New type of cavitation instability with peculiar frequency characteristic in liquid rocket inducer

In a liquid rocket turbopump with low pressure and thin wall tanks, cavitation inevitably occurs at the inducer which is installed at the inlet of the turbopump. The cavities at each blade oscillate periodically, and the turbopump sometimes becomes unstable, called cavitation instability. Rotating cavitation is one of the common types of cavitation instability, in which the cavities at each inducer blade oscillate respectively and appear to propagate from blade to blade. However, the new type of cavitation instability which does not follow the conventional principle has been observed in our experiment with an inducer. Converting the frequency in the inertial frame to that in the rotational frame and organizing it in the cavitation number, it was found that the frequency of unsteady cavitation increases as the cavitation number decreases, and this is the peculiar point of this instability. Additionally, during the cavitation instability, a few numbers of backflow vortex cavities were observed and moved sub-synchronously from blade to blade. In addition, the tip leakage vortex cavitation on each blade also propagates like sub-synchronous rotating cavitation but in different propagation speed from that of the backflow vortex cavities. As a result, it was supposed that the new instability is one of the types of sub-synchronous rotating cavitation related to backflow vortex cavitation, tip leakage vortex cavitation, and interaction between cavity propagation and inducer blade.


Introduction
Liquid rocket turbopump, which feeds propellant to combustion chamber at high pressure, is required to be lightweight to increase payload capacity of the launch vehicle.Then the low-pressure tank with thin wall is used, and the liquid rocket turbopump is driven with high rotation speed.An axial flow impeller, called inducer, is installed at the inlet of the liquid rocket turbopump to avoid decreasing suction performance of the main impeller.Therefore, the inducer is always driven with cavitation.The cavitation begins unsteady oscillation under certation condition, causing a phenomenon called "cavitation instability," in which the turbopump becomes unstable.One of the conventional cavitation instabilities is rotating cavitation (R.C.), which cavities on each inducer blade seems to be propagate [1].R.C. can be classified into three conventional types: super-synchronous rotating cavitation (super-S R.C.), synchronous rotating cavitation (sync R.C.), and sub-synchronous rotating cavitation (sub-S R.C.), according to the propagation velocity of cavity.It was reported that some accidents were caused by cavitation instability such as LE-7 engine of H-II rocket [2] and space shuttle main engine [3], then clarifying the mechanism of cavitation instability is required.For the unsteady characteristics of cavitation instabilities in inducer, it is known that super-S R.C. cause super-synchronous shaft vibration of approximately 1.05 -1.3 times the shaft rotation, and sub-S R.C. also cause sub-synchronous shaft vibration of approximately 0.8 -0.9 times the shaft rotation [2,4].Since the propagation speed of sync R.C. is the same as inducer rotation speed, the shaft vibration synchronized with the inducer rotation speed [5].These types of R.C. are the forward rotating cavitation, i.e., the propagation direction of cavity is the same as the direction of inducer rotation.In addition, another type of instability has also been observed.For example, backward rotating cavitation, in which the propagation direction of cavity is actually opposite of the direction of inducer rotation [6].Furthermore, Hashimoto reported that the shaft vibration caused by cavity clouds of backflow vortex occurred after head breakdown [7].
At the same time, fundamental characteristics of frequency of unsteady sheet cavitation, namely sheet/cloud cavitation, arising on the suction surface of a hydrofoil have been studied.Le [8] and Pham [9] showed that the natural oscillation characteristic of sheet/cloud cavitation is that Strouhal number based on cavity length remains constant.Kjeldsen [10] and Lohrberg [11] presented that the break-off frequency decreases with decreasing the cavitation number.
In our research group, numerical and experimental investigation of cavitation instability were conducted.Iga numerically reproduced some types of cavitation instabilities, namely super-S R.C., sync R.C., sub-S R.C., cavitation surge, and rotating-stall cavitation, in a three-bladed cyclic cascade and explained the propagation mechanism by unified rule of cavity oscillation and assumption of existence of latent rotating stall [12].Kondo suggested that there is a lower limit to the inducer rotation speed which a super-S R.C. can occur [13].Yokoi observed cavitation behavior during the occurrence of super-S R.C. from axial direction using a centrifugal pump [14].Tamura reported that the nondimensional frequency of tip leakage vortex cavitation in the occurrence of R.C. can be organized by the cavitation number through the experiments using an inducer at several inducer rotation speed [15].However, we found the peculiar cavitation instability which does not follow fundamental characteristics of unsteady cavitation, i.e., the frequency of cavity oscillation increases when cavitation number decreases [16].In this study, the occurrence range, the frequency characteristics, and visualization result of new cavitation instability arising in inducer were reported.

