## Abstract

It is well known that cavitation breakdown, which is a phenomenon in which the pump head suddenly drops with a decrease in the inlet cavitation number, occurs in turbopumps. Especially in cryogenic pumps, cavitation breakdown occurs at a lower inlet cavitation number than that of ordinary fluids such as water. This phenomenon is referred to as a thermodynamic effect, as Stepanoff reported. The thermodynamic properties of the working fluid affect the sizes of the cavitation elements, which affect cavitation breakdown; therefore, experimental flow visualization is an effective approach to realize a more efficient and more reliable cryogenic pump. In 2010, the author and colleagues developed the world's first test facility to enable the visualization of cavitation on a rotating inducer in both cryogenic and ordinary fluids. At that time, only two reports on the flow visualization of a rotating cryogenic impeller had been published: one on flow visualization in liquid hydrogen by NASA in 1967 and the other on flow visualization in liquid nitrogen by the Japan Aerospace Exploration Agency (JAXA) in 2010. The present facility employs a triple-thread helical inducer with a diameter of 65.3 mm and a rotation rate of up to 8000 rpm, with both liquid nitrogen and water available as working fluids. Unsteady visualization experiments for cavitation on an inducer in liquid nitrogen and water have revealed the characteristics of tip vortex cavitation, backflow vortex cavitation, and cavitation element size based on comparisons between cryogenic fluids that exhibit a stronger thermodynamic effect and ordinary fluids such as water.

## 1 Introduction

Cavitation is one of the most difficult phenomena to consider when developing turbopumps for high-performance liquid-fueled rocket engines. As the cavitation number σ decreases, cavitation inception occurs in low-pressure regions, cavitation regions gradually expand, and cavitation breakdown (which is the loss of a pump head due to cavitation) ultimately occurs. Here, σ is expressed as follows:
$σ=ps,in−pV12ρLWin2=ps,in−psat(Tin)12ρLWin2$
(1)
where ps,in is the inlet static pressure, pv is the saturation pressure that corresponds to the inlet static temperature Tin, and Win is the inlet velocity on the rotating blade coordinate system. To realize a large thrust and a high specific impulse (ISP: the momentum obtainable from unit mass propellants), high-pressure combustion is necessary; thus, the turbopump must stably discharge a propellant into a combustor at sufficiently high pressure. In addition, to realize a large payload, a small and lightweight turbopump is necessary; thus, the turbopump must be operated at a high rotation rate to realize nominal performance. Furthermore, to realize a large payload, lightweight propellant tanks that have thin walls are important because the tanks constitute almost 90% of the volume of the whole rocket. Hence, the propellant storage pressure should be sufficiently low for thin tank walls, and the turbopump must stably suction a propellant from the tank at this sufficiently low pressure. Under these requirements, cavitation is inevitable in the turbopump. Thus, the combination of an inducer at the upstream position and a centrifugal pump at the downstream position is the optimal setting for the turbopump. The inducer is typically a multithread axial helical inducer. It has a lower pressure ratio than a centrifugal pump; however, it is more durable against cavitation breakdown than centrifugal pumps. On the other hand, centrifugal pumps have a higher pressure ratio; however, they are more sensitive to cavitation, and cavitation breakdown occurs more easily. Therefore, first, the inducer compresses a propellant and ceases cavitation. Next, the centrifugal pump compresses the propellant to nominal pressure. Even in the inducer, cavitation breakdown occurs in cases of excessive cavitation. Figure 1 shows the relationship between the head coefficient ψ and σ under a constant flow coefficient ϕ. Here, ψ and ϕ are as follows:
$ψ=pt,out−pt,inρLUin2=pt,out−pt,inρLR2ω2$
(2)
$ϕ=VinUin=VinRω$
(3)
$Win=Uin2+Vin2$
(4)

where pt,in and pt,out are the inlet and outlet total pressures, respectively; ρL is the liquid density; Vin is the average inlet axial velocity; Uin is the tangential velocity of the blade tip of the inducer on the fixed coordinate; R is the inducer radius; and ω is the angular velocity in radians/s. In Fig. 1, although cavitation occurs with decreasing σ, ψ initially remains at the same value as that prior to cavitation. As σ decreases further, ψ rapidly attains its maximum value, after which it abruptly decreases and “cavitation breakdown” occurs. To utilize the high value of ψ that is attained only in a narrow σ region, cavitation should be allowed, but the unstable phenomena that are caused by cavitation must be controlled (e.g., cavitation surge and rotating cavitation). Numerous studies [13] related to this field have been conducted, but these studies have focused on water pumps.

Fig. 1
Fig. 1
Close modal

Worldwide, high-performance rockets utilize cryogenic propellants, such as liquid oxygen, liquid hydrogen, and liquid methane. Cryogenic fluids exhibit the “thermodynamic effect,” while ordinary fluids, such as water, exhibit only a minimal thermodynamic effect. As shown in Fig. 1, the thermodynamic effect is a phenomenon in which a cryogenic pump maintains a pump head at a low cavitation number at which an ordinary fluid pump would exhibit cavitation breakdown. This phenomenon was identified by Stahl et al. [4], who notably discussed Stepanoff's B factor. The mechanism of the thermodynamic effect is explained as follows: when cavitation nucleates and grows, the surrounding liquid supplies a latent heat, which causes its temperature and saturation pressure to decrease, thereby suppressing cavitation growth further. If an inducer can be designed with consideration of the thermodynamic effect, higher discharge and lower suction inducers can be realized compared with those of conventional inducers that were designed based on knowledge of and experience with water pumps. In particular, to clarify the thermodynamic effects, exact comparisons of the cavitation processes in cryogenic and ordinary fluids are useful under the same geometry of the flow fields, including the supply and discharge pipes, and the same fluid-dynamic conditions, e.g., the same ϕ and σ values. Hence, it is important to understand physical and fluid-dynamic unsteady phenomena through, for example, the performance of unsteady visualization experiments. However, when the author and colleagues planned a test facility to enable the visualization of cavitation on a rotating inducer in both cryogenic and ordinary fluids in 2009, only one report relevant to visualization of cryogenic cavitation had been published. This is because the development of a test facility enabling visualization that can be used with cryogenic fluids is difficult due to the low-temperature embrittlement of metals and plastics, defrosting visualization windows, and thermal insulation requirement. Furthermore, the development of a test facility that can be used for both cryogenic and ordinary fluids—that is, a single test facility that can be used for cryogenic fluid and for ordinary fluid, without the intention of combining the two types of fluid—is extremely difficult. This is because thermal expansion over a wide range of temperatures causes stress in components and misalignment between the inducer and its casing wall. In addition, it is difficult to achieve the temperature control of bearings for only cryogenic or ordinary fluid alone. Taking these challenges into consideration, development of this facility was initiated.

The world's first historic achievement in this area was accomplished in 1969 by Ball et al. of NASA, U.S. [5], who made a helical inducer rotate with a transparent cylindrical casing in a large liquid hydrogen tank and visualized it via an observation window with an ordinary (not high-speed) video camera. Unfortunately, the visualizations were unclear. Hence, the cavitation that was generated around the inducer was identified, but its mechanism was not clarified.

