Enhancement of the Thermodynamic Suppression Effect of Cavitation in a Hydrofoil With a Slit

In fluid machinery, where a liquid is used as the working fluid, cavitation often occurs when the flow accelerates and pressure decreases under the saturated vapor pressure. Cavitation is a high-speed gas–liquid two-phase flow that causes undesirable phenomena such as noise, erosion, and decline in the performance of the fluid machinery. Although the operation of fluid machinery should be avoided under cavitation conditions, downsizing and lightweight fluid machinery such as a rocket pump are forced to operate under these conditions because of the high rotation speed. Thus, several studies have focused on cavitation in the field of rocket pumps.

Cavity volume is suppressed in cryogenic fluids, such as liquid hydrogen and oxygen, which are used as propellants for liquid rockets; This suppression phenomenon is called the thermodynamic suppression effect of cavitation [1]. When the temperature around the cavity region decreases locally because of the latent heat of evaporation and the saturated vapor pressure decreases, the cavity volume becomes small owing to the suppression of evaporation and the decrease in the inner pressure of the cavity bubble with the decrease in local temperature. Here, the cavity shrinks further with an increase in evaporation. This is referred to as the self-suppression effect of cavitation. The thermodynamic suppression effect occurs not only in cryogenic fluids but also in hot water and refrigerants, and the suppression effect becomes stronger under the condition of a higher fluid temperature in the respective fluid because of the thermophysical properties, gas–liquid density ratio, gradient of the saturated vapor pressure against temperature, and so on. Franc et al. [2] conducted experiments using an inducer with a refrigerant as the working fluid and demonstrated that the cavity length was more suppressed in the refrigerant compared to that in water, and the length was more suppressed at higher mainstream conditions than that at lower temperatures.

It cannot be distinguished based solely on observation if a cavity is suppressed to its volume or if the cavity originally had the specific volume. Measuring the temperature inside the cavity is the only approach to determine if a cavity is suppressed because the thermodynamic suppression effect is attributed to the temperature drop inside the cavity. Therefore, several studies have focused on measuring the temperature in cavitating flows. For cryogenic flows, Hord et al. measured the temperature in the cavitation flow of liquid hydrogen and nitrogen in a Venturi tube [3] around a hydrofoil [4] and ogives [5] using a thermocouple. Niiyama et al. measured the temperature in a cavitating flow around a hydrofoil in liquid nitrogen using a diode sensor [6]. For a fluid at room temperature, Fruman et al. measured the temperature in the cavitating flow of refrigerant R-114 in a Venturi tube [7] and Watanabe et al. measured the temperature of a fluoric solvent in a convergent–divergent nozzle [8]; both studies used thermocouples on the wall surface. For hot water, Tagaya et al. measured the temperature in a cavitating flow around a hydrofoil from 100 to 140 °C using a thermistor on the hydrofoil surface [9]. Petkovsek et al. measured the unsteady temperature distribution in a cavitating flow in a Venturi tube at 95 °C [10], and Hosbach measured the temperature distribution of a cavitating flow in a microchannel [11] using infrared thermography from outside of the flow channels. Thus, temperature measurements were performed under various conditions ranging from cryogenic to high temperatures, and for internal and external flow fields. However, the temperature measurements were not conducted directly in the vapor phase inside a cavity, but on a wall, such as by installing a sensor on a surface inside the wall or by performing measurements from outside the sidewall in the case of thermography, which measures the temperature of a liquid phase. Further, the heat entering from the wall cannot be considered negligible during sensor measurements on a wall surface because ensuring complete heat insulation is difficult.

In our previous study, we developed a high accuracy technique for measuring the temperature inside a cavity by inserting a thermistor probe directly into the cavity from the sidewall [12]. The measurement technique showed that the temperature reduction inside the cavity ΔT was ∼1.3 °C at T_∞ = 140 °C [13], ∼0.3 °C at T_∞ = 80 °C, and 0.05 °C at T_∞ = 20 °C [14] in water under supercavitation conditions. In addition, a prediction method was developed for the vapor temperature in an unsteady cavitating flow using a temperature probe with a long duration time constant by solving inversely the lumped capacitance model [15].

