotation angles of Figs.6 were θr=0°. The b and B2 show an axial length from the hub and an outlet flow passage width respectively and its definitions are illustrated in Fig.7. It was found in Figs.6 that the fluid flowed out mainly from the pressure surface region of the blades, although separation regions were observed on the suction surface of the blade. There were vortexes in the separation region near the suction surface of the blade. The fluid near the trailing edge of the blade flowed out along the suction surface of the blade. These flow conditions near the hub were similar to each impeller TypeA, B and C, however the main flow condition of open impeller TypeB and C showed the slight difference from that of TypeA. Therefore, the influence of the back shroud shape difference on the axial cross sectional flow condition was small.Figures8(a)-(c) show relative velocity vectors on the cylindrical plane at the radial position r=45mm. The flow rate and rotation angle were the same with those of Figs.6. The rotational direction of the impeller was right direction of the paper. It could be found in Figs.8(a)-(c) that the flow conditions near the suction surface(SS) of the blade were complex

Figure. 5

 5 Figure.5 Comparison of performance curves

  ) and (b) show axial cross sectional shear rate distributions in the impeller near the hub(b/B=0.18). The flow rate, rotation angle and rotational direction of the impeller were the same with those of Figs.6. The shear rate was high at the leading edge, trailing edge of the blade and volute tongue for both TypeB and C. The shear rates at the edge of the back shroud were relatively large for both TypeB and C, however, the tendency of the shear rate distributions between TypeB and C was similar. Therefore, it is necessary to consider making a round at the edge of the back shroud to reduce the shear loss of the open impeller.

Figure. 8

 8 Figure. 8 Relative velocity vectors on cylindrical plane (Qd, r=45mm)

Figure. 7 Figure. 6

 76 Figure. 7 Definition of b and B (a) TypeA

3 .

 3 The flow conditions near the pressure surface of the blade for TypeA was more uniform than that for TypeB and C. The interaction between the main flow and flow behind the back shroud was generated for TypeB and C. 4. The shear rates at the edge of the back shroud were relatively large for both TypeB and C. Therefore, it is necessary to consider making a round at the edge of the back shroud to reduce the shear loss of the open impeller. (b) TypeC

Figure. 9

 9 Figure. 9 Shear rate distributions (Qd, b/B=0.18)

Table . 1

 . Primary dimensions of impeller


Satoshi Kawasaki
email: kawasaki.satoshi@jaxa.jp

Masaharu Uchiumi

Takashi Shimura

Yuka Iga

One-dimensional Analysis Method for Cavitation Instabilities of a Rotating Machinery

Keywords:

Rotating cavitation is an important problem, which makes it difficult to design reliable rotating machines. In the present study, a simple analysis method that tried to evaluate the cavitation instabilities of a rotating machinery by using one-dimensional system analysis software was attempted. In this method, cavitation compliance and mass flow gain factor are distributed in each flow path of the inducer.

Analysis results show that cavitation instabilities, including rotating phenomena, exist. With the evolved analysis model, effects of various parameters on the eigenvalues of the system were investigated. Analysis results agreed with inducer test results qualitatively. Furthermore, by the analysis considered whirl motion of the rotor, effects of it on cavitation instabilities were investigated.

.

INTRODUCTION

Rotating cavitation was identified about 40 years ago [START_REF] Kamijo | An Experimental Investigation of Cavitating Inducer Instability[END_REF]. It is one of the most important problems in high-speed rocket turbopumps. Theoretical and numerical analyses (two-and three-dimensional) of rotating cavitation have been performed to elucidate the instability mechanism [START_REF] Tsujimoto | A Theoretical Analysis of Rotating Cavitation in Inducers[END_REF] [START_REF] Iga | Numerical Analysis of Cavitation Instabilities Arising in the Three-Blade Cascade[END_REF]. Computational fluid dynamics (CFD) is very useful, but too many resources are needed to analyze whole systems including suction lines, discharge lines and inducers parametrically.

Therefore, establishment of a one-dimensional analysis method to analyze fluid systems, including dynamic characteristics of cavitating inducer, was attempted.

A method to analyze pump dynamic characteristics by cavitation compliance, mass-flow gain factor and dynamic gain has been reported [START_REF] Brennen | The Dynamic Transfer Function for a Cavitating Inducer[END_REF] [START_REF] Shimura | A Study on Dynamic Characteristics of Liquid Oxygen Pumps for Rocket Engines (1st Report[END_REF]. This analysis method has been used to analyze one-dimensional phenomena such as surge phenomenon in the past. For example, POGO, which is an auto-oscillation where the thrust oscillation of rocket propulsion system including turbopumps is coupled with the natural frequency of the rocket structure, has been analyzed using this method [START_REF] Rubin | Prevention of Coupled Structure-Propulsion Instability (POGO)[END_REF] [START_REF] Ujino | POGO Prevention of H-2 Launch Vehicle[END_REF]. In the present study, this one-dimensional analysis method was further developed in order to analyze rotating cavitation too, using one-dimensional multi-domain system analysis software AMESim [START_REF] Nakazawa | System Modeling & Simulation[END_REF]. This software has been used to analyze system stability of a self-balancing type axial-thrust balancing-system in the past [START_REF] Shimura | Stability of an Axial Thrust Self-Balancing System[END_REF]. In this software, physical systems are described by connecting sub-models which transfer energy to each other. Cavitation compliance and mass-flow gain factor were distributed to each flow path of an inducer, and evaluation of the possibility of analyzing rotating phenomena such as rotating cavitation by this one-dimensional analysis method was attempted. Rotating cavitation is closely related to whirling motion of the rotor assembly and a study of non-linear modeling of excitation force and rotor whirling has been reported [START_REF] Uchiumi | Excitation Force Modeling of Rotating Cavitation and Rotor Vibration Analysis of a Turbopump Rotor, 16th Cavitation Symposium[END_REF]. Whirling motion was also accounted for by variable flow area orifices which connect with the next flow path in the present analysis. The variation of leakage flow rate through the tip clearance of each blade is simulated by the motion of the variable flow area orifices. With the improved analysis model, effects of various parameters on the eigenvalues of the system were investigated. Analysis results showed that rotating phenomena existed and agreed with inducer test results qualitatively. This analysis method is considered to be able to contribute to the development of a dynamic design method for turbopumps [START_REF] Uchiumi | Integrated Design Method of a Rocket Engine Turbopump Sub-system for Suppressing Rotor Lateral Vibration[END_REF] and to be applicable to the analysis of other rotating machineries.

METHODS 1.1 Modeling of a three-bladed inducer

Figure 1 shows a modeling image of a three-bladed inducer. It is assumed that rotating cavitation is caused by the characteristics of fluid motion in the flow paths divided by inducer blades. Hence, the three-bladed inducer is represented by three simplified pipes and cavity oscillation is considered to occur inside the each pipe. Leakage flows from tip clearances are described by variable flow area orifices connecting the middle of the path and the inlet of the next path. The areas of the orifices are changed harmonically and the oscillation phase of each orifice is given according to the number of blades. The steady flow rate of the leakage flow is given by the mean flow area of the variable orifice and dynamic flow rate caused by rotor whirl is given by the amplitude and phase of flow area variation. The flow rate of the next path is changed by the leakage flow rate. Figure 2 shows inducer tip clearance variation caused by whirling motion of a three-bladed inducer. In the present model, moving coordinate rotating with the inducer is applied. The number of each blade is assigned in series in the direction of counter rotation, as shown in Fig. 2. The inducer tip clearance varies in the sequence of path 1 first, path 3 second and path 2 third in the case of super-synchronous whirling motion.

Figure 3 shows a schematic diagram of the whole analysis model used in the present study. The model represents the blowdown type test loop. The test inducer and the run tank are connected by the inlet pipe model considering the inertia. The inducer and the catch tank are connectted by the discharge pipe model considering the inertia and the orifice for flow rate control is located between the discharge pipe and the catch tank. In the inducer model, each flow path is divided into a front part and a rear part. The inducer delivery head is divided into two and distributed to the front part and the rear part, respectively. In this study, the head was equally divided in half tentatively. Furthermore, the inertia of each flow path was divided at the location that occurs cavity oscillation and distributed.

In this model, the cavitation compliance and mass flow gain factor are distributed to the three paths and calculated by the inlet pressure of the each path. The definition of cavitation compliance and that of mass flow gain factor are shown in Fig. 3. Cavitation compliance is defined as the cavity volume change rate to the inlet pressure change. Mass flow gain is determined as the cavity volume change rate to the inlet flow change. The flow difference between upstream and downstream of the cavity is obtained by time derivation of cavity volume. In this model, inducer is represented by the parallel paths so that the cavitation compliance of whole inducer is represented by summation of those three paths and the mass flow gain factor is equal to each three paths in In the analysis model using the one-dimensional multi-domain system analysis software, the fluid system is described by connecting fluid sub-models such as tanks, variable orifices, pipes, etc. Values of cavitation number, head coefficient, etc. are calculated by function sub-models and transmitted to each other through signal lines.