While the author and colleagues were developing the present test facility, other researchers accomplished the second achievement. In 2010, Watanabe et al., at the Japan Aerospace Exploration Agency (JAXA), Japan [6], made a ϕ150-mm helical inducer rotate in a cavitation tunnel in liquid nitrogen, which almost has the same intensity of the thermodynamic effect as that of liquid oxygen; they visualized the inducer via a transparent polycarbonate cylindrical casing with a high-speed video camera. The visualized movies were very clear, revealing that the cavitation around the rotating inducer consisted of many tiny bubbles. They also identified detailed structures corresponding to many kinds of cavitation around the inducer. Unfortunately, their test facility exclusively used liquid nitrogen, and direct comparison with well-understood water cavitation was desired.

The author and colleagues built the world's first test facility to enable the visualization of cavitation on a rotating inducer in both cryogenic and ordinary fluids in 2010. The author and colleagues made a ϕ65.3-mm helical inducer rotate in prototype version 1 of the present cavitation tunnel in both liquid nitrogen and water and successfully visualized the inducer with a high-speed video camera via a transparent cylindrical casing that was composed of quartz glass, employing a mechanism to align the axes between the inducer and the casing in both liquid nitrogen and water. At that time, it was difficult to control the experimental conditions for liquid nitrogen. Thus, the initial experiments in liquid nitrogen were conducted without control features; however, their conditions were measured, and experiments in water were then conducted under the same conditions as those in the previous liquid nitrogen experiments. After a continuing process of trial and error, in 2012, the author and colleagues [7,8] published their results of visualization experiments of cavitation on a rotating inducer in both liquid nitrogen and water in test facility version 1.

In 2013, the author and colleagues improved this facility, as shown in Fig. 2(a); some controlling devices for liquid nitrogen were added, such as a nitrogen cooler and a high-performance vacuum pump, making it possible to control σ and ϕ even in liquid nitrogen. Using this improved facility, the author and colleagues conducted visualization experiments focusing on backflow vortex cavitation. In 2015, the author and colleagues [9,10] published their results of visualization experiments of backflow vortex cavitation on a rotating inducer in liquid nitrogen, high-temperature water, and ambient temperature water in test facility version 2.

Fig. 2
Fig. 2
Close modal

In 2016, the author and colleagues further improved this facility, as shown in Fig. 2(b). First, a volute shape was modified for accurate measurement of the total pressure difference through the inducer. Second, a Coriolis flowmeter was introduced for the accurate measurement of flow rate. Third, a vacuum chamber enclosing the mechanical portion was introduced to stabilize pressure inside the channel in an ordinary fluid. In addition, some miscellaneous modifications were applied to maintain the experimental conditions for both cryogenic and ordinary fluids. Therefore, the controllability of σ and ϕ was much better, and measurements were much more accurate even at small ϕ. Then, the author and colleagues successfully visualized backflow vortex cavitation on a rotating inducer in both cryogenic and ordinary fluids in the same flow channel under the same values of σ and ϕ, and the visualized cavitation was analyzed. Using this test facility version 3, the author and colleagues [11] published their results of the visualization experiment to validate the theoretical analyses performed by the author.

In this paper, the structures of the cavitation tunnel that enable the visualization of cavitation on a rotating inducer in both cryogenic and ordinary fluids are first introduced; then, the features of cryogenic cavitation, which strongly exhibit the thermodynamic effect, are expounded based on the visualization results of cavitation compared with water cavitation under the same geometric and hydrodynamic conditions using test facility versions 2 and 3.

## 2 Cavitation Tunnel With Visualization for Both Cryogenic and Ordinary Fluids

### 2.1 Overall Structure of the Cavitation Tunnel With Visualization for Both Cryogenic and Ordinary Fluids.

Figure 2 presents photos and schematics of the developed cavitation tunnels. The upper panel shows the improved cavitation tunnel version 2 used from 2013 to 2016 [10], and the lower panel shows the improved cavitation tunnel version 3 used from 2017 to 2020 [11]. These two tunnels were located at Tokyo Institute of Technology. Currently, next-generation cavitation tunnel version 4 is being developed at another site, due to the personnel transfer of the author.

All these cavitation tunnels were pump test loops, i.e., the tunnels were closed-loop systems, and the tested pumps recirculated the working fluid by themselves; tests could be conducted over an arbitrary time period. The setup included two stainless steel tanks with an inner volume of 600 ℓ, a test section with flow visualization, a flowmeter, a flow control valve, pipes connecting all components, and sensors.

In this system, the working fluid is sucked by the inducer from the supply tank and discharged via the flow control valve to the catch tank, which removes cavitation bubbles. Because the flow control valves produce a large number of cavitation bubbles, the incoming working fluid to the inducer may contain many such bubbles if the discharge working fluid from the flow control valve directly enters the inducer. These two tank systems work effectively in maintaining the incoming working fluid as a pure liquid. The catch tank separates the cavitation bubbles from the working fluid by using the buoyancy force, and this tank returns a liquid with fewer cavitation bubbles from the bottom of the tank to the supply tank. The supply tank also separates the cavitation bubbles from the working fluid by using the effect of the buoyancy force, and this tank provides pure liquid from the bottom of the supply tank to the inducer.

Each tank should be anchored on the floor for earthquake resistance; thus, some bellows are required. As shown in Fig. 2, the inducer visualization section is rigidly connected to the supply tank, and the flow control valve and Coriolis flowmeter are rigidly connected to the catch tank. Therefore, one bellows is needed between the inducer visualization section and the Coriolis flowmeter, and another bellows is required between the supply and catch tanks. These made-to-order bellows are composed of SUS316 L.

The material of each part was carefully selected such that fluids of various temperatures could be used, from liquid nitrogen to high-temperature water. For the metal parts, materials with a face-centered cubic lattice structure were used because they are insusceptible to low-temperature embrittlement. O-rings with an outer shell made of Teflon sheet and an internal spring structure composed of stainless steel were used because they remain elastic at temperatures from 20 K to 500 K. For thermal insulation, boards that were composed of foamed polystyrene and filler composed of foamed urethane were used because they are less susceptible than other insulative materials to low-temperature embrittlement.

### 2.2 Tested Inducer.

Figure 3 shows the tested inducer. It is an NAL-TR696-series inducer, i.e., a triple-thread axial helical inducer with a diameter of 65.3 mm that was designed by Kamijo et al. [12] at the National Aerospace Laboratory of Japan (NAL)2 in 1982. The best-performing inducer of the series was employed as an oxygen turbopump of the second-stage rocket engine (LE-5) of Japan's space launch rocket (H-II). In 1985, Yamada et al. In 1985, Yamada et al. [13] made another rotate at the designed rotational rate of 16,500 rpm in a cavitation tunnel in liquid nitrogen. They visualized a flow field that was immediately upstream of the inducer; however, unfortunately, they did not visualize a flow field around the inducer.