Oscillation phenomena called cavitation instabilities occur when cavitation occurs in an axial-flow impeller called an inducer in a liquid rocket turbopump. These oscillation phenomena are undesirable because they can cause axial whirling, pulsation of the working fluid, or even fatal oscillations of the engine system. Therefore, suppressing these cavitation instabilities is essential to increase the reliability of rocket turbopumps. Thus far, many researchers have developed methods to suppress cavitation instabilities in rocket turbopumps. Our research group proposed a method to suppress the instabilities by cutting off a part of an inducer blade, which is then referred to as a slit inducer. The slit inducer can be combined with conventional suppression methods on the casing, such as stabilization devices [16], expansion of the casing diameter [17], J-grooves [18], baffle plates [19,20], and swirl breakers [21,22]. The effectiveness of slits in suppressing cavitation and cavitation instabilities has been confirmed in our previous studies. Iga et al. [23] analyzed the flow field from a two-dimensional numerical analysis of a three-blade flat-plate cyclic cascade with slits and showed that the slits can suppress cavitation instabilities via a mechanism in which the jet flows from the pressure side to the suction side through the slit, thereby breaking the regularity of the cycle of unsteady cavitation. Kobayashi et al. [24] conducted experiments using a single hydrofoil with a slit and showed that the slit reduces the cavity volume and changes the unsteady characteristics, which is the nondimensional frequency of cavitation. Kanamaru et al. [25] conducted experiments with different slit positions in an inducer and demonstrated that symmetric slits at the throat suppress the instability phenomenon. Kowata et al. [26] determined the optimal position and shape of the slit using a three-dimensional numerical simulation, suppressing cavitation instabilities without reducing the pump head as much as possible. Ishikawa et al. [27] showed that the suppression mechanism in the optical position and shape involves two cavities separated by the slit becoming two independent short cavities, followed by the oscillation of the cavity being drastically suppressed.

Although we confirmed the suitability of the slit inducer for suppressing the cavitation oscillation, there is a need to demonstrate that the slit does not inhibit the thermodynamic suppression effect of cavitation if it is installed in the inducer of a rocket turbopump. Therefore, in this study, the thermodynamic suppression effect of cavitation is examined in an experiment involving a single hydrofoil with and without a slit in a hot-water cavitation tunnel as a preliminary step toward investigating a slit inducer. The cavity occurrence region and the variation of cavity length are investigated by changing the mainstream water temperature T_∞ between 50 and 90 °C and the angle of attack of the hydrofoil from 8 to 14 deg. Additionally, the temperature drops and the distribution inside the cavity are measured under T_∞ = 90 °C to investigate the small difference in the thermodynamic suppression effect that does not appear in the cavity length. The temperature inside the cavity is measured directory by our direct temperature measurement method [14] and the vapor temperature in unsteady cavitation is estimated using our unsteady temperature estimation method [15]. From the results, the cavity suppression amount in the region downstream of the slit inside the cavity is examined under the condition of unsteady cavitation in the hydrofoil with a slit.

A high-temperature water cavitation tunnel at the Institute of Fluid Science, Tohoku University was used as the cavitation tunnel for the experiment [14]. An overview of the cavitation tunnel is shown in Fig. 1. The tunnel includes a settling tank, pressure tank, test section, and circulation pump. The pressure tank is connected to a vacuum pump through a vacuum tank, compressor, and atmospheric release valve for the pressure control of the entire experimental apparatus. The mainstream velocity is controlled by the rotational speed of the circulation pump, which sends water at 0.5 m³/min. The working fluid is water, and the chlorine is removed using a filter. The temperature of the working fluid can be changed from room temperature to 140 °C using electric heaters.

In this study, the mainstream temperature T_∞ was changed at 50 and 90 °C. The Reynolds numbers Re in each mainstream temperature were kept constant at 7.38 × 10⁵ by changing the mainstream velocity U_∞ to 13.6 and 8.0 m/s at each T_∞ condition for eliminating the promoting effect of the occurrence of cavitation in increasing Re, which is called the cavitation scale effect. At that time, Froude numbers are 25 and 15 at each U_∞, but the influence is negligible because of no free surface. Then, only the thermodynamic suppression effect is compared against the different mainstream temperatures T_∞. The dissolved oxygen content in the working fluid was measured using a dissolved oxygen meter, and the dissolved oxygen content was ∼30% under atmospheric conditions.