Analysis inducer and conditions

The model analyzed was a three-bladed inducer for a 100-ton class rocket engine hydrogen turbopump. The rotational speed is 42500/60=708 Hz and the mass flow rate is 37.3 kg/s at nominal operating point. Nominal tip clearance is about 0.5% of the inducer tip radius. Unless otherwise mentioned, tip clearance at each blade was constant at a nominal value.

The cavitation compliance and mass flow gain factor were calculated by the approximate expressions, which were obtained by nondimensionalized test results in the past [START_REF] Shimura | Geometry Effects in the Dynamic Response of Cavitating LE-7 Liquid Oxygen Pump[END_REF]. This test had been carried out by generating sinusoidal perturbation in the pump flow. Figure 4 shows the test results and the approximate expressions on the CASE 1 inducer. In the CASE 1 inducer, super-synchronous rotating cavitation was clearly observed. Dimensionalized values were given by substituting operating rotational speed, cavitation number, etc., in the process of calculation. Pump dynamic gain was assumed to be zero in the present analysis because rotating cavitation occurs in the region of relatively large cavitation number. The fluid system analysis was conducted with single-phase flow. The working fluid was liquid hydrogen and the fluid properties (density, viscosity, compressibility, etc.) are given by polynomial approximation.

RESULTS AND DISCUSSION

Modal analyses of cavitation instabilities

In order to confirm the existence of oscillation modes caused by cavitation, frequency domain analysis of the system was carried out. Figure 5 shows the eigenvalues of the system for various cavitation numbers σ on the complex plane. When the eigenvalue of the oscillation mode has a positive real part, this mode is unstable. The ellipse line represents the iso-frequency line and the radial line represents the iso-damping ratio line. In some oscillation modes (MODE-A, MODE-B and MODE-C), the natural frequencies and the damping ratios changed with the cavitation number. If the eigenvalue of oscillation mode is not influenced by cavitation, the natural frequency and the damping ratio are constant with change of cavitation number. Therefore, it is considered that these modes are related to cavitation phenomena.

Figure 6 shows comparison with the natural frequency and damping ratio of MODE-A, MODE-B and MODE-C. In the case of MODE-A and MODE-B, the natural frequencies and the damping ratios decreased with decreasing cavitation number. Meanwhile, in the case of MODE-C, natural frequency also decreased with decreasing cavitation number. However, the damping ratio was not much changed with decreasing cavitation number.

To investigate these oscillation modes caused by cavitation, modal analysis was carried out. The results of modal analysis for MODE-A, MODE-B and MODE-C are shown in Fig. 7

MODE-A

The flow rate fluctuation of MODE-A propagated in the order of path 1, path 3 and path 2 as shown in Fig. 7(a).

In the present analysis model, moving coordinate system rotating with the inducer was applied, and the inducer rotated in the order of path 1, path 3 and path 2 as shown in Fig. 1. Since the fluctuation of the flow rate propagated in the same direction as that of inducer rotation, it seems to propagate at higher speed than inducer rotation speed in the fixed coordinate system. In this case, the propagating frequency in the fixed coordinate system changed from 1.62 to 1.20 times the rotation frequency with decreasing cavitation number σ from 0.06 to 0.02. In the reference paper [START_REF] Tsujimoto | Observation of Oscillating Cavitation of an Inducer[END_REF], the propagating frequency had changed from 1.25 to 1.10 times the rotation frequency with decreasing cavitation number σ from 0.06 to 0.045. Although there is quantitative difference of the propagation frequency, the tendency of the frequency change to the variation of cavitation number qualitatively agrees with the phenomenon of super-synchronous rotating cavitation which had been observed in the inducer test [13][14]. The damping ratio of MODE-A became negative with lower cavitation number as seen in Fig. 6 Propagation of the flow rate fluctuation was similar to that of MODE-A. However, the direction of propagation was opposite to that of MODE-A. Therefore, the flow rate fluctuation seems to propagate at lower speed than the inducer rotation speed in the fixed coordinate system. In this case, the frequency of the flow rate fluctuation in the fixed coordinate system changed from 0.38 to 0.80 times the rotating frequency with decreasing cavitation number σ from 0.06 to 0.02. These results suggest that MODE-B represented the mode of sub-synchronous rotating cavitation (SB-RC), which had also been observed in the inducer test [START_REF] Tsujimoto | Observation of Oscillating Cavitation of an Inducer[END_REF] [START_REF] Goirand | Vinci Fuel Turbopump Testing Activity Achievements, 4th International Conference on Launcher Technology "Space Launcher Liquid Propulsion[END_REF]. Furthermore, the result of theoretical analysis [START_REF] Tsujimoto | A Theoretical Analysis of Rotating Cavitation in Inducers[END_REF] indicated that both SP-RC and SB-RC modes exist, in agreement with the present result.

Table 1 shows the damping ratios of MODE-A and MODE-B in cases with or without tip clearance. The damping ratio of MODE-B was slightly smaller than that of MODE-A in the case with tip clearance. In the case without tip clearance, which means no leakage flow in inducer, the damping ratios of both modes were the same and larger negative than those in the case with tip clearances. This result suggests that leakage flow through the tip clearance affects the difference of damping ratio between the SP-RC mode and the SB-RC mode and that the leakage flow enables both modes to improve the stability.

MODE-C

Effects of cavitation compliance and mass flow gain factor on eigenvalues

Effect of cavitation compliance

Figure 8 shows the variations of eigenvalues due to the cavitation compliance at cavitation number σ =0.04. Cavitation compliance changed from 1/4 to 4 times the value obtained by approximate expression. Natural frequencies of the SP-RC mode, the SB-RC mode and the CS mode decreased with increasing cavitation compliance. By contrast, damping ratios increased with increasing cavitation compliance.

It has been indicated that the increase of cavitation compliance has a stabilizing effect on low frequency oscillation in the system as shown by another stability analysis for the pump loop system [START_REF] Shimura | A Study on Dynamic Characteristics of Liquid Oxygen Pumps for Rocket Engines (1st Report[END_REF]. This result agrees with the effect of cavitation compliance for the CS mode by the present analysis.

Effect of mass flow gain factor

Figure 9 shows the variations of eigenvalues due to the mass flow gain factor at cavitation number σ =0.04. The mass flow gain factor changed from 1/4 to 4 times the value obtained by approximate expression. Unlike the effect on cavitation compliance, natural frequencies of the SP-RC mode, the SB-RC mode and the CS mode were almost constant with increasing mass flow gain factor and damping ratios decreased with increasing mass flow gain Enlargement of the housing diameter at the inducer inlet has been known to be one way to suppress rotating cavitation. It has been reported test results showed that the mass flow gain factor decreased due to this treatment [START_REF] Shimura | Geometry Effects in the Dynamic Response of Cavitating LE-7 Liquid Oxygen Pump[END_REF]. This result is agreement with the effect of mass flow gain factor for rotating cavitation mode by the present analysis.

Step response by time domain analysis

In order to simulate the instability phenomenon caused by the oscillation modes of the inducer, especially rotating cavitation, time domain analysis was carried out in the present model. Table 2 shows eigenvalues of the oscillation modes under the analysis conditions (σ =0.03, Cb/Cb0=3, Mb/Mb0=1), where the damping ratio of CS mode is positive but these of SP-RC mode and SB-RC mode are both negative. Consequently, it is considered that instability phenomenon caused by the rotating cavitation modes appears more strongly than that by cavitation surge mode. The step change of pressure in the run tank was given and then the response of the system was observed. The upper graph in Fig. 10(a) shows the time histories of the tank pressure, the inlet pressure of inducer and the pressures of three paths. The lower graph in Fig. 10(a) shows the frequency analysis results of the fluctuations of these pressures by FFT.

After the step change of pressure in run tank, low frequency fluctuations of the inlet pressure of the inducer and the pressures of the three paths occurred with the same phase. The frequencies of the pressure fluctuations were slightly less than 30 Hz, which approximately corresponded to the natural frequency of the CS mode. Hence, the oscillation phenomenon of these low frequency fluctuations is assumed to be the CS mode. According to the positive damping ratio of the CS mode (see Table 2), the amplitude of the fluctuation gradually decreased with time. After the fluctuation of the CS mode Since these fluctuations propagated in the order of path 1, path 3 and path 2, the oscillation phenomenon of these high frequency fluctuations is assumed to be the SP-RC mode. According to the negative damping ratio of the SB-RC mode (see Table 2), amplitude of the fluctuation was exponentially increased with time, and then the fluctuations diverged.