Fig. 3
Fig. 3
Close modal

Then, the author and colleagues installed the tested inducer in the cavitation tunnel with visualization capability, as shown in Fig. 2, and performed visualization tests. The inducer is composed of an Inconel superalloy, and its linear thermal expansion coefficient is 12.6 × 10−6 1/K. This value is sufficiently small compared with that of polycarbonate (which NASA and JAXA have used), of which the linear thermal expansion coefficient is 70 × 10−6 1/K; however, the effect of the thermal expansion/shrinkage should be considered. Table 1 was shown in Ref. [10] and is reprinted for the reader's convenience. Table 1 specifies the inducer diameter, the inner diameter of the casing, and the tip clearance under the temperatures that the author and colleagues tested. The maximum value of the tip clearance is 0.56 mm in liquid nitrogen, and the minimum is 0.49 mm in water at 330 K. The values of the ratio (the tip clearance)/(the span width of the blade) at the maximum and minimum tip clearances are 0.025 and 0.022, respectively; thus, the results reflect this error of 0.003. This error is much smaller than that of a polycarbonate casing.

Table 1

Tip clearance at different temperatures

Temperature, KInducer diameter, mmCasing diameter, mmHub diameter, mmBlade span, mmTip clearance, mmRatio of the tip clearance to the blade spanFluid
7865.1466.2720.1322.500.560.025Liquid nitrogen
273.1565.3066.3020.1822.560.500.022
29065.3166.3120.1822.560.500.022Water
33065.3566.3220.1922.580.490.022Water
Temperature, KInducer diameter, mmCasing diameter, mmHub diameter, mmBlade span, mmTip clearance, mmRatio of the tip clearance to the blade spanFluid
7865.1466.2720.1322.500.560.025Liquid nitrogen
273.1565.3066.3020.1822.560.500.022
29065.3166.3120.1822.560.500.022Water
33065.3566.3220.1922.580.490.022Water

### 2.3 Test Section With Flow Visualization.

Figure 4(a) shows a cutaway of test section versions 1 and 2 with flow visualization of cavitation around the rotating inducer, and Fig. 4(b) shows that of test section version 3. By the aforementioned control of the tip clearances, visualization tests are conducted at various temperatures of both liquid nitrogen and water. To maintain the alignment of the axes between the inducer and the casing as materials expand or shrink with a change in temperature over a wide range in liquid nitrogen and water, the casing is supported by the aforementioned O-rings in the radial direction. With this mechanism, the O-ring elasticity absorbs the changes in the diameters of both the inducer and the casing with thermal expansion or shrinkage; thus, the tip clearance remains uniform in the circumferential direction at any temperature.

Fig. 4
Fig. 4
Close modal

Some pressure taps penetrate the casing, and the casing is set in a liquid-tight rectangular pool. The casing and supply pipe from the supply tank are surrounded by the working fluid at the same temperature; thus, they can be regarded as adiabatic walls. The liquid-tight pool has four side faces, on which three types of sidewalls can be mounted. The first type is a visualization window. For defrosting, each window is composed of a double-glazed window. The inner window is composed of polycarbonate, the outer window is composed of acrylic resin, and the layer between the windows is always evacuated by an oil-lubricated rotary vane vacuum pump. The second type is a lighting window. Because a lighting angle is restricted due to the thickness of a double-glazed window, a single-glazed window composed of polycarbonate is more suitable for lighting. The third type is a metal wall, to which sensors can be attached. Hereinafter, the described sensors (e.g., pressure transducer and resistance temperature detector (RTD)) are assumed to be mounted on a metal wall. Figure 4 shows an example of the sidewall arrangement. In this case, one sidewall is a hybrid window (the lower part is a visualization window, and the upper part is a lighting window), another sidewall is a visualization window, and the other two sidewalls are metal walls for sensors.

Illumination is very important to clearly record cavitation around the rotating inducer blades. Based on trial and error, illumination in two directions is effective: one from the same face of a high-speed video camera to record the reflected light and another from a side face of a high-speed video camera to record the refracted light. Thus, one of the best configurations is the placement of one lighting window at the top next to the double-glazed visualization window and the other lighting window on a side face, as shown in Fig. 4. A high-speed video camera requires a sufficient luminous flux density for a short exposure time to record a still image of rotating cavitation with a high rotation rate. Thus, two halogen lights (300 W) were used to gain a sufficient luminous flux density. During illumination, strong infrared light from the halogen light reached the acrylic resin and sometimes melted it. Thus, these lights were switched on only during a recording. Light-emitting diode (LED) lights, which emit less infrared light, did not work well for this purpose because they emitted light with flickering. This may have occurred because the tested LED lights were driven by alternating-current (AC) electricity and the performance of the AC-direct-current (DC) converters was not sufficient. It is possible that LED lights driven by pure DC electricity are better for use with a high-speed video camera.

The inducer is set atop the shaft, which is composed of Inconel superalloy and supported by two bearings. In practical rockets and tests at JAXA and NAL, bearings for cryogenic fluids are used; however, they cannot be used in water. Thus, ordinary waterproof and grease-lubricated bearings are used for the present cavitation tunnel. In this case, when a test in liquid nitrogen is performed, the upper bearing must remain at room temperature. Hence, two thermostatic electric heaters were installed: one is for heating the outer race at a flange that is close to the outer race, and the other is for heating the inner race on the rotating shaft just below the inner race. The temperature sensor for controlling the heaters was installed at the center of the rotating shaft beneath the upper bearing. The electricity for the heater and the signal from the sensor are transmitted via slip rings between the rotating shaft and connecting wires. By using this mechanism, both bearings operate normally.

A rotating pump impeller transfers mechanical energy to the working fluid. Part of the transferred mechanical energy is converted to flow energy (static pressure change per unit volume flow rate), but the remaining part still constitutes rotating kinetic energy (dynamic pressure change per unit volume flow rate), which results in a downstream frictional loss. A volute is located downstream of a rotating pump impeller; it converts kinetic energy (∝ dynamic pressure) into flow energy (∝ static pressure), and its performance is evaluated by the static pressure recovery ratio η
$η=ps,out−ps,inpd,in−pd,out$
(5)

In test facility versions 1 and 2, the value of η for the volute was almost 0 (i.e., almost no work was done as a volute). In test facility version 3, a volute was designed by finding a suitable volute channel shape to maximize η simulated by ansysfluent 17.2. The value of η for the volute was 0.59 at the design point.

For a shaft seal of the rotating shaft, gland packing composed of graphite-reinforced Teflon (John Crane C1065) available at various temperatures from 20 K to 500 K was used. This packing restricts liquid leakage but does not restrict gas leakage. To adjust σ according to the rocket operating conditions, lower than atmospheric pressure is required in tests with water at temperatures below 373 K. In the case of test facility versions 1 and 2, air leaking through the gland packings contaminates the inner flow channel of the test facility. To degas the working fluid, the tested inducer first raises the pressure in the volute above atmospheric pressure to prevent air from entering the test facility. Then, a vacuum pump connected to the tanks operates to degas the working fluid. Therefore, it took a long time to prepare experiments in test facility versions 1 and 2. In the case of test facility version 3, as shown in Fig. 4(b), a full mechanical portion, e.g., a rotating shaft, gland packing, and bearings, among other components, is enclosed in a vacuum chamber. The rotating power is transmitted across the wall of the vacuum chamber by the coupling of two face-to-face magnets (BPRO MGC10010P-20SD and MGC10010P-20AD). The bulkhead of the vacuum chamber between the magnets is composed of polycarbonate. The outer side of the magnetic coupling is driven via accelerated timing belts by an inverter-controlled 3.7-kW AC motor. The inducer can rotate in the range of 0–8000 rpm according to the test conditions.