The test section of the cavitation tunnel is shown in Fig. 2. The test section is a rectangular channel (20 mm wide, 30 mm high, and 330 mm long) and has an observation window in the cross-sectional direction through which the appearance of cavitation on the test body is observed. Cavitation is visualized using a high-speed video camera (FASTCAM Mini AX50 type 170 K-M-8GB, PHOTRON Ltd.) with a frame rate of 4500 fps and a shutter speed of 1/100,000 s. The test body is a NACA0009 hydrofoil, which has a chord length of 30 mm and a span width of 20 mm; further, it is made of SUS303. It should be noted that the blockage ratio for this hydrofoil is 13.9% at the angle of attack 8 deg and 24.2% at 14 deg, which is rather large compared to a conventional cavitation tunnel. Two hydrofoils were used in this study: the NACA0009 hydrofoil with and without a slit. The geometry of the slit is illustrated in Fig. 3. The center of the slit position on the suction side of the hydrofoil is 12 mm, the slit position on the pressure side is 3 mm from the leading edge of the chord, and the slit width is 5 mm. The slit geometry is determined from a previous study [24].

where p_∞, p_sat, and ρ_∞ are the mainstream pressure, saturated vapor pressure, and density of the liquid at the mainstream temperature, respectively. p_∞ used to define the cavitation number is measured using a pressure transducer installed 110 mm upstream from the center of the hydrofoil. When operating the cavitation tunnel, the pressure is increased from the low-pressure condition because the condition of incipient cavitation varies widely, and the desinent condition has reproductivity [28,29].

A direct measurement method for the temperature inside the cavity is used in this study. In this method, a temperature probe is inserted inside the cavity [14]. The temperature was measured using a high-strength thermistor (No. 13 Fμ, SEMITEC Co., Ltd.) with a 0.5 mm diameter; this was also used in a previous study [15]. A schematic of the thermistor used in this study is shown in Fig. 4. The thermistor is inserted into an SUS304 pipe (diameter = 1.59 mm) to prevent bending and installed from the side wall of the tunnels. The positions of the three thermistors are shown in Fig. 5. Two are placed on the suction side of the hydrofoil at the cavity temperature, and one is placed upstream of the hydrofoil on the pressure side at the mainstream temperature. We believe that the influence of the temperature probe on cavitation is small because we found that additional cavitation does not occur around the temperature probe when a sheet cavity length is shorter than the position of the temperature probe in a cavitation condition [14]. The sampling frequency of the digital multimeter was 1100 Hz. Each thermistor was calibrated using a handmade temperature calibration device [15].

The expanded uncertainty of the temperature probe used in this study and the time constant are 17 mK and 50 ms, respectively. The time constant is large, and therefore, this probe cannot follow the time variation of the temperature in the unsteady cavitating flow such as sheet/cloud cavitation, where cloud cavities are released cyclically at roughly every 20 ms and the probe is exposed alternately to vapor and liquid. The vapor temperature in unsteady cavitation is estimated using the unsteady temperature estimation method devised by our group [15], wherein the vapor temperature in the unsteady cavitation is estimated using the inverse solution of the heat balance equation of the temperature probe based on the lumped capacitance model.

Cavitation patterns on the NACA0009 hydrofoil with and without slit were visualized at α = 6–14 deg and compared between mainstream temperatures T_∞ = 50 and 90 °C. The typical aspects of each cavitation pattern for a hydrofoil without a slit are shown in Fig. 6. In this hydrofoil, three cavitation patterns were observed within the range of the present a, based on a decrease in cavitation number σ: attached sheet cavitation (Fig. 6(a)), sheet/cloud cavitation (Fig. 6(b)), and supercavitation (Fig. 6(c)). Figures 7(a) and 7(b) show cavitation pattern maps of hydrofoils without a slit at T_∞ = 50 and 90 °C, respectively. The desinent condition at α = 14 deg in T_∞ = 50 °C cannot be detected in the operation in this cavitation tunnel. The cavitation desinent point moves slightly to a lower $σ$ at 90 °C in Fig. 7(b) compared to that at 50 °C, as shown in Fig. 7(a), which implies that the thermodynamic suppression effect at 90 °C is larger than that at 50 °C. The occurrence σ range of the attached sheet cavitation becomes large and the occurrence of the sheet/cloud cavitation is suppressed at α = 8 and 10 deg at T_∞ = 90 °C.