SB-RC mode was not observed in this simulation. However, if the step size or the step direction of pressure given in run tank was changed, SP-RC mode appeared in some cases. As seen in Table 2, the difference of the damping ratio between the SP-RC mode and the SB-RC mode is not so large, so that it is considered that the determination of the fluctuation mode of rotation cavitation was not affected by this difference of the damping ratio.

Response to the whirling motion of the rotor

In actual rotating machinery, whirling motion of the rotor often occurs. In order to investigate the effect of such whirling motion of the rotor on cavitation instabilities, the simulation accounted for the whirling motion of rotor was carried out by time domain analysis. The method used to simulate the whirling motion of the rotor was presented in section 1.1. Figure 11(a) and (b) show the time histories of the pressure of each of the three paths and FFT results with forward whirl and backward whirl, respectively. The whirl amplitude was 2% of the nominal tip clearance and the cavitation number, the cavitation compliance and the mass-flow gain factor were same as these of step response analysis in section 2.3. The whirl frequency was 108 Hz, as the same as the natural frequency of the SP-RC mode and the SB-RC mode. When the forward whirling motion was inputted, pressure fluctuation occurred and grew. The frequency of the fluctuation was a little over 100 Hz and the fluctuation propagated in the order of path 1, path 3 and path 2. Therefore, the fluctuation phenomena with forward whirl is assumed to be the SP-RC mode. On the other hand, when the backward whirling motion was inputted, pressure fluctuation with frequency of 108 Hz also occurred and grew. However, the fluctuation propagated in the order of path 1, path 2 and path 3. Therefore, the fluctuation phenomenon with backward whirl is assumed to be the SB-RC mode.

Furthermore, in order to investigate the effect of whirl frequency and duration time of whirling motion, time domain analysis was carried out, whirling motion with a frequency of 50 Hz from 0 to 0.01 sec being inputted. After 0.01 sec, whirling motion was stopped and tip clearance at each blade was kept constant at the nominal value. Figure 12 (a) and (b) show the time histories of pressure of each of the three paths and FFT results with forward whirl and backward whirl, respectively. With both forward and backward whirl, pressure fluctuations with a frequency of 105 Hz occurred and the propagating direction of the pressure fluctuation was the same as that of the whirl direction. Therefore, the fluctuation phenomena caused by forward whirl and backward whirl are assumed to be the SP-RC mode and the SB-RC mode, respectively. In both cases, after the whirling motion was stopped, pressure fluctuations grew and finally diverged. Although the damping ratio of the SP-RC mode was less negative than that of the SB-RC mode, fluctuation of SP-RC mode was not replaced by that of the SB-RC mode.

The results indicate the whirling motion of rotor affects the determination of the propagating direction of the rotating cavitation mode.

CONCLUSIONS

A simple one-dimensional analysis method to investigate the dynamics of the fluid system of the rotating machinery was established. Applying this method to a rocket turbopump inducer, the existence of cavitation instabilities such as rotating cavitation and cavitation surge were confirmed and the following conclusions were obtained.

(1) The method to distribute cavitation compliance and mass flow gain factor to the paths between the blades enables the simulation of the cavitation instability of rotating phenomena, such as rotating cavitation, even by using the one-dimensional analysis method. (2) The effects of whirling motion of the rotor on cavitation instabilities were investigated. The results suggested that whirling motion of the rotor affects the propagating direction of fluctuations caused by rotating cavitation mode. (3) The effects of cavitation compliance and mass flow gain factor on the eigenvalues of the system were investigated. The results, such as the stabilizing effect on cavitation surge by increase of cavitation compliance and suppressing effect on rotating cavitation by decrease of mass flow gain factor, agree with the results obtained by past experimental and analytical studies qualitatively. (4) It is considered that rotor vibration related to the cavitation instability of rotating phenomena can be analyzed by coupling with rotor motion analysis and the present analysis. (5) The present calculation is completed within 1 minute per case by a typical personal computer. This method of analysis is quite useful for carrying out many parametric studies.

As the analysis results, it is confirmed that the cavitation compliance and mass flow gain factor are affect to the cavitation instabilities. However, it is not possible to calculate the cavitation compliance and mass flow gain factor by using the present analysis. Therefore, it is very important to investigate and model the characteristics of these factors such as the effect of flow rate and the frequency characteristics. By using these characteristics models, it will be possible to simulate the cavitation instabilities accurately and helpful to understand the mechanism of the cavitation instabilities.

Figure 3

 3 Figure 3 Schematic diagram of the analysis model for inducer test loop

Figure 1 Figure 2

 12 Figure 1 Modeling image of three-bladed inducer

Figure 4 Figure 5

 45 Figure 4 Approximate formulas of cavitation compliance and mass flow gain factor used by the present analysis

  (b). Hence, the unstable fluctuation is possibly caused when the inlet pressure of the inducer decreases. The results suggest that MODE-A represents the mode of super-synchronous rotating cavitation (SP-RC).

Figure 7 Figure 6 Figure 7

 767 Figure 7 Flow rate fluctuations of three-bladed inducer

Figure 7 (

 7 Figure 7(c) shows the flow rate fluctuation of MODE-C. In the lower graph, all lines of amplitude overlap. The flow rate fluctuation in each path occurs in the same phase with the same amplitude. The result suggests that MODE-C represents the mode of cavitation surge (CS).

Table 1 Figure 8

 18 Figure 8 Effects of cavitation compliance on eigenvalues (σ =0.04)

Table 2 Figure 10

 210 Figure10Step response of the system (σ =0.03, Cb/Cb0=3, Mb/Mb0=1)

Figure 9

 9 Figure 9 Effects of mass flow gain factor on eigenvalues (σ =0.04)

  Figure 10(b) shows the high frequency fluctuations in detail. The pressure of each of the three paths fluctuated with the different phases. The frequencies of the pressure fluctuations were about 110 Hz, which approximately corresponded with the natural frequency of rotating cavitation modes (SP-RC or SB-RC).

Figure 11

 11 Figure 11 Response to the 108 Hz whirling motion of rotor: 105 Hz is same as the eigenvalue of rotating cavitation mode (σ =0.03, Cb/Cb0=3, Mb/Mb0=1)

Figure 12 50

 12 Figure 12 Response to the 50 Hz whirling motion of rotor (σ =0.03, Cb/Cb0=3, Mb/Mb0=1)

NOMENCLATURE

  Ai : inducer inlet area (m 2 ) Cb : cavitation compliance in each path (m4 s 2 /kg) Cb* : Nondimensional cavitation compliance in each path, Cb* =Cb / (2πDt Ai / nb ρ Ut 2 ) Dt : inducer tip diameter (m) fn : natural frequency (Hz) Mb : mass flow gain factor in each path (s) Mb* : Nondimensional mass flow gain factor in each path, Mb* =Mb / (πDt / nb Ut) nb : number of inducer blades (number of inducer paths) Pin : pressure at inducer path inlet (Pa) Pv : vapor pressure (Pa) Qin : flow rate at inducer path inlet (m/s) Qout : flow rate at inducer path outlet (m/s) t : time (s) Ut : tip speed of inducer (m/s) Vc : cavity volume (m 3 ) δ : difference per time interval ∆Pf : pressure rise in front part by inducer head (Pa) ∆Pr : pressure rise in rear part by inducer head (Pa) ζ : damping ratio ρ : density (kg/m 3 ) σ : cavitation number, σ = 2(Pin -Pv)/( ρUt 2 ) SUBSCRIPTS 0 : value obtained by approximate expression in Fig. 4 total : total value of whole inducer


Mikhail V Timoshevskiy

Konstantin S Pervunin
email: pervunin@itp.nsc.ru

Dmitriy M Markovich

Kemal Hanjalić

Control of cavitating flow over a 2D hydrofoil by active mass injection through a slot channel in its surface

Keywords: Cavitation, partial cavities, flow control, active mass injection, high-speed imaging, PIV, guide vane modell

We studied cavitating flow over the suction side of a symmetric 2D foil -a scaled-down model of high-pressure hydroturbine guide vanes (GV) -in different cavitation regimes at several attack angles. High-speed imaging was used to analyze spatial patterns and time dynamics of the gas-vapor cavities, as well as for evaluating the characteristic integral parameters. A hydroacoustic pressure transducer was employed to register time-spectra of pressure fluctuations behind the hydrofoil and, thereby, determine dedicated frequencies of unsteady regimes. A PIV technique was applied to measure the velocity fields and its fluctuations, which were compared for the free and forced flow conditions. The active flow control was implemented by means of a continuous liquid supply with different flow rates through a slot channel located in the GV surface at the distaence of 60% of the chord length from the foil leading edge. It was found that the active mass injection does not influence the primary flow upstream of the slot channel position absolutely. At small angles of incidence, the injection flow at velocities in the range between zero to 0.76 of the mean bulk velocity was observed not to practically influence the distributions of turbulent characteristics so that the global difference is only between the free and forced flow conditions. For cavitation-free and cavitation inception cases, the active mass injection was shown to make the flow turbulence structure more developed and the wake past the GV section more intense. However, the active flow control system considered also allows a favorable and efficient flow manipulation, especially at the regimes with developed gas-vapor cavities. Moreover, the active flow management makes it possible to reduce substantially the amplitude or totally suppress the periodic cavity length oscillations and pressure pulsations associated with them.