### 2.4 Liquid Nitrogen Cooler, Electric Heater, Vacuum Pump, and Gas Cylinder.

The condition that should be controlled initially is the cavitation number σ, as shown in Eq. (1).

Pin was measured by an absolute pressure transducer (GE PMP5073-TCA3CBH0PA-0P200KPAW). The method that is used to adjust σ to the target value depends on the working fluid. In cases in which liquid nitrogen is used, stationary pressurization is prohibited by laws and regulations in Japan; thus, pin is set close to atmospheric pressure. Tin approaches the saturation temperature that corresponds to the atmospheric pressure in uncontrolled states. In other words, σ approaches 0. Therefore, a liquid nitrogen cooler reduces Tin and, thus pv, to adjust σ to the positive target value. In cases in which water is used, no cavitation occurs at the value of Pin that corresponds to the atmospheric pressure and the value of Tin that corresponds to the ambient temperature because σ is too large. Therefore, the use of a water heater increases Tin and, thus, pv, and the use of a vacuum pump decreases pin to adjust σ to the target value.

### 2.5 Rotation Rate Sensors.

The condition that should be controlled next to the cavitation number σ is the flow coefficient ϕ, as shown in Eq. (3).

First, the inducer is rotated at a constant angular velocity of ω. In the case of test facility versions 1 and 2, a function generator (Wavetek model 19) produces a signal (e.g., 100-Hz square-wave pulse for 6000 rpm); the signal is then sent to a stroboscope (Sugawara TYPE MS-230A); and the 100-Hz pulse light lights a rotating inducer. If the inducer rotates at exactly 6000 rpm, it appears stationary to the experimenter; otherwise, it appears to slowly rotate. The experimenter controls the inverter to retain the stationary appearance. This system is like one of Rube Goldberg's devices. In the case of test facility version 3, the rotation rate was measured by an optic rotation sensor (Satotech DT-2230), and a pulse signal was transmitted to a PC.

### 2.6 Flow Rate Control Valve.

After the rotation rate is set at the target value, the flow control valve gradually closes from the fully opened state toward the completely closed state to adjust ϕ to the target value.

The flow rate control valve is a 3-in. full-bored ball valve (FUJIKIN UBV-21J10R-KALX), which is a valve using a pivoted stainless ball with a 3-in. bore supported by seats composed of Teflon. When the pivoted ball is set for full opening, the bore is aligned to the flow direction, and this arrangement achieves the maximum flow rate because the minimum pressure loss is very close to zero. Instead, where the pivoted ball is set to be fully closed (turned 90 deg from fully open), the flow channel is completely closed by the pivoted ball, and this arrangement achieves a zero flow rate. In cases in which liquid nitrogen is used, seats hold the pivoted ball too tightly by thermal shrinkage, and it is difficult to rotate the pivoted ball precisely. Therefore, a worm-gear mechanism is used to allow precise valve adjustment even in liquid nitrogen.

### 2.7 Pitot Tube Flowmeter.

In the case of test facility versions 1 and 2, a Pitot tube flowmeter was installed at a flange between the supply tank and the double pipe to measure the flow rate into the visualization test section, as shown in Fig. 2(a). The Pitot tube consists of a total pressure hole and a static pressure hole. A differential pressure sensor (Validyne DP303-36-N-3-S-4-A) measures the dynamic pressure by connecting one port of the pressure sensor to the total pressure and another port to the static pressure to improve accuracy. However, because the Pitot tube detects the dynamic pressure, the following limitations arise:

1. The Pitot tube detects the dynamic pressure at the center of a cylindrical inlet pipe, and an average flow rate is calculated based on the assumption that the flow is uniform with an ideal velocity distribution. However, a practical velocity distribution may be nonideal; in addition, backflow vortices (a reverse flow from the inducer tip clearance) sometimes reach the Pitot tube flowmeter at a small ϕ.

2. The Pitot tube detects the dynamic pressure, which is proportional to the squared velocity; thus, the value is very small at a small ϕ. However, the accuracy of a differential pressure sensor is defined as a constant value irrespective of the measured pressure value. Therefore, the accuracy of ϕ worsens as ϕ decreases. Even in cases of ideal velocity distribution, the accuracy of ϕ is ±0.002 at the maximum flow rate of ϕ = 0.17 and ±0.006 at the design flow rate of ϕ = 0.1 but ±0.05 at a small flow rate of ϕ = 0.01. Hence, the accuracy of ±0.05 exceeds the measured value ϕ = 0.01 at a small flow rate ϕ.

In summary, measurements made with the Pitot tube flowmeter are unstable and unreliable, especially at a small flow rate ϕ.

### 2.8 The Coriolis Flowmeter.

In the case of test facility version 3, a Coriolis flowmeter (CA080A41SC22CA2115) was used. The Coriolis flowmeter detects the periodicity and phase delay of the two oscillating U-shaped channels in which the working fluid flows. The periodicity is proportional to the density of the working fluid, and the phase delay is proportional to the mass flow rate of the working fluid. The Coriolis flowmeter calculates and displays the volume flow rate from the measured mass flow rate, the measured density, and the cross section of the channel. The following advantages are provided:

1. These measured mass flow rates and densities do not affect the velocity distribution in a flow channel because both periodicity and phase delay are properties of the two oscillating U-shaped channels.

2. The stability and accuracy are independent of the measured value of the flow rate in a measurable flow rate range, again because both periodicity and phase delay are properties of the two oscillating U-shaped channels. Thus, the measurement of the flow rate is stable and accurate even with a small flow coefficient ϕ.

3. No calibration is necessary by users in the case of a change in density or temperature over the measurable density and temperature ranges. Hence, no calibration is necessary by users even when the working fluid is switched between liquid nitrogen and water.

However, the following defects exist:

1. The pressure loss is greater than that of the Pitot tube. This narrows the operable range of flow coefficients ϕ. Figure 5 shows the relationship between the head coefficient ψ and flow coefficients ϕ performance curve of the tested inducer under no-cavitation conditions; the detailed physical meaning will be explained below. Here, the plotted data range, corresponding to an operable range for each test facility, is focused on. In Fig. 5(a), the maximum ϕ is approximately 0.14 because test facility versions 1 and 2 used a Pitot tube flowmeter, the pressure loss of which is very small. Instead, in Fig. 5(b), the maximum ϕ is approximately 0.12 because test facility version 3 used the Coriolis flowmeter, and its pressure loss is relatively large. However, in Fig. 5(a), the minimum ϕ is approximately 0.06 because the Pitot tube flowmeter is unstable and unreliable at a small ϕ. In contrast, in Fig. 5(b), the minimum ϕ is 0 because the Coriolis flowmeter is stable and accurate even with a small ϕ.