Figures 8(a) and 8(b) show the cavitation pattern maps for the hydrofoil with a slit at T_∞ = 50 °C and 90 °C, respectively. In the hydrofoil with a slit, the cavitation desinent point moves slightly to a lower σ at 90 °C compared to that at 50 °C. Comparing hydrofoils with and without a slit at T_∞ = 50 °C revealed that the desinent point showed an increasing trend with an increase in the angle of attack in the hydrofoil without the slit (Fig. 7(a)); however, it decreased at α = 14 deg in the hydrofoil with a slit (Fig. 8(a)), which is attributed to the effect of only the slit. The desinent cavitation point at which the increased desinent cavitation number decreases moves to a lower angle of attack at T_∞ = 90 °C compared to that at 50 °C in the hydrofoil with the slit, i.e., α = 12–10 deg (Fig. 8(b)). The thermodynamic suppression effect on the desinent cavitation was also enhanced in the hydrofoil with a slit at a higher angle of attack.

Cavitation on the hydrofoils was visualized using a high-speed video camera and the cavity length was estimated. For sheet/cloud cavitation, the sheet cavity lengths were measured immediately before the release of the cloud cavity and averaged over ten break-off cycles. For attached sheet cavitation and supercavitation, which are quasi-steady cavitation, the cavity length was defined by averaging for a time during 1 s. The error bars in each figure show the maximum and minimum cavity lengths among the ten cycles in sheet/cloud cavitation, and the maximum and minimum cavity lengths throughout the duration in attached sheet cavitation and supercavitation. The cavity length was estimated on a horizontal line.

Figures 9,–11 compare cavity lengths between T_∞ = 50–90 °C. The difference corresponds to the increase in the thermodynamic suppression effect from 50 to 90 °C. Figures 9–11 show the results for α = 8, 12, and 14 deg, respectively. As shown in Figs. 9(a), 10(a), and 11(a), the suppression of the cavity length by the thermodynamic effect decreases with an increase in the angle of attack. At α = 14 deg in Fig. 11(a), there is no significant difference between the cavity lengths at 50 and 90 °C for σ < 2.0 although the cavity length at 90 °C is shorter than that at 50 °C for σ > 2.0. In the hydrofoil with a slit (Figs. 9(b), 10(b), and 11(b)), the suppression of cavity length by the thermodynamic effect decreases at α = 12° and later increases at α = 14 deg. In addition, as shown in Figs. 9 and 10, the suppression amount is approximately the same between the hydrofoils with and without slits at α = 8 and 12 deg. At α = 14 deg (Fig. 11(b)), the cavity length at T_∞ = 90 °C becomes largely shorter than that at T_∞ = 50 °C for σ < 2.0. The characteristics of the suppression of cavity length at a lower σ region at α = 14 deg is opposite between hydrofoils with and without a slit, where the suppression effect is not observed in the hydrofoil without the slit and becomes obvious in the hydrofoil with the slit. In the σ condition under 2.0, sheet/cloud cavitation occurs in both hydrofoils with and without the slit at α = 14 deg, which is shown in the cavitation map presented in Figs. 7 and 8. The promotion effect of the slit on thermodynamic suppression effect for cavity length appears under the condition of unsteady sheet/cloud cavitation at a high angle of attack condition, which is, α = 14 deg for the present NACA0009 hydrofoil. Further, there was no weakening effect of the slit on the thermodynamic suppression at α = 8 deg and 12 deg in the NACA0009 hydrofoil. The above results of thermodynamic suppression effect in the cavity length are summarized as shown in Table 1.

Equation (2) can be estimated using the measured temperature drop inside a cavity in a flow field ΔT = T_∞ − T_cav. The Δσ estimated using ΔT at 90 °C does not indicate the decrease in s at T_∞ = 90 °C from 50 °C; however, it is the decrease from the condition in which there is no suppression, which may lower the temperature beyond 50 °C. In this study, Δσ was estimated at two positions: Pb_up, which is at the slit outlet, and Pb_dwn, which is at the downstream of the slit outlet, as shown in Fig. 5.