INTRODUCTION

Unsteady cavitation phenomena occurring within the ducts of real hydropower systems (especially, on guide vanes, runner blades and in draft tubes of hydroturbines) are known to be one of the main sources of instabilities that cause significant vibrations, overloads and, consequently, fatigue damages. These might lead to unstable operating conditions, reduced efficiency and safety problems. Therefore, in full-scale conditions all types of instabilities are typically suppressed as much as possible. Apart from omnipresent erosion, cavitation is not regarded as dangerous as long as its dynamics is stable. Preventing or at least diminishing and controlling cavitation and other causes of instabilities in full-scale hydropower systems is nowadays a vital and relevant problem.

Presently, two main classes of partial cavity instabilities are distinguished [START_REF] Callenaere | The cavitation instability induced by the development of a re-entrant jet[END_REF]: intrinsic and system. If instability originates in the cavity itself, it is of intrinsic type. If it is caused by interaction of the cavity with other cavities or elements of the hydraulic system, especially inlet and outlet ducts, it is system dependent. For example, cloud cavitation occurs because of the development of an intrinsic instability -a re-entrant jet underneath the sheet cavity (e.g., see [START_REF] Kubota | Unsteady structure measurement of cloud cavitation on a foil section using conditional sampling technique[END_REF][START_REF] Kawanami | Mechanism and control of cloud cavitation[END_REF][START_REF] Pham | Investigation of unsteady sheet cavitation and cloud cavitation mechanisms[END_REF]). The thickness of the re-entrant jet and its velocity are determined by an adverse pressure gradient. If the pressure gradient is strong enough to force the reentrant jet to impinge upon the cavity detachment region (i.e., close to the cavity leading edge), the attached cavity interface breaks up over the whole foil span and a cloud cavity appears, which is then convected downstream by the primary flow.

The most general case of system instabilities is the so-called cavitation surge that prevails for relatively long and thin cavities (typically when the cavity length is about 75-100% of a foil chord) on hydrofoils [START_REF] Callenaere | The cavitation instability induced by the development of a re-entrant jet[END_REF][START_REF] Watanabe | Theoretical analysis of transitional and partial cavity instabilities[END_REF][START_REF] Kawakami | An assessment of the influence of environmental factors on cavitation instabilities[END_REF]. According to Callenaere et al. [START_REF] Callenaere | The cavitation instability induced by the development of a re-entrant jet[END_REF], the mechanism of this instability is as follows. A minor pressure drop at a cavity location makes its volume increase and, as a result, hydraulic resistance grows. In order to counterbalance this, the flow decelerates at the inlet duct, which makes the local pressure gradually grow and the cavity volume decreases. When the cavity reaches its minimum length, local velocity on the contrary increases maximally that leads again to a local pressure drop. Afterwards, the process repeats and the system as a whole oscillates at a certain frequency. Thus, when cavitation surge conditions are fulfilled, the mechanism of the cavity length pulsations is driven by propagation of pressure waves along the test channel.

The control of cavitation inception and development, its suppression or avoidance can be achieved by both the active and passive methods. Presently, there are several promising approaches of active management of cavitating flow based on different underlying principles. The most prevailing are the control of cavitation nuclei by means of ultrasound or electrolysis (e.g. [START_REF] Chatterjee | Towards the concept of hydrodynamic cavitation control[END_REF][START_REF] Chatterjee | Some investigations on the use of ultrasonics in travelling bubble cavitation control[END_REF]), injection or suction of basic fluid or various solutions through a body surface [START_REF] Chang | Tip vortex cavitation suppression by active mass injection[END_REF] and forced transition of a boundary layer to turbulence by imposing external oscillations or vibrations [START_REF] Franc | Unsteady attached cavitation on an oscillating hydrofoil[END_REF]. In all passive methods cavitation manipulation is implemented by means of changing the wall properties in one or another way. These are selection of the surface material with certain characteristics [START_REF] Leger | Examination of the flow near the leading edge of attached cavitation. Part 1. Detachment of two-dimensional and axisymmetric cavities[END_REF], creation of irregular [START_REF] Coutier-Delgosha | Effect of wall roughness on the dynamics of unsteady cavitation[END_REF] or distributed [START_REF] Kawanami | Mechanism and control of cloud cavitation[END_REF][START_REF] Ph | The effects of a tripped turbulent boundary layer on vortex shedding from a blunt trailing edge hydrofoil[END_REF] roughness and application of compliant coatings [START_REF] Akcabay | Cavity induced vibration of flexible hydrofoils[END_REF].

The paper aims at the investigation of partial cavity flows around a scaled-down model of guide vanes (GV) of a high-pressure turbine at different attack angles and cavitation numbers and control of the cavitation by means of continuous mass injection through a spanwise slot channel in the foil surface. We report here on high-speed visualization of spatial structure and dynamics of attached and cloud cavities and their integral characteristics, a comparison of frequency spectra of pressure pulsations near the profile made by a hydrophone and PIV measurements of the spatial distributions of the mean velocity and its turbulent fluctuations around the cavitating GV model.

EXPERIMENT

The experiments were carried out in the Cavitation tunnel in Kutateladze Institute of Thermophysics SB RAS. Its description as well as details on the experimental conditions and measurement techniques applied can be found in [START_REF] Yu | High-speed visualization and PIV measurements of cavitating flows around a semi-circular leading-edge flat plate and NACA0015 hydrofoil[END_REF]. Cavitation number σ = (P in -P V )/(ρU 0 2 /2), where P in and P V are the static pressure at the test section inlet and the water vapor pressure, respectively, ρ is water density and U 0 is the mean (bulk) flow velocity, was controlled by varying the mean flow velocity, which was chosen to achieve a range of targeted cavitation regimes. Three attack angles have been considered in tests, α = 0, 3 and 9°.

The test foil of C = 100 mm chord length was a scaled-down model of a Francis turbine guide vane (GV), which is a two-dimensional body (Figure 1 and Table 1), with a blunt trailing edge of 1.31 mm height. It was made of brass with the mean surface roughness of 1.5 µm. The maximum width of GV is H max = 0.2206C measured at the distance of x max = 0.4396C from the leading edge. The GV rotating axis is slightly shifted to its trailing edge and is located at the distance of 0.54C from the foil leading edge. The aspect ratio (the ratio of the foil span to its chord) was AR = 0.8 that is very close to the one in some practical full-scale conditions. In order to provide the flow control, the hydrofoil was manufactured with a spanwise slot channel in its surface (Figure 1) through which the liquid from a bypass circuit was supplied directly to the test section along the GV surface. The slot nozzle exit of 0.6 mm height and 70 mm width was located at the distance of 0.6C from the foil leading edge so that the liquid was injected across the whole foil span, except for the end caps of 5 mm width. The height of the backward facing step formed by the slot and its wall was 0.8 mm. The nozzle chamber inside the GV profile had a helical geometry with a contraction towards the exit (Figure 1a).

Table 1. Points of the generatrix of the guide vane model at different cross-sections. y up and y down are the transversal coordinates of these points at the upper (with slot) and lower (w/o slot) surfaces with respect to the foil chord (zero line). The GV thickness H(x) = y upy down . The GV shape is a result of a cubic spline approximation of these values.

x The injected liquid was delivered into the nozzle chamber through a channel of 13 mm inner diameter in the hydrofoil axis by a pump or owing to the pressure gradient. The maximum flow rate through the slot channel was 0.44 l/s, which corresponded to the maximum velocity of the slot wall jet U in = 10.6 m/s. The measurement uncertainty of the injection velocity was 0.2 m/s. In the branch circuit, the liquid was taken close to the centrifugal pumps of the main facility loop and was partly supplied to the test section and the rest of it was returned via a bypass line back into the facility so that the total liquid volume was constant.