2. The Coriolis flowmeter has a larger volume and larger mass and is more expensive than the Pitot tube flowmeter.

Fig. 5
Fig. 5
Close modal

### 2.9 Pressure Sensor.

The head coefficient ψ is obtained from Eq. (2) under the specified experimental conditions at the cavitation number σ and flow coefficient ϕ. First, the static pressure head (ps,outps,in) is measured. In test facility versions 1 and 2, each static pressure ps,in or ps,out is measured by an absolute pressure transducer (Kyowa PHS-10KA) and transmitted to a PC; then, the differential pressure is calculated by the PC. In test facility version 3, each static pressure ps,in or ps,out is measured by an absolute pressure transducer (PMP5073-TCA2CAH0PA-0P200KPAA). At the same time, the differential pressure (ps,outps,in) is also measured by a differential pressure transducer (PMP5073-TCA3CBH0PA-0P200KPAW) to improve accuracy. The head effect (i.e., the effect of the height at each pressure port) is corrected by the PC.

### 2.10 Temperature Sensor.

A platinum RTD and type T thermocouples are used to measure the temperatures of the working fluid. Because the thermocouple detects the temperature difference between the hot and cold junction, a compensating contact is required to improve the accuracy. During liquid nitrogen experiments, all thermocouples always have compensating contact with liquid nitrogen at atmospheric pressure, and during water experiments, all thermocouples always have compensating contact with ice water at atmospheric pressure.

In test facility versions 1 and 2, a platinum RTD was used to measure the inlet temperature Tin, and four type T thermocouples were used to measure the temperatures of the working fluid near the casing wall. Before the liquid nitrogen experiments, both the RTD and compensated thermocouples were calibrated by liquid nitrogen at atmospheric pressure. Before the water experiments, both the RTD and compensated thermocouples were calibrated by ice water at atmospheric pressure. Here, the liquid nitrogen temperature was assumed to be the corresponding saturation temperature of the measured atmospheric pressure, and the ice water temperature was assumed to be 273.15 K.

In test facility version 3, type T thermocouples were used to measure the inlet temperature Tin and wall temperatures. Before liquid nitrogen experiments, compensated thermocouples were calibrated by liquid nitrogen. Here, the liquid nitrogen temperature was measured by a Lake Shore Cryotronics platinum RTD (PT-102-14 L) with a calibration certificate in the range of 14 K-325 K. Before the water experiments, compensated thermocouples were calibrated by water. Here, the water temperature was also measured by the same platinum RTD (PT-102-14 L).

### 2.11 High-Speed Video Camera.

In test facility versions 1 and 2, a high-speed video camera (Photron FASTCAM SA5) recorded the visualization view of the cavitation. Its recording rate is 12,000 frames per second (fps), its recording resolution is 896 × 704 pixels, and its exposure time is 9.8 μs.

In test facility version 3, a high-speed video camera (Phantom Miro LC310) recorded the visualization view of the cavitation. Its recording rate is 5158 fps, its recording resolution is 896 × 704 pixels, and its exposure time is 9.8 μs.

## 3 Characteristics of Cavitation on a Rotating Inducer in Liquid Nitrogen and Water

### 3.1 Performance Curve of ψ − ϕ for the Tested Inducer.

Figure 5 shows the performance curve of the tested inducer, i.e., the relationship between the head coefficient ψ and flow coefficient ϕ under no-cavitation conditions. For each experiment, one symbol can be placed on the ψ − ϕ diagram at a certain opening degree of the flow rate control valve at a certain rotation rate of the inducer. Experiments were performed as the flow rate control valve was gradually closed from fully open to completely closed. The performance curve is identified by the geometric shape of a pump impeller and flow channel and is independent of the rotation rate under incompressible conditions. The no-cavitation conditions involve nearly complete incompressibility, but cavitation conditions are very compressible conditions because the vapor phase is much more compliant than pure liquid; thus, the cavitation conditions deviate from the performance curve. Generally, a performance curve of an axial helical inducer is a downward-sloping curve as ϕ increases; the maximum ψ is achieved at ϕ = 0, and the minimum ψ is exhibited at the maximum ϕ. Based on these inducer characteristics, increasing ϕ results in decreasing ψ. However, decreasing ψ results in a decrease in ϕ by the pressure loss of the flow channel. That is, the negative feedback loop works; therefore, both ϕ and ψ are stable at certain values on the performance curve. Figure 5(a) shows data obtained in test facility version 2, and Fig. 5(b) shows data obtained in test facility version 3. These tendencies are the same; however, as mentioned in the previous section, test facility version 2 had more shortcomings than test facility version 3, and the data in Fig. 5(a) are more scattered than the data in Fig. 5(b).

Table 2 shows the differences between test facility versions 2 and 3.