Test facility
The photograph of tested inducer named THK inducer, which is designed based on the three-bladed inducer of liquid oxygen turbopump of Japanese launch vehicle, is illustrated in Figure . 1 [15].The design flow coefficient   is 0.075.The inlet tip diameter   is 152 mm.The inlet hub diameter is 38 mm.The experiments were performed at the closed-loop water cavitation tunnel at Kakuda Space Center of Japan Aerospace Exploration Agency (JAXA).A schematic diagram of the tunnel is shown in Figure .2. The facility consist of test section, flow straightener, pressurization-depressurization piston, vacuum pump, vacuum tank, deaeration tank, control valve and motor.The temperature of working fluid, water, is kept for constant by heat exchanger.The mean flow rate was controlled by main flow control valve and measured by turbine flow meter and ultrasonic flow meter.The inducer was driven by a DC motor through the multiplying gear box.The inducer rotation speed was adjusted constant.A pressurizationdepressurization piston is installed upstream of test section and controls the mainstream static pressure.In this study, pressure fluctuation at the inlet, middle, and outlet of the inducer on the casing, and shaft vibration were measured to investigate the cavitation instability.The pressure fluctuations were measured by strain gauge type pressure transducer (PGM-50KE, Kyowa Electronic Instrument Co. Ltd., Japan) flush mounted on the casing.The pressure sensors were located 0 (inlet), 43.0 (middle), and 75.3 (outlet) mm from the hub of the inducer, respectively, shown in Figure .3 [12].The shaft vibration was measured by eddy current type displacement sensor (VS-020L, Shinkawa Electric Co. Ltd., Japan).These sensors are each installed in two directions with 90° phase in the circumferential direction.These data were recorded with an FM data recorder and conducted spectrum analysis by a fast Fourier transform (FFT) analyzer (DS-3000, Ono Sokki Technology Inc., Japan).The sampling frequency was 10 kHz.

Procedure
Two types of experiment, decompression and visualization experiment, were conducted.In decompression experiment, the inducer casing was made of steel, and the tip clearance is 0.75 mm.The inducer rotation speeds  were set to 4000, 5500, 6000, 7000, and 7500 rpm at mean flow rate ratio /  = 0.98 where  is flow coefficient.In visualization experiment, the inducer casing was made of transparent acrylic to observe the cavity aspect from side by a high-speed camera (FASTCAM SA5, Photoron Co. Ltd., Japan), and the tip clearance is 0.60 mm.The frame ratio was 10000 frame per second, and the shutter speed was 1/25000 second.The flow was lighted by metal halide lamps (HVC-SL, Photoron Co. Ltd., Japan).The  was set to 6000 rpm at /  = 1.00.
In decompression experiment, the upstream mean pressure was gradually decreased.In other word, the cavitation number  which express the margin to the occurrence of cavitation gradually reduced.The  is defined as follow: where   ,   , ,   present inlet static pressure of the inducer, saturated vapor pressure, density of water, and inflow relative velocity at the inducer blade tip, respectively.The head coefficient  which illustrates how the pump compress the pressure of the fluid, defined as follow is used to examine the occurrence condition of new cavitation instability.
where   ,   ,   ,  present outlet static pressure of the inducer, inlet axial velocity, outlet axial velocity, and gravitational acceleration.To evaluate the characteristics of cavitation instability, the frequency of unsteady cavitation   was used.In visualization experiment,  was keep constant when target cavitating phenomenon was occurring.The   was determined by image processing based on MATLAB Image Processing Toolbox (The MathWorks Inc.).The process is shown in Figure .4. First, the area near the center of the rotation axis was cut and combined because the observed cavity region is different from the actual cavity region at greater distance from the rotation axis.The cut width of image  is defined as  =   sin(/) where  presents the number of images to create a combined image.Second, the combined image was binarized by the adaptive thresholding method.The threshold value is 1.The extracted cavity operation was performed on binarized image including the stripe pattern created by the reflection of light.Finally, the cavity area on each inducer blade for each flow path was determined.After this operation was repeated, and the time variation of cavity area was obtained.Then the FFT analysis was performed on it.The frequency resolution is rough, 3.15 Hz, because the cavity area is only acquired for each inducer revolution, and less than 40 revolutions were visualized.