In tests, local pressure measurements were also carried out near the GV foil by means of a hydroacoustic pressure transducer -a hydrophone Brüel&Kjaer Type 8103 (the sensor diameter -9.5 mm, sensitivity -24.6 µV/Pa, frequency spectrum -from 0.1 Hz to 180 kHz, resolution accuracy -+1.5 dB for 0.1 Hz and -6 dB for 100 kHz) -that was flushmounted into the sidewall of the test section. It was positioned at the center of the test section in the vertical direction and 80 mm shifted behind the GV trailing edge. The sidewall at which the hydrophone was fixed is a black-colored duralumin plate of 10 mm width that is vibrationally isolated by rubber pads from the resting part of the test section and the experimental facility as a whole.

The velocity fields were measured by a ''POLIS'' PIV-system consisting of a double-pulsed Nd:YAG Quantel EVG00200 laser (wavelength 532 nm, repetition rate 15 Hz, pulse duration 10 ns, pulse energy 200 mJ), a ImperX B2020 CCD-camera (14 bits per pixel, matrix resolution 2048x2048 pixels) equipped with Nikon AF Nikkor 50 mm f/1.4D lens and an optical lowpass filter (bandpass edge at 570 nm), as well as a synchronizing processor. The PIV-system was operated via a computer using ''ActualFlow'' software. The current measurements were carried out at the sampling rate of 4 Hz. The thickness of the laser light sheet formed by a cylindrical lens to illuminate tracer particles was about 0.8 mm in the measurement plane that coincided with the central longitudinal plane of the test section parallel to its larger sidewalls. The distance between the camera and the laser sheet was 574 mm. The size of the measurement area was approximately 124x124 mm. Since in a cavitating flow the micro-size vapor bubbles can act as tracers, their contaminating effect on the PIV measurements was reduced by adding fluorescent tracers (average size 10µm, wavelength range 550-700 nm) into the working liquid.

In order to enhance the quality of the registered images, the data were subjected to a two-step preprocessing. First, the mean two-frame intensity field over 5000 initial image pairs was subtracted from each image pair. This allows to compensate the nonuniformity of the image exposure, independently for the first and second frames, as well as to remove glares and shadows in the images. As a result, the intensity of both frames becomes quite similar, improving the accuracy of the velocity vectors calculation in further processing. Secondly, all image pairs were masked to remove from the calculations the areas corresponding to the foil and the shadow.

The velocity fields were calculated using the iterative cross-correlation algorithm with a continuous window shift and deformation and 75% overlap of the interrogation windows. In addition, during the correlation step the local particle image concentration was accounted for in the correlation analysis only in those interrogation areas where the particle concentration level is above the given number of particles. The threshold value for seeding particles concentration was set to be 5 tracers per 32x32 pixel area. The tracers were regarded as the convolution of a Gaussian mask with 1 pixel radius and initial image over 5x5 pixels window, with the correlation threshold being 0.7. The sub-pixel interpolation of a cross-correlation peak was performed over three points, using one-dimensional approximation by the Gaussian function. In order to have a relatively large dynamic range (the span between the maximum and minimum velocity), the initial size of the interrogation window was chosen to be 64x64 pixels, but it was subsequently reduced so that the final interrogation window was 8x8 pixels, which provided high enough spatial resolution. The obtained instantaneous velocity vector fields were validated with the three procedures applied successively: the peak validation with the threshold 2.0, the adaptive median filter over 7x7 nodes and the cluster validation with the coefficient of 50 (for more details refer to [START_REF] Yu | High-speed visualization and PIV measurements of cavitating flows around a semi-circular leading-edge flat plate and NACA0015 hydrofoil[END_REF]).

RESULTS AND DISCUSSION

We present below some results of the experimental investigation of cavitating flows over the GV foil for two attack angles, α = 3° and 9° at different regimes defined by the cavitation number. The presentation begins with a discussion of selected images from highspeed visualization and measured pressure fluctuations, followed by distributions of the mean velocity and the rms of the streamwise velocity fluctuations.

Visualization and pressure measurements

At α = 3°, cavitation is initiated right behind the leading edge on the suction side of the GV profile in form of traveling bubbles (Figures 2-a.1 and 2-a.2). At σ = 0.95, initially transitional cavity that is characterized by relatively small clouds (compared to cloud cavitation regimes) shedding from the cavity closure region is transformed to a typical traveling bubble cavitation as soon as the injection through the slot is started. This leads to a decrease in the dimensions of the cavitation area on the GV surface. However, pressure fluctuations for both cases are almost identical and reveal no effect of injection (Fig. 2-a.3). This influence detected visually is probably linked with the existence of a shear layer past the backward facing step produced by the slot and its wall, which leads to an abrupt reduction of the shear stresses once the injection begins. Weakened shear stresses cannot anymore produce strong spanwise vortices (like in the case without injection) in the centers of which an intense evaporation occurs and, thus, cavitation is suppressed.

When the cavitation number is slightly decreased down to 0.89, the partial cavity on the GV section becomes unsteady and starts to oscillate periodically in the free flow case (Figure 2-b.1). This is also proved by the pressure measurements. The main maximum that is clearly seen in Figure 2 For a higher incidence angle α = 9°, the injection influence is less pronounced. At some regimes (e.g. see Figure 3), the active flow control has no influence on the cavity characteristics and dynamics, though under certain conditions cavitation area becomes shorter and flow dynamics change (Figures 3-a.1 and 3-a.2). Moreover, the injection sometimes leads to a reverse effect -an increase in pressure fluctuations, for example in Figure 3-a.3 a peak appears at St ≈ 0.07. This occurs when the flow rate through the slot is too low. In such a case, the auxiliary flow is choked periodically because its momentum is sufficient to overcome the adverse pressure gradient only in those moments when the pressure inside the nozzle chamber becomes higher than that at the GV trailing edge.

Otherwise, the flow through the slot is blocked for a while.

Velocity distributions

Figures 4 and 5 display selected profiles of the streamwise component of the time-averaged velocity (plotted as the normalized velocity defect, (U -U 0 )/U 0 ) and the rms of the streamwise component of its turbulent fluctuations ũ at several locations along the GV model. In order to facilitate a direct comparison, the profiles are plotted jointly for free and forced flow conditions only for α = 3° as at this attack angle the injection effect is more pronounced. Typical attached cavity interfaces (obtained from visual data) for the developed cavitation regimes are depicted in the figures as solid and dashed lines for the free and forced flow conditions, respectively. The reference point (the origin of coordinate axes) coincides with the leading edge of the GV section in the measurement plane at α = 0°. As seen, the injection does not affect the profiles of the mean and fluctuating velocities upstream of the slot channel position at all (cf. the profiles at sections x/C = 0.1 and 0.4 in Figures 4 and5). Moreover, the injection flow rate corresponding to the range of the injection velocities from U in /U 0 = 0 to 0.76 was found not to practically influence the distributions of these characteristics at α = 3° (not shown in the images) so that the global difference is only between the free and forced flow conditions. Thus, below only these two cases are considered, without taking into account the values of U in /U 0 .

For the single-phase flow (Figure 4-a), the injection leads to a decrease in the mean velocity just above the foil trailing edge (x/C = 1) from U/U 0 = 0.41 down to 0.15 (∆U/U 0 = 0.26). Besides, in the GV wake (x/C > 1) a double growth of the turbulent velocity fluctuations up to ũ/U 0 = 0.23 occurs at the distance of y/C = 0.024 from the GV surface when the injection is started (Figure 5a), while their amplitude remains almost the same close to the wall. In case of traveling bubble cavitation at σ = 1.06 (Figures 4-b and5-b), the active mass injection causes the cavitation disappearance and the profiles of free flow ones. Thus, for these two flow regimes the active mass injection makes the flow turbulence structure more developed and the wake past the GV section more intense. Transitional cavitation (Figure 4-c) causes substantial changes in the velocity distributions for the free flow conditions. The boundary layer thickness increases from 0.03C at x/C = 0.7 and 0.06C at x/C = 1 (cavitation-free and cavitation inception cases, Figure 4-a andb) up to 0.04C and 0.09C at the same sections, respectively. The mean velocity is noticeably reduced in the near-wall region (y/C < 0.05) in comparison with those for higher cavitation numbers so that its minimum reaches 0.43U 0 at x/C = 0.7 and 0.28U 0 at x/C = 1. Turbulent fluctuations at the same time grow within the boundary and mixing layers up to ũ/U 0 = 0.21 at x/C = 0.7 and 0.17 at x/C = 1. The mass injection results in the modification of the cavitation pattern to traveling bubbles, a decrease of the boundary layer thickness to 0.011C at x/C = 0.7 and 0.035C at x/C = 1 and a reduction of the velocity fluctuations down to ũ/U 0 = 0.14 at x/C = 0.7 and their equalization about the value of ũ/U 0 = 0.2 at x/C = 1 for the same cavitation number σ = 0.95. Consequently, at the regimes of transitional cavitation the active flow control system allows a favorable and efficient flow manipulation.