Table 2

Differences between test facility versions 2 and 3

Test facility version 2Test facility version 3
FlowmeterPitot tubeOVAL Coriolis flowmeter CA080A41SC22CA2115
Consists of a total pressure hole and a static pressure hole.Measures mass flow rate directly and very stably.
Measures the dynamic pressure by the difference between total and static pressures.Compatible with both cryogenic and ordinary fluids without calibration if calibration is performed once.
Is affected by nonuniform and unsteady flow rate, e.g., a backflow vortex reaches the Pitot tube at a small flow rate.The accuracy of mass flow rate is ±0.1% of the indicated value. It corresponds to the accuracy of ϕ of ±0.006 for any flow rate and fluids.
The accuracy of ϕ is ±0.002 at the maximum flow rate at ϕ = 0.17 and ±0.006 at the design flow rate at ϕ = 0.1 but ±0.05 at a small flow rate at ϕ = 0.01; i.e., the value may be negative at a small flow rate.
In total, measurement is unstable and unreliable.
VolutePrimitive-cylindrical-shaped voluteOptimized-shaped volute
Has a static pressure recovery ratio η of almost 0.Was designed with ansysfluent 17.1.
Has a static pressure recovery ratio η of 0.59 at the design point.
Vacuum chamber for mechanical portionNo vacuum chamberVacuum chamber
Because gland packings were used, in cases where the inner pressure in the test facility is lower than atmosphere pressure (water cases), air bubbles contaminate the test facility.Covers the mechanical portion.
Keeps the inner pressure in the test facility lower than atmosphere pressure (water cases) without contamination by air.
The concentration of dissolved oxygen was 4.8 mg/L, as measured by an optical dissolved oxygen sensor: WTW-W2FD350 Multi3510IDS.
The driving force is transmitted across the chamber wall by using magnet couplings.
High-speed video camera Pressure sensors Temperature sensorsPhotron FASTCAM SA5Phantom Miro LC310
12,000 frames∕s, 896 × 704 pixels, digital shutter speed of 1∕102,000 s.Validyne DP303-36-N-3-S-4-AThe accuracy is ±175 Pa.Krone KDM30-200 kPaA-AThe accuracy is ±500 Pa.Kyowa PHS-10KAThe accuracy is ±2 kPa.SUNX DP2-22ZThe accuracy is ±2 kPa.The accuracy of ψ is ±0.01, and the accuracy of σ is ±0.006. However, these values are unstable at small ϕ.Platinum resistance temperature detectorsType T thermocouples5158 frames/s, 896 × 704 pixels, digital shutter speed of 1/102,041 s.PMP5073-TCA2CAH0PA-0P200KPAAThe accuracy is ±200 Pa.PMP5073-TCA3CBH0PA-0P200KPAWThe accuracy is ±80 Pa.The accuracy of ψ is ±0.01, and the accuracy of σ is ±0.0016. These values are stable even at a small ϕ.Type T thermocouplesCalibrated by Lake Shore Cryotronics platinum resistance temperature detector TP-12-2NQ-Y-2-P07 (PT-102-14 L with a calibration certificate in a range of 14 K–325 K. The accuracy is ±5 mK at 77 K).
Test facility version 2Test facility version 3
FlowmeterPitot tubeOVAL Coriolis flowmeter CA080A41SC22CA2115
Consists of a total pressure hole and a static pressure hole.Measures mass flow rate directly and very stably.
Measures the dynamic pressure by the difference between total and static pressures.Compatible with both cryogenic and ordinary fluids without calibration if calibration is performed once.
Is affected by nonuniform and unsteady flow rate, e.g., a backflow vortex reaches the Pitot tube at a small flow rate.The accuracy of mass flow rate is ±0.1% of the indicated value. It corresponds to the accuracy of ϕ of ±0.006 for any flow rate and fluids.
The accuracy of ϕ is ±0.002 at the maximum flow rate at ϕ = 0.17 and ±0.006 at the design flow rate at ϕ = 0.1 but ±0.05 at a small flow rate at ϕ = 0.01; i.e., the value may be negative at a small flow rate.
In total, measurement is unstable and unreliable.
VolutePrimitive-cylindrical-shaped voluteOptimized-shaped volute
Has a static pressure recovery ratio η of almost 0.Was designed with ansysfluent 17.1.
Has a static pressure recovery ratio η of 0.59 at the design point.
Vacuum chamber for mechanical portionNo vacuum chamberVacuum chamber
Because gland packings were used, in cases where the inner pressure in the test facility is lower than atmosphere pressure (water cases), air bubbles contaminate the test facility.Covers the mechanical portion.
Keeps the inner pressure in the test facility lower than atmosphere pressure (water cases) without contamination by air.
The concentration of dissolved oxygen was 4.8 mg/L, as measured by an optical dissolved oxygen sensor: WTW-W2FD350 Multi3510IDS.
The driving force is transmitted across the chamber wall by using magnet couplings.
High-speed video camera Pressure sensors Temperature sensorsPhotron FASTCAM SA5Phantom Miro LC310
12,000 frames∕s, 896 × 704 pixels, digital shutter speed of 1∕102,000 s.Validyne DP303-36-N-3-S-4-AThe accuracy is ±175 Pa.Krone KDM30-200 kPaA-AThe accuracy is ±500 Pa.Kyowa PHS-10KAThe accuracy is ±2 kPa.SUNX DP2-22ZThe accuracy is ±2 kPa.The accuracy of ψ is ±0.01, and the accuracy of σ is ±0.006. However, these values are unstable at small ϕ.Platinum resistance temperature detectorsType T thermocouples5158 frames/s, 896 × 704 pixels, digital shutter speed of 1/102,041 s.PMP5073-TCA2CAH0PA-0P200KPAAThe accuracy is ±200 Pa.PMP5073-TCA3CBH0PA-0P200KPAWThe accuracy is ±80 Pa.The accuracy of ψ is ±0.01, and the accuracy of σ is ±0.0016. These values are stable even at a small ϕ.Type T thermocouplesCalibrated by Lake Shore Cryotronics platinum resistance temperature detector TP-12-2NQ-Y-2-P07 (PT-102-14 L with a calibration certificate in a range of 14 K–325 K. The accuracy is ±5 mK at 77 K).

### 3.2 Types of Cavitation on a Rotating Inducer Organized by ϕ − σ.

Figure 6 presents snapshots of the visualization view of the cavitation around the rotating inducer in water at 295 K obtained by test facility version 3. In Ref. [10], the author and colleagues presented the same kind of figure obtained by test facility version 2. The reader can compare Fig. 6 in this paper and Fig. 5 in Ref. [10]. In addition, there are two types of cavitations on a rotating inducer in both water and liquid nitrogen, as mentioned in Refs. [711].

Fig. 6
Fig. 6
Close modal

One type of cavitation occurs in a tip region close to the leading edge. This type of cavitation is called tip vortex cavitation. The pump blades transfer mechanical energy to the working fluid downstream; thus, pressure increases on the downstream side of each blade and decreases on the upstream side of each blade. Therefore, the downstream side of each blade is called the pressure surface, and the upstream side of each blade is called the suction surface. There is a large pressure difference across each blade; via the tip region, leak flow occurs from the pressure surface to the suction surface. This leak flow makes a vortex (the “tip vortex”) at the corner of the tip on the suction surface. The pressure along the axis of the tip vortex is lower than that along the surroundings. Where this pressure is less than the local saturation pressure, liquid is vaporized, and tip vortex cavitation occurs.

Another type of cavitation occurs in the suction surface of a rotating inducer. This type of cavitation is called backflow vortex cavitation. As with the tip vortex, leak flow occurs. Then, the leak flow produces an axial-reverse whirling flow along with the cylindrical casing. The axial-reverse whirling flow goes upstream, collides against the main flow, and is ultimately pushed back to the suction surface of the inducer. The axial-reverse whirling flow has a tangential velocity, but the main flow has almost no tangential velocity; thus, a circular shear flow forms along the contact surface between the axial-reverse whirling flow and the main flow. In a circular shear flow, several secondary vortices form due to the Taylor instability [14]. These secondary vortices are called “backflow vortices.” The backflow vortices rotate around themselves and revolve around the inducer axis. The revolving rate is slower than the rotation rate of the inducer. The movement and position of the backflow vortices were theoretically explained and experimentally validated in Ref. [11] in detail; the backflow vortices' movement depends on the flow coefficient ϕ and is independent of the cavitation number σ. In the same manner as the tip vortex cavitation, the pressure along the axis of each backflow vortex is lower than that of the surroundings. Where this pressure is less than the local saturation pressure, liquid is vaporized; then, backflow vortex cavitation occurs.

The physical meaning of the decreasing σ is determined through the decreasing pin, the increasing pv, and/or the increasing Win; cavitation occurs easily and extensively. As shown in Figs. 6(a) and 6(b), with decreasing σ for the same ϕ, the thickness of each tip vortex cavitation event increases. Additionally, as shown in Figs. 6(a) and 6(c), the thickness of each backflow vortex cavitation event increases with decreasing σ for the same ϕ, although the number of backflow vortex cavitation events, revolving velocity, and revolving diameter do not change with decreasing σ for the same ϕ.