Occurrence range and frequency characteristics of cavitation instabilities
The occurrence range of R.C. and new type of cavitation instability is presented in Figure .5. The super-S R.C., occurred when the head was stable region.When sync R.C. occurred, the head was temporarily dropped.The reason why is presumably that sync R.C. has unevenness of cavity length and the longest cavity covered the throat [17].The sub-S R.C. occurred just before the head is decreased.On the other hand, the new instability occurred in the operating condition where the head was decreasing by about 1 -10 %.Under this condition, the head is smaller for smaller flow rate, and the performance curve had a positive slope.It is well known that the positive slope of the performance can cause a pump to become unstable.This is because when the flow rate increases/decreases at a certain moment, the positive slope causes an increasing/decreasing pressure by the compressing effect of the pump, which performed as exciting force to further increase/decrease the flow rate [18].This feedback process is what causes the cavitation surge.  = 3  −   (5) It was generally known that this harmonic component is generated in R.C. [19].In other word, either component (i) or (ii) is the actual phenomenon; the other is a byproduct of interaction between actual phenomenon and the blade passing frequency.In this section, the component (i) is treated as cavitating phenomenon due to the visualization result noted the Sec.3.2.
To examine the propagation direction of cavity, the phase and coherence analysis have been performed using cross-correlation of signals from two pressure transducers and shaft displacement sensors configurated with 90° phasing.The results are shown in Figure .7. For axial instability such as cavitation surge, the phase difference is 0°.For circumferential instability, the phase difference is ±90°, which positive means the actual propagation direction is the same as the direction of inducer rotation and the negative means the opposite.The peak of new cavitation instability was /  = 0.73, 2.27 at  = 0.038 .The phase differences between two pressure transducers were 54° at /  = 0.73 and −104° at /  = 2.27 with both value of coherence near unity.In the analysis of shaft vibration at /  = 2.26, the phase difference was −55° where the value of coherence was also near unity.In contrast, when the phase difference of shaft vibration was 26° at /  = 0.73, the value of coherence was 0.46.Then the component (ii) presents a rotating phenomenon due to the results of both phase analysis of pressure and shaft vibration, and it was supposed that the component (i) also presents a rotating phenomenon while it does not cause shaft vibration.Therefore, the new instability is rotating instability that propagates in the same direction as the inducer rotation direction.The frequency of component (i) in the stationary system obtained from the waterfall diagram was converted to the frequency in the rotational system.This converted frequency is the frequency of cavity oscillation observed from the inducer blade.The result of each inducer rotation speed condition is presented in Figure .8. It was shown that the   increased as  decreased.This frequency characteristic is completely different from the fundamental characteristic of cavitation.It was also found that the   is higher with the higher inducer rotation speed.The result of nondimensionalizing   with   is shown in Figure .9. It was also found that the   /  is independent of inducer rotation speed.These trends are the same as the conventional R.C.; i.e., super-S R.C. and sub-S R.C. [15].Therefore, it was supposed that the new instability is also a cavity oscillation.