When transition to unsteady regimes occurs (Figure 4-d ande), the boundary layer widens up to 0.07C at x/C = 0.7 and 0.14C at x/C = 1 for the free flow conditions. The mass injection does not, however, influence the transversal size of the boundary layer but somewhat reduces the mean flow velocity inside it. So, close to the wall ∆U/U 0 reaches 0.3 at x/C = 0.7 and 0.1 at x/C = 1 for σ = 0.89 and 0.2 and 0.16 in the same cross-sections for σ = 0.84 (see Figure 4-d ande). In addition, under the forced conditions, the turbulent fluctuations are in general lower (typically about 0.08U 0 ) in comparison with the free flow conditions. Further, the active flow management makes it possible to totally suppress the periodic cavity length oscillations for certain conditions as for instance in the case of σ = 0.89 (Figures 4-d and5-d). Hence, the continuous liquid supply through a slot channel in the GV section surface is an effective tool for flow management.

CONCLUSIONS

High-speed visual imaging and PIV velocity measurements provided information on the main features of the cavitation incipience and development, the cavity spatial structure and its dynamics, as well as distributions of the mean and turbulent characteristics of the flow over a modified scaled-down model of symmetric guide vanes of high-pressure turbines (GV) with a slot spanwise channel in its surface at several attack angles. The results under free flow conditions are compared with those under the forced flow conditions for different cavitation regimes (from singlephase flow to unsteady cavitation). It was found that the active mass injection does not influence the primary flow upstream of the slot channel position absolutely. At small angles of incidence, the injection flow at velocities in the range between zero to 0.76 of the mean bulk velocity was observed not to practically influence the distributions of turbulent characteristics so that the global difference is only between the free and forced flow conditions. For cavitation-free and cavitation inception cases, the active mass injection was shown to make the flow turbulence structure more developed and the wake past the GV section more intense. Thus, in these cases the use of the injection is undesirable.

However, the active flow control system considered also allows a favorable and efficient flow manipulation, especially at the regimes with developed gas-vapor cavities. Moreover, the active flow management makes it possible to reduce substantially the amplitude or totally suppress the periodic cavity length oscillations and pressure pulsations associated with them. Hence, the continuous liquid supply through a slot channel in the GV section surface is an effective tool for flow management. Unfortunately, the spatial resolution of the performed velocity measurements is insufficient to distinguish the shear layer at the slot position and, therefore, some conclusions made throughout the paper must remain only conjectures so far. This makes it necessary to carry out additional experiments in future with a higher spatial resolution.

Figure 1 .

 1 Figure 1. (a) A 3D scheme and (b) photograph of the fabricated GV profile.

  -b.3 at St = fC/U 0 = 0.21 corresponds to the frequency of the attached cavity pulsations. At this regime, liquid injection results in stabilization of the attached cavity dynamics (Figure 2b.2), changing the partial cavity regime to transitional one like that shown in Figure 2-a.1. At the same time, the peak in the pressure fluctuations spectrum at St = 0.21 vanishes completely (Figure 2-b.3). When the cavitation number is further reduced to σ = 0.84, the active mass injection does not already change the cavity unsteady behavior (Figures 2-c.1 and 2-c.2) but the amplitude of the pressure pulsations diminishes about three times (cf. the peak heights in Figure 2-c.3).

Figure 2 .

 2 Figure 2. Instantaneous images of partial cavities (top view) on the suction side of the GV model at α = 3° (1) without and (2) with mass injection through the slot channel and (3) comparison of the spectra of pressure fluctuations measured by the pressure transducer for the same flow conditions when: (a) σ = 0.95, U in /U 0 = 0.79 (transitional cavitation/traveling bubbles), (b) σ = 0.89, U in /U 0 = 0.74 (unsteady cavity/ transitional cavitation) and (c) σ = 0.84, U in /U 0 = 0.8 (unsteady cavitation in both cases). For unsteady regimes, half-images indicating the phases of the attached cavity evolution when it has (left) the shortest (just before its detachment) and (right) maximal lengths are shown together. the mean and fluctuating velocities at x/C = 1 become quite close to each other. The one for the free flow remains unchanged, whereas that for the forced flow is accompanied by a reduction of the fluctuations amplitude (Figure 5-b) and approaches the former. However, at this regime the velocity fluctuations are still higher for the forced flow conditions compared to the

Figure 3 .

 3 Figure 3. Instantaneous images of partial cavities (top view) on the suction side of the GV model at α = 9° (1) without and (2) with mass injection through the slot channel and (3) comparison of the spectra of pressure fluctuations measured by the pressure transducer for the same flow conditions when: (a) σ = 2.11, U in /U 0 = 0.27 (transitional cavitation/sheet cavity) and (b) σ = 1.61, U in /U 0 = 0.65 (unsteady cavitation in both cases). For unsteady regimes, half-images indicating the phases of the attached cavity evolution when it has (left) the shortest (just before its detachment) and (right) maximal lengths are shown together.

Figure 4 .

 4 Figure 4. Downstream evolution of the streamwise component of the mean velocity over the suction side of the GV model for (a) cavitation-free flow at σ = 1.3, U in /U 0 = 0.65; (b) traveling bubbles/cavitation-free flow at σ = 1.06, U in /U 0 = 0.65; (c) transitional cavitation/traveling bubbles at σ = 0.95, U in /U 0 = 0.66; (d) unsteady cavity/transitional cavitation at σ = 0.89, U in /U 0 = 0.74 and (e) unsteady cavitation in both cases at σ = 0.84, U in /U 0 = 0.76. α = 3°. ▲ -w/o injection, ◇ -with injection. Solid and dashed lines represent interfaces (extracted from visual data) of the attached cavities of maximum size on the GV section for the free and forced flow conditions. s denotes the position of the slot channel in the foil surface. The flow direction is from the left.

Figure 5 .

 5 Figure 5. Downstream evolution of the streamwise turbulence intensity (rms-values) over the suction side of the GV model for (a) cavitation-free flow at σ = 1.3, U in /U 0 = 0.65; (b) traveling bubbles/cavitation-free flow at σ = 1.06, U in /U 0 = 0.65; (c) transitional cavitation/traveling bubbles at σ = 0.95, U in /U 0 = 0.66; (d) unsteady cavity/transitional cavitation at σ = 0.89, U in /U 0 = 0.74 and (e) unsteady cavitation in both cases at σ = 0.84, U in /U 0 = 0.76. α = 3°. ▲ -w/o injection, ◇ -with injection. Solid and dashed lines represent interfaces (extracted from visual data) of the attached cavities of maximum size on the GV section for the free and forced flow conditions. s denotes the position of the slot channel in the foil surface. The flow direction is from the left.

ACKNOWLEDGMENTS

The research was funded by a grant from the Russian Scientific Fund (Project No. 14-29-00203) through Novosibirsk State University.


Linyuan Jia

Yuchun Chen

Validation of a Physically Enhanced Sub Idle Compressor Map Extrapolation Method

Keywords:

The objective of this study is to validate the physically enhanced compressor map extrapolation (PECME) method brought forward by Zachos. The essential steps for this method are introduced and carried out in conjunction with the compressor zero speed curve to extrapolate the Allison 250 engine compressor map to the sub idle region. The power extrapolation engine sub idle performance simulation model and the evaporation controlled combustion efficiency model are used in the engine sub idle performance simulation model. The complete compressor map is then used in the engine performance simulation model and the engine starting performance is simulated and validated with the test data. The deviations of the rotational speeds are analyzed, showing that the PECME method is practical and valid. A case study is also carried out by simulating the Allison 250 engine operating process from engine starting to 200Nm load. The simulation results are well matched with the test data which proves that the whole methodology introduced in this paper is able to provide high fidelity simulation results in a wide range of engine operation.

INTRODUCTION

Gas turbine engine starting performance simulation is an inevitable section for engine performance analysis and starting control schedule design. The component maps based model is regarded as the most practical gas turbine engine starting performance simulation model due to the detailed information it provides. However, component maps for sub idle region are usually unavailable. Therefore, several map extrapolation methods were developed. In 2000, Kurzke developed a tool named Smooth C for compressor map extrapolation based on the estimated zero speed line and some principles deducted from incompressible similarity laws [START_REF] Kurzke | How to Get Component Maps for Aircraft Gas Turbine Performance Calculations[END_REF][START_REF] Riegler | Some aspects of modeling compressor behavior in gas turbine performance calculations[END_REF]. A different method was developed by Sexton in 2001, in which the exponents of incompressible similarity laws were modified in order to take the compressible effects into consideration [START_REF] Sexton | A method to control turbofan engine starting by varying compressor surge valve bleed[END_REF]. In 2007, Sexton's method was simplified by Gaudet [START_REF] Gaudet | Development of a Dynamic Modeling and Control System Design Methodology for Gas Turbines[END_REF][START_REF] Gaudet | A simple sub-idle component map extrapolation method[END_REF], which was then validated by Hönle in 2013 [START_REF] Hönle | Experimental validation of a sub-idle compressor map extrapolation. 21st ISABE conference[END_REF]. In 2002, Jones compared the existing map extrapolation methods and suggested that the zero speed line be either calculated, measured or estimated to transform extrapolation into interpolation and to obtain more reliable results [START_REF] Jones | Extrapolation of compressor characteristics to the low-speed region for sub-idle performance modelling[END_REF]. Following Jones' suggestion, in 2007, Howard determined the zero speed curve using the generic coefficients obtained from 3D CFD simulation [START_REF] Howard | Sub-idle modelling of gas turbines: Altitude relight and windmilling[END_REF]. The curve was used to generate the compressor sub idle maps. In 2010, Zachos improved Howard's work by employing the locked rotor corrected torque characteristics, thus converting the mathematical map extrapolation method into a physically enhanced interpolation method with the ability to obtain a more accurate sub idle compressor map [START_REF] Zachos | Gas turbine sub-idle performance Validation of a Physically Enhanced Sub Idle Compressor Map Extrapolation Method -8 modelling: altitude relight and windmilling[END_REF][START_REF] Aslanidou | A physically enhanced method for sub-idle compressor map generation and representation[END_REF]. Some remarkable engine performance simulation results were provided by Zachos to show the capability of the PECME method [START_REF] Zachos | Gas turbine sub-idle performance Validation of a Physically Enhanced Sub Idle Compressor Map Extrapolation Method -8 modelling: altitude relight and windmilling[END_REF]. However, the map extrapolation results along with the engine simulation results are not validated due to the insufficient data available for sub-idle operation. In 2013, the Allison 250 engine compressor locked rotor characteristics, in forms of pressure ratio and torque curves, were measured by Hönle [START_REF] Hönle | Experimental validation of a sub-idle compressor map extrapolation. 21st ISABE conference[END_REF], which makes it possible to extrapolate the compressor map to sub idle region, by employing the PECME method. Therefore, this method can then be validated with test data. In this paper, the sub idle compressor map of the Allison 250 engine was generated based on locked rotor characteristic measurement data provided by Hönle [START_REF] Hönle | Experimental validation of a sub-idle compressor map extrapolation. 21st ISABE conference[END_REF]. The completed compressor map was then used to simulate the starting performance of the Allison 250 engine. Simulation results were then validated with engine starting test data. Some special concerns regarding the validation method, such as the engine sub idle performance modelling and test data correction method are also discussed in this paper.

MAP EXTRAPOLATION

The PECME method was developed described in detail in reference [START_REF] Zachos | Gas turbine sub-idle performance Validation of a Physically Enhanced Sub Idle Compressor Map Extrapolation Method -8 modelling: altitude relight and windmilling[END_REF]. During this section, the essential steps of this method are introduced.

NOMENCLATURE

Greek symbols

C

Specific heat capacity [J/(kg The first step of the map extrapolation method is to rearrange the compressor map by introducing the corrected torque to avoid the discontinuity of efficiency in sub idle region. The most convenient way to derive corrected torque is to use the flow coefficient, corrected rotating speed and the efficiency, as these parameters are already well described in the conventional compressor maps. Using the Euler's equation in conjunction with the corrected parameters of a turbo machinery, the quasi-dimensional torque group can be derived using Equation [START_REF] Kurzke | How to Get Component Maps for Aircraft Gas Turbine Performance Calculations[END_REF].

      1/ 30 1/ kk cor std cor c cor c Tq T Cm N       (1)
The Allison 250 engine compressor map in terms of corrected mass flow, corrected torque and pressure ratio as functions of beta values and corrected rotating speed is shown in Figure 1.

(a) Pressure ratio (b) Corrected torque Figure 1. Compressor map rearrangement with corrected torque and pressure ratio The second step is to generate the compressor zero speed line by conducting a locked rotor study. In Zachos' study, the single stage zero speed line was generated followed by the stage stacking method to get the zero speed line of the whole compressor. In this paper, the Allison 250 engine compressor zero speed line was deduced from the locked rotor characteristic measurement data provided by Hönle [START_REF] Hönle | Experimental validation of a sub-idle compressor map extrapolation. 21st ISABE conference[END_REF]. With the zero speed line available, the beta lines are expanded by matching each point on the lowest above idle curve to the zero speed curve. And the final step of this method is to interpolate the speed lines between the zero speed curve and the lowest above idle curve. As is mentioned by Zachos, the equations used for beta lines extrapolation and speed lines interpolation can be either linear or polynomial. In this study, a linear algorithm used for both.

The complete extrapolated compressor map is shown in Figure 2.

VALIDATION

The validation of the PECME method can be carried out by comparison between Allison engine starting performance simulation results and test data. During this section, some details regarding the validation method, such as the engine starting simulation model and the test data correction method, are initially discussed. Then the Allison engine starting performance simulation results using the extrapolated map are presented and validated with the test data.

Engine sub idle performance simulation

The so-called component map based engine performance simulation model in conjunction with the complete extrapolated compressor map is employed in this study to simulate the starting performance of the Allison engine. However some adjustments are required before this model can be used in sub idle region properly. Two main adjustments of the model, i.e. the power extrapolation model and the evaporation controlled combustion efficiency model, are discussed in this section.

Power extrapolation method

The component maps based engine simulation model should be used in the whole engine starting sequence for a complete validation of the extrapolated map. However, in starting simulation, it is found that the component maps based model encounters extremely poor convergence at the very beginning of the engine starting process, i.e. power turbine rotational speed close to zero. Thus, the so-called power extrapolation method was used at the very beginning of the engine starting process, to avoid model iteration divergence.

The power extrapolation method was developed by Walsh and Fletcher [START_REF] Walsh | Gas Turbine Performance[END_REF] and was utilized by Gaudet [START_REF] Gaudet | Development of a Dynamic Modeling and Control System Design Methodology for Gas Turbines[END_REF] to build up the dynamic model of gas turbine engine starting sequence. This method is based on the assumption that the power of turbine machinery is proportional to the cube of its rotational speed.

33

ref ref P P NN  (2)
Thus, the compressor power and the turbine power can be extrapolated along the speed line using Equation 2. With the starter power known, the acceleration power can be calculated by employing Equation 3. Furthermore, the spool acceleration/deceleration rate can also be deduced from the acceleration/deceleration power and the rotor's moment of inertia using Equation 4.

acc
        (4)

Sub idle combustion efficiency

Engine combustion efficiency is mainly influenced by three factors: evaporation rates, mixing rates and reaction rates. However, the dominative factor varies according to the combustor working conditions. Although the combustion efficiency is rarely determined by three factors at the same time, it is inevitable that two or more factors participate in determining the overall efficiency where the combustor works in transition from one region to another [START_REF] Lefebvre | Fuel effects on gas turbine combustion[END_REF].

A combustion efficiency model which is practical under a wide range of working conditions was developed by Lefebvre [START_REF] Lefebvre | Fuel effects on gas turbine combustion[END_REF].

In this model, the efficiencies controlled respectively by reaction, evaporation and mixing were calculated separately. The final combustion efficiency was obtained by multiplying the three efficiencies. Due to the that the mixing rates tend to be only limiting at conditions where the level of combustion efficiency is close to 100 percent, it is often preferable to express combustion efficiency as the product of the reaction efficiency and the evaporation efficiency [START_REF] Lefebvre | Fuel effects on gas turbine combustion[END_REF]. The reaction controlled efficiency and the evaporation controlled efficiency are defined in Equation 5and Equation 6 respectively.

1.3 b 0.002 / 400 1 b b cr b a Pt V exp T exp f m                        f m   f m   b a   b a f m   f m b a f m b a   b a f m b a       f m   f m   f m   f m (5) 6 b 2 36*10 1 b ce b b a Pt V exp T D f m                     b   b T D f m   T D f m   b b a   b b a T D f m   T D f m b b a T D f m b b a   b b a T D f m b b a       T D f m   T D f m   T D f m   T D f m (6)
The Sauter mean diameter (D in Equation 6) of the Allison 250 engine was given by Sutton in reference [START_REF] Sutton | Analytical Fuel Property Effects-Small Combustors[END_REF]. The fuel evaporation coefficient (λ in Equation 6) is interpolated from the data provided by Lefebvre [START_REF] Lefebvre | Fuel effects on gas turbine combustion[END_REF].