Decreasing ϕ causes an increase in the head coefficient ψ based on the characteristics of the tested inducer, as shown in Fig. 5; thus, the leak flow strengthens compared with the main flow. As shown in Figs. 6(a) and 6(b), with decreasing ϕ for the same σ, the direction of the tip vortex axis is turned upstream, and the region where tip vortex cavitation occurs extends from not only the leading edge but also other areas. Furthermore, as shown in Figs. 6(a) and 6(c), with decreasing ϕ for the same σ, the number of backflow vortex cavitation events decreases, the revolution diameter around the inducer axis decreases, the height of backflow vortex cavitation events increases, and the thickness of each backflow vortex cavitation event increases.

### 3.3 Movements of Tip Vortex Cavitation.

Figure 6 shows several snapshots. Next, in Fig. 7, unsteady phenomena are considered.

Fig. 7
Fig. 7
Close modal

Figure 7 shows movements of tip vortex cavitation in liquid nitrogen and water. The flow coefficient ϕ is 0.085 in water. Although ϕ is missing due to sensor error in liquid nitrogen, it can be assumed to be close to 0.085 for the following two reasons:

1. As mentioned in Ref. [11], overall flow structures (e.g., the number, size, direction of cavitation) can be determined by ϕ and are independent of the working fluid.

2. The overall flow structures are similar between liquid nitrogen and water.

First, the position of the tip vortex cavitation is the focus. As mentioned in Sec. 3.2, the leak flow from the inducer blade tip makes the tip vortex and the tip vortex cavitation form at the low-pressure region in the tip vortex. Thus, the tip vortex cavitation is fixed at the tip region close to the leading edge. However, the thickness of this cavitation changes over time; for example, the tip vortex cavitation in water, which is on the left blade at θ = 60 deg, is thick at θ = 60 deg, thin at θ = 84 deg, and thick again at θ = 108 deg. This is because the inlet pressure fluctuates due to cavitation on a rotating inducer. In liquid nitrogen, more interesting phenomena occur. The tip vortex cavitation is separated and unified with time. For example, the tip vortex cavitation in liquid nitrogen, which is on the left blade at θ = 0, is anchored to the inducer blade tip and gradually separates from the inducer blade tip at θ = 48.6 deg. Another tip vortex cavitation grows from the inducer blade tip at θ = 83.5 deg, and the first separated tip vortex cavitation ultimately connects to the new tip vortex cavitation at θ = 107.8 deg. This phenomenon is physically the same as the thickness change because local evaporation occurs at a local pressure less than the local saturation pressure (as a function of the local temperature), and local condensation occurs at a local pressure greater than the local saturation pressure. In other words, this phenomenon indicates that the tip vortex cavitation is convected and occurs and/or disappears locally.

### 3.4 Movements of Backflow Vortex Cavitation.

Figure 8 shows the movements of backflow vortex cavitation in liquid nitrogen and water. The flow coefficients ϕ are 0.085 in both cases. The cavitation numbers σ are 0.03 in liquid nitrogen and 0.07 in water; these values are smaller than those in Fig. 7. Therefore, the backflow vortex cavitation in Fig. 8 is more distinct than that in Fig. 7.

Fig. 8
Fig. 8
Close modal

In both liquid nitrogen and water, the revolving rate of backflow vortex cavitation around the inducer axis is slower than the rotation rate of the inducer. As described in Ref. [11], the revolving rate of backflow vortex cavitation around the inducer axis is determined by ϕ. Let us compare the movements of each backflow vortex cavitation in liquid nitrogen and water in Fig. 8. Backflow vortex cavitation in liquid nitrogen and water, in front of the inducer axis on the center inducer blade at θ = 0, is focused on here. The left inducer blade catches up with the backflow vortex cavitation at θ = 24 deg in both fluids, and it overtakes the backflow vortex cavitation at θ = 48 deg. The upper part of the backflow vortex cavitation rides the left inducer blade, and the lower part of the backflow vortex cavitation enters between the blades. The upper part of the backflow vortex cavitation still revolves around the inducer axis at a rate slower than the rotation rate of the inducer. Furthermore, another incoming inducer blade catches up again with the backflow vortex cavitation at θ = 168 deg in both fluids, and it overtakes the backflow vortex cavitation at θ = 192 deg. Because of the three-bladed inducer, when the inducer rotates 120 deg, the incoming blade catches up with a backflow vortex cavitation if the backflow vortex remains stationary. In this case, when the inducer rotates 144 deg, the incoming blade catches up with the revolving backflow vortex cavitation; thus, the backflow vortex cavitation revolves 24 deg, and the revolving rate is 1/5 of the rotation rate of the inducer. Moreover, the other incoming inducer blade catches up again with the backflow vortex cavitation at θ = 336 deg in both fluids.

Figure 9 shows the movements of backflow vortex cavitation in liquid nitrogen and water at a flow coefficient ϕ of 0.07. As shown in Figs. 6(a) and 6(c), with decreasing ϕ for the same cavitation number σ, the number of backflow vortex cavitation events decreases, the revolution diameter around the inducer axis decreases, the height of backflow vortex cavitation increases, and the thickness of each backflow vortex cavitation increases; these characteristics can be verified based on a comparison between Figs. 8 and 9.

Fig. 9
Fig. 9
Close modal

Here, focus on a backflow vortex cavitation in liquid nitrogen and water, located in front of the inducer axis on the center inducer blade at θ = 90 deg. The left inducer blade catches up with the backflow vortex cavitation in both fluids at θ = 90 deg, and it then overtakes the backflow vortex cavitation at θ = 135–138 deg. Another incoming inducer blade catches up again with the backflow vortex cavitation at θ = 270 deg in both fluids. The next incoming inducer blade catches up again with the backflow vortex cavitation at θ = 405–408 deg in both fluids. In a detailed analysis, when the inducer rotates 160 deg, the incoming blade catches up with the revolving backflow vortex cavitation; thus, the backflow vortex cavitation revolves 40 deg, and the revolving rate is 1/4 of the rotation rate of the inducer. It is confirmed that the revolving rate in Fig. 9 at ϕ = 0.07 is greater than that in Fig. 8 at ϕ = 0.085, as mentioned in Ref. [11].

### 3.5 Collapsing Backflow Vortex Cavitation on the Pressure Surface of the Inducer Blade.

Figure 10 shows the appearance of collapsing backflow vortex cavitation when an incoming inducer blade overtakes the cavitation. At θ = −42 deg and −45 deg, backflow vortex cavitation appears on the center inducer blade. At θ = 0, the left inducer blade catches up with the backflow vortex cavitation. The inducer blade transfers mechanical energy to the working fluid downstream; thus, pressure increases on the pressure (downstream) surface of the inducer blade, and pressure decreases on the suction (upstream) surface of the inducer blade. Therefore, the lower part of the backflow vortex cavitation entering below the pressure surface condenses and disappears in the region marked with a white arrow at θ = 42–45 deg in both liquid nitrogen and water. However, the upper part of the backflow vortex cavitation riding over the suction surface still exists at θ = 42–45 deg in both liquid nitrogen and water.