Figure 12 shows the cavity suppression amount Δσ at T_∞ = 90 °C and α = 12 deg in the upstream probe Pb_up in (a) and downstream probe Pb_dwn in (b), respectively. Triangle plots indicate the cavity on the hydrofoil without a slit, and the circle plots indicate that on the hydrofoil with a slit. The outlined plots show unsteady sheet/cloud cavitation, wherein the vapor temperature inside the cavity is estimated using the unsteady temperature estimation method [15], while the filled plots show quasi-steady cavitation, such as attached sheet cavitation and supercavitation. The values of the temperature drop ΔT under the uncertainty of the present measurement method need to be considered as zero; however, they should still be plotted to understand cases in which the thermodynamic suppression effect can be considered as zero or the cavity does not pass the probe at the probe location even though the two scenarios cannot be distinguished. Figure 12 indicates that there is a distribution in the temperature inside the cavity and the upstream temperature is higher than that downstream because Δσ in Pb_up in Fig. 12(a) is larger than that in Pb_dwn in Fig. 12(b). In addition, the amount of suppression in the hydrofoil with the slit (circle) decreases compared to that in the hydrofoil without the slit (triangle) in all σ regions for both Pb_up in (a) and Pb_dwn in (b). The suppression amount increases under the condition of unsteady cavitation (outlined) in the hydrofoil with a slit (circle) in the downstream region of the slit, as shown in Pb_dwn in (b).

The variation of the temperature drops inside the cavity ΔT from Pb_up to Pb_dwn is plotted in Fig. 13 for hydrofoils with and without a slit to investigate what happened in the downstream region of the slit. As shown in Fig. 13, ΔT had a negative slope in almost all cases. This implies that the decreased temperature inside the cavity recovers when flowing from Pb_up to Pb_dwn. The recovery of the decreased temperature was attributed to the heat transfer from the liquid phase to the gas phase around the cavity surface region. However, a few cases have a positive slope, which is the case in the unsteady cases (outlined) in the hydrofoil with a slit (circle). In this case, the temperature inside the cavity decreased further with the flow from Pb_up to Pb_dwn. The increase in the region with a low temperature inside the cavity indicated the promotion of the thermodynamic suppression effect. The cause of the additional temperature drop was considered to be the evaporation or adiabatic expansion of the cavity. Cases that have a positive slope in Fig. 13 correspond to cases in which the suppression amount Δσ increases in the downstream region of the slit in Fig. 12(b).

The gradient of the temperature drop ΔT between Pb_up and Pb_dwn shown in Fig. 13 is estimated as gradΔT and sorted against σ in Fig. 14. A negative value of gradΔT implies that the temperature drop upstream of Pb_up recovers with the flow of Pb_dwn. However, a positive value of gradΔT implies that the decreased temperature upstream of Pb_up decreases more as it flows to Pb_dwn. Subsequently, in the cavity with a higher gradΔT, the low-temperature region spreads inside the cavity. An increase in gradΔT corresponds to an increase in the suppression effect. For the overall variation of gradΔT in both hydrofoils with and without the slit shown in Fig. 14, gradΔT is close to zero in the higher σ region, where the temperature probes are considered to not be covered by the cavity, or the temperature drop is considered to be very small. The cavity developed with a decrease in σ, and then, the temperature drop in the upstream region inside the cavity increased and the temperature recovery in the downstream region inside the cavity increased. Subsequently, gradΔT decreased according to the decrease in s in the lower σ region. Contrary to the overall variation, in the hydrofoil with a slit, gradΔT increased according to the decrease in s under the condition of unsteady cavitation (outlined circle). The gradΔT becomes finely positive at the minimum σ during the condition of unsteady cavitation in the hydrofoil with a slit. At σ of the positive gradΔT, the low-temperature region may spread around Pb_dwn and the thermodynamic suppression effect remains in the hydrofoil with a slit. Therefore, the promotion effect of the slit on the thermodynamic suppression of cavitation under the condition of unsteady cavitation increased with a decrease in σ.

The mechanism of the increase in the promotion effect of the slit on thermodynamic suppression based on the decrease in σ is schematically shown in Fig. 15. The unsteady cavitation that occurred under the present flow conditions was sheet/cloud cavitation, wherein the break-off cycle is repeated, causing the sheet cavity to develop and break and the cloud cavity to be released (Fig. 6(b)); the sheet cavity develops again. The upper row in Fig. 15 