Test data correction

An Allison 250-C20B engine was installed on the engine test bed at the Institute for Flight Propulsion, in Technical University of Munich, see Figure 3. The test bed setup is introduced in detail in [START_REF] Kerler | Helicopter Engine-in-the-Loop Test Setup[END_REF]. The measured parameters include the temperature and pressure at the ambient condition, the compressor outlet, the gas generator outlet and engine exhaust nozzle. Also the fuel mass flow rate, combustor outlet total pressure, rotational speeds and the torque on power turbine are measured. Since the engine performance depends partially on the ambient conditions which usually differ according to the weather, the altitude and other conditions, it is necessary to correct the test data to standard conditions before they can be used to validate the simulation results. The correction equations can be found in [START_REF] Kurzke | Model based gas turbine parameter corrections[END_REF] and [START_REF] Xiaochun | Aero Engine Principles[END_REF]. Furthermore, the measured temperature is always timedelayed compared with the real data due to the thermo inertia effect of the thermocouple, which must be considered to ensure the comparability between simulation results and test data. A method to correct the temperature test data by calculating the thermo inertia effect was introduced by Liu [START_REF] Liu | Time-resolved heat transfer characteristics for steady turbulent flow with step changing and periodically pulsating flow temperatures[END_REF]. In this method, the thermocouple is treated as a lump of material heated at uniform temperature, based on the assumption that the diameter of the bead thermocouples is very small and the thermo conductivity of the thermocouple material is very high. Therefore, the lump heat capacity method was used to describe the heat balance of the thermocouple conjunction (Equation 7).

  tc tc tc g tc tc tc tc dT h S T T V C dt

   (7)
The calculation of thermocouple heat transfer coefficient tc h is essential to obtain the corrected temperature from Equation 7.

An empirical equation was provided in reference [START_REF] Bejan | Heat transfer handbook[END_REF] to calculate the heat transfer coefficient:

  / ,?

Validation

The Allison 250 turboshaft engine starting performance was simulated employing the extrapolated compressor map and the simulation model introduced above. The rotational speed simulation results of both gas generator and power turbine are shown in Figure 5 in comparison with test data. The deviations of gas generator rotational speed and power turbine rotational speed in some critical points are calculated and listed in Table 1 and Table 2 respectively. As can be seen in Table 1, the maximum deviation of the gas generator rotational speed is around 23% at 30% the design speed. While the maximum deviation of the power turbine rotational speed, as is provided in Table 2, is around 22.7% at 15% the design speed. The maximum deviations are both detected during the period with the most dramatic acceleration, i.e. from 40s to 50s. Most of the relative deviations of both the gas generator and the power turbine rotational speeds increase with the decreasing rotational speed. However, no such trend is found among the absolute deviations. The larger deviation at lower rotational speed is mainly caused by smaller denominators in deviation calculation. The simulated gas generator operating line during the engine starting process is shown in Figure 6. The lowest gas generator operating speed is around 10% of the design speed. For lower speed, the power extrapolation method is employed. Although the power extrapolation method was used at the beginning of the starting process, the operating line covers most of the sub idle speed lines. So the sub idle map validation is sufficient and valid.

CASE STUDY

In this section, the Allison 250 engine operating process from engine starting to 200Nm load is simulated and the simulation results are presented as a case study. The simulation results are again compared with test data to show the capability of the whole methodology introduced above. The operating process is described in Table 3. As is shown in Figure 7, the rotational speeds simulation results match the test data well. The deviations at steady state are smaller than 5 percent, while the deviation of transient state is relatively larger. For instance, the deviation of phase 3 reaches 17 percent. This is due to the insufficient modelling of some transient effects such as the delay of the actuator in response to the commands of the engine controller. Figure 8 Validation of a Physically Enhanced Sub Idle Compressor Map Extrapolation Method -5

shows the total pressure at the power turbine in-and outlet.

The deviations for both steady state and transient state are smaller than 5 percent, making for a good match. The temperature simulation results are all corrected using the method introduced in section 2.2 to match the test data.

The total temperature at the compressor outlet and the gas generator outlet are shown respectively in Figure 9 and

CONCLUDING REMARKS

The Allison 250 engine compressor map was extrapolated employing the physically enhanced map extrapolation method. Then the complete compressor map was used in the engine starting performance simulation model to obtain the engine starting performance. The simulation results were validated with engine starting test data. The absolute deviations of the rotational speeds of both the gas generator and the power turbine are smaller than 7.5% and 5% of the design speeds respectively. A case study of the Allison engine operating process from engine startup to 200Nm load is also provided and compared with the test data. The deviations of the total temperature at compressor outlet and the total pressure at power turbine inand outlet are smaller than 5%. The deviation of the gas generator exahaust temperature is within 6%. The validation and case study illuminate that the methodology introduced in this paper is practical and can provide resonable simulation results in a wide range of engine operation. The sub idle combustion model and test data correction method introduced in this paper can also be a guide for further gas turbine engine performance simulation.

Figure 2 .

 2 Figure 2. Compressor map extrapolation result

Figure 3 .

 3 Figure 3. Allison 250 engine test bed in LFA, TUM.Since the engine performance depends partially on the ambient conditions which usually differ according to the weather, the altitude and other conditions, it is necessary to correct the test data to standard conditions before they can be used to validate the simulation results. The correction equations can be found in[START_REF] Kurzke | Model based gas turbine parameter corrections[END_REF] and[START_REF] Xiaochun | Aero Engine Principles[END_REF]. Furthermore, the measured temperature is always timedelayed compared with the real data due to the thermo inertia effect of the thermocouple, which must be considered to ensure the comparability between simulation results and test data. A method to correct the temperature test data by calculating the thermo inertia effect was introduced by Liu[START_REF] Liu | Time-resolved heat transfer characteristics for steady turbulent flow with step changing and periodically pulsating flow temperatures[END_REF]. In this method, the thermocouple is treated as a lump of material heated at uniform temperature, based on the

  7 can be not only used to correct the test data but also used to calculate the simulation results. In this study, Equation7was used in each time step during simulation to calculate the measured temperature which is actually the temperature of the thermocouple conjunction. A comparison between simulation results with and without considering the thermo inertia is shown in Figure4. It is obvious that the simulation results considering the thermo inertia match the measured data much better.

Figure 4 .

 4 Figure 4. Influence of thermocouple inertia on measured data.

Figure 5 .

 5 Figure 5. Rotational speed

Figure 6 .

 6 Figure 6. Gas generator operating line

Figure 7 .Figure 8 .

 78 Figure 7. Rotational speed

Figure 10 .

 10 Figure10. The largest deviations of the temperature at these two stations are within 5 percent and 6 percent respectively, illuminating that the data correction method is practical and valid.

Figure 9 .Figure 10 .

 910 Figure 9. Total temperature at compressor outlet

Table 1 .

 1 Deviations of gas generator rotational speed

	Time [s]	Test	Simulation	Deviation Absolute	Relative [%]
	27	1.73	1.46	0.27	15.65
	30	13.57 14.04	0.47	3.47
	35	23.90 21.45	2.44	10.22
	40	32.12 24.75	7.37	22.95
	50	52.07 54.52	2.46	4.72
	54	58.25 63.02	4.77	8.18
	70	58.84 61.05	2.21	3.75
	100	58.64 60.53	1.89	3.22
	125	59.07 59.71	0.65	1.10

Table 2 .

 2 Deviations of power turbine rotational speed

	Time [s]	Test	Simulation	Deviation Absolute	Relative [%]
	40	1.67	1.96	0.29	14.88
	45	5.38	6.49	1.11	17.07
	50	14.84 19.20	4.36	22.72
	54	28.23 32.81	4.58	13.95
	70	61.13 60.67	0.46	0.76
	100	74.76 77.47	2.71	3.50
	125	77.35 80.14	2.79	3.49

Table 3 .

 3 Engine test operating process

	Phase Time [s]	Operations
	1	0-419	Standby
	2	420-1089	Startup and steady running
			at idle rating
	3	1090-1134 Power turbine torque step
			change to 50Nm and back
	4	1135-1316 steady running at idle rating
	5	1317-1618 Power turbine accelerating
			to 100% design speed and
			steady running
	6	1619-2472 Accelerating to flight mode
			and steady running
	7	2473-2646 Power turbine torque
			increases to 200Nm and
			steady running

Validation of a Physically Enhanced Sub Idle Compressor Map Extrapolation Method -2

ACKNOWLEDGMENTS

Hereby, the author would like to thank Dr. Wolfgang Erhard and Martin Kerler from the Institute for Flight Propulsion, Technical University of Munich for the test data they provided and their kind suggestions.