Fig. 10
Fig. 10
Close modal

### 3.6 Appearance of Cavitation Between Inducer Blades at Cavitation Breakdown.

Figure 11 shows the appearance of cavitation that accumulated between the inducer blades at cavitation number σ = 0.01 and flow coefficient ϕ = 0.00 in liquid nitrogen. At this time, the head coefficient ψ = 0.00, meaning that cavitation breakdown occurred. As shown in Fig. 11, the working fluid goes through the tip clearance from the pressure (downstream) surface to the suction (upstream) surface, and cavitation occurs on the suction surface. Hence, the flow paths between the inducer blades are completely filled with cavitation bubbles. At this time, the inducer blade simply agitates the liquid nitrogen and no longer works as a pump.

Fig. 11
Fig. 11
Close modal

### 3.7 Sizes of Cavitation Elements Between Liquid Nitrogen and Water.

Figures 711 show the appearance of liquid nitrogen cavitation compared with that of water cavitation. Liquid nitrogen cavitation consists of a crowd of smaller-scale (foggy) cavitation elements compared with water (foamy) cavitation in any circumstance, such as tip vortex cavitation and backflow vortex cavitation. These results have confirmed appearances such as “foggy” and “foamy,” as reported in Ref. [10], in a range of 0.01 ≤ σ ≤ 0.1 and 0 ≤ ϕ ≤ 0.09. When the same volume of cavitation occurs, the surface area of the finer liquid nitrogen cavitation is greater than that of water cavitation; thus, heat transfer between liquid nitrogen and its cavitation events must be more active than heat transfer between water and its cavitation events, which may be related to the thermodynamic effect. Further studies are currently in the planning phase.

## 4 Conclusions

In liquid nitrogen, which is representative of cryogenic fluids, and water, which is an ordinary fluid, the results of the presented unsteady visualization experiments for cavitation on a rotating inducer elucidated the characteristics of tip vortex cavitation and backflow vortex cavitation.

In both cryogenic and ordinary fluids, the following common characteristics were clarified. The physical meaning of the decreasing cavitation number σ was determined through the decreasing inlet static pressure pin, the increasing saturation pressure pv, and/or the increasing inlet velocity Win; cavitation occurs easily and extensively. With decreasing σ for the same flow coefficient ϕ, the thickness of each tip vortex cavitation event increases. Additionally, the thickness of each backflow vortex cavitation event increases with decreasing σ for the same ϕ. Even in these cases, it was confirmed that the number of backflow vortex cavitation events, revolving velocity, and revolving diameter do not change with decreasing σ for the same ϕ, as predicted in the theoretical analyses related to movements of backflow vortex cavitation on a rotating inducer, as reported in the previous literature by the author and colleagues.

Decreasing ϕ causes an increase in the head coefficient ψ based on the characteristics of the tested inducer; thus, the leak flow strengthens compared with the main flow. With decreasing ϕ for the same σ, the direction of the tip vortex axis is turned upstream, and the region where tip vortex cavitation occurs extends from not only the leading edge but also other areas. The tip vortex cavitation is convected and occurs and/or disappears locally at a local pressure less/greater than pv. Furthermore, with decreasing ϕ for the same σ, it was also confirmed that the number of backflow vortex cavitation events decreases, the revolution diameter around the inducer axis decreases, the height of backflow vortex cavitation events increases, and the thickness of each backflow vortex cavitation event increases, as predicted in the theoretical analyses related to movements of backflow vortex cavitation on a rotating inducer that were reported in the previous literature by the author and colleagues. Backflow vortex cavitation is convected and occurs and/or disappears locally.

In addition, it was confirmed that the inducer works correctly when cavitation bubbles enter between the inducer blades condenses and disappears and that cavitation breakdown occurs when the flow paths between the inducer blades are completely filled with cavitation bubbles.

However, the following different characteristics between liquid nitrogen and water were also clarified. Liquid nitrogen cavitation always consists of a crowd of smaller-scale (foggy) cavitation elements compared with water (foamy) cavitation in all cases, such as tip vortex cavitation and backflow vortex cavitation in a range of 0.01 ≤ σ ≤ 0.1 and 0 ≤ ϕ ≤ 0.09, as reported in the previous literature by the author and colleagues. In addition, each tip vortex cavitation event and each backflow vortex cavitation event in liquid nitrogen are thinner than those in water.

The author and colleagues are developing this visualization technology for cryogenic cavitation and aim to perform visualization experiments for cavitation on inducers in liquid hydrogen to develop a fuel pump for liquid-hydrogen-fueled aircraft and liquid hydrogen-transporting pumps to support a hydrogen society.

## Acknowledgment

The author acknowledges all the contributing former students at Tokyo Institute of Technology, Mr. Yuto Kurishita, Dr. Satoshi Kitano, Mr. Atsuhiro Tsunoda, Mr. Ichiro Takagi, Mr. Yuhei Sato, and Mr. Takuya Yukawa for their enthusiastic preparation and performance of the experiments, as well as the Precision and Manufacturing Center at Tokyo Institute of Technology for fabricating and adjusting the inducer section. The author is also grateful to Professor Takao Nagasaki (in the Department of Mechanical Engineering, Tokyo Institute of Technology) for providing opportunities to conduct this series of studies, Dr. Naoki Tani and Dr. Kazuki Niiyama (at JAXA) for their technical advice, and Professor Toshinori Watanabe (in the Department of Aeronautics and Astronautics, The University of Tokyo) for his support. In addition, the author thanks Dr. Shigeyuki Tomimatsu (at DMW Corporation) and Professor Teiichi Tanaka (Vice-president at Kumamoto National College of Technology) for their recommendations.

## Funding Data

• IHI Corporation (The Social Cooperation Program; Funder ID: 10.13039/501100004907).

• The Iwatani Naoji Foundation (Iwatani Science and Technology Research Grants; Funder ID: 10.13039/501100007656).

• The Ebara Hatakeyama Memorial Foundation (The Research Grant).

• The Japan Aerospace Exploration Agency (JAXA) (Collaborative Research) (Funder ID: 10.13039/501100004020).

• JSPS KAKENHI (Grant No. JP 18K03922; Funder ID: 10.13039/501100001691).

## Nomenclature

• $pd,in, pd,out$ =

inlet or outlet dynamic pressure

•
• $ps,in, ps,out$ =

inlet or outlet static pressure

•
• $psat(T)$ =

saturation pressure as a function of $T$

•
• $pt,in, pt,out$ =

inlet or outlet total pressure

•
• $pv$ =

vapor pressure

•
• $R$ =

•
• $Tin$ =

inlet temperature

•
• $Uin$ =

tangential velocity of the inducer tip

•
• $Vin$ =

inlet velocity in the inducer axial direction

•
• $Win$ =

inlet velocity on the rotating inducer blade coordinate system

•
• $η$ =

static pressure recovery ratio

•
• $θ$ =

rotation angle of the inducer

•
• $ρL$ =

liquid density

•
• $σ$ =

cavitation number

•
• $ϕ$ =

flow coefficient

•
• $ψ$ =

•
• $ω$ =

angular velocity of the inducer

## Footnotes

2

NAL is currently part of the Japan Aerospace Exploration Agency (JAXA).

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