Observation of cavitation in the occurrence of the instability
Figure 10 shows the cavitation observed from side of the inducer.The small backflow vortex cavitation and long tip leakage vortex cavitation were observed.There appeared to be five backflow vortex cavities, the moving velocity of these backflow vortex cavities was about   = 0.84  to the direction of opposite to the inducer rotation in the rotation frame according to the analysis using Photoron FASTCEM Viewer.Then it is though that the component (i) is due to the moving of backflow vortex, namely   ≃ 5 × (1 − 0.84)  = 0.80  .However, this hypothesis cannot explain the increasing the   as  decreased, since the propagation velocity of backflow vortex in rotational frame on inducer to the direction opposite to the inducer rotation becomes faster due to the reduction of the swirling radius of backflow vortex as the flow rate decreases [20].Although Hashimoto reported that the similar shaft vibration with a component of 2.5  is caused by the backflow vortices moving into the inducer blade pass [7], the difference between this phenomenon and instability which we found is that there is no mode transition.In other words, shaft vibration found by Hashimoto shows a jump in the dominant frequency from 2.5  to 5.4  over time, while ours shows a continuous frequency change.However, the backflow vortex cavity cut by the inducer blade was observed.It was suggested that the shaft vibration in the present experiment was also related to the interference between the cavity cloud of backflow vortex and the inducer blade.The cavity aspect of tip leakage vortex cavitation obtained by image processing is illustrated in Figure .12.In the new instability, the cavity was observed splitting or uniting.Normally, a cavity on the blade oscillates periodically in the conventional mode cavitation instability such as super-S R.C. shown in Figure .12 (b).Therefore, it was found that not just one cavity on the blade continues to oscillate while the instability is occurring.Since it was also identified, as shown in Figure .10, the cavity was cut by the inducer blade, it was indicated that the instability is a complex sub-synchronous phenomenon induced by the whirling of backflow vortex and multiple cells, and the interaction between cavities and inducer blades.

Conclusion
In the present study, the characteristics of the new cavitation instability were reported.The results were summarized as follows.1.The new instability occurred when the head is decreasing at lower cavitation number where cavitation surge normally occurred.2. The new instability is a local instability since the frequency of unsteady cavitation was not influenced by inducer rotation speed.However, frequency of the unsteady cavitation increased as the cavitation number decreased.3. The new instability is a kind of sub-synchronous oscillation which is related to backflow vortex cavitation, multiple cells, and the interaction between cavities and inducer blade edge at the inlet of flow path.
= |  −   | (3) where   ,   present frequency in stationary frame, frequency of inducer rotation.The   indicate the oscillation frequency of cavitation in rotation frame.The   is determined from the waterfall diagram obtained by FFT analysis of pressure fluctuation and shaft vibration.The frequency resolution is 0.60 Hz.
(a) Pressure fluctuation at the inlet.(b) Pressure fluctuation at the middle.(c) Pressure fluctuation at the outlet.(d) Shaft vibration.

Figure. 8
Figure. 8 Frequency of unsteady cavitation in the occurrence of the cavitation instabilities.

Figure. 9
Figure. 9 Normalized frequency of unsteady cavitation in the occurrence of cavitation instabilities.

Figure. 10
Figure. 10 Cavity aspect observed from the side where  = 6000rpm, /  = 1.00,  = 0.033.The cavity area fluctuation and the spectrum analysis of the tip leakage vortex cavitation are shown in Figure.11.It was found that cavity on each blade oscillated with a phase lag and propagated in the opposite direction of the inducer rotation, in the order of blade 1, blade 2, and blade 3. The   of each cavity was 28.1 Hz, which is   = 0.28  and the cell number is one, then   = 0.72 .Therefore, this propagation characteristic of the tip vortex cavitation is similar to that of sub-S R.C., and the value of   is also comparable to that of sub-S R.C. and super-S R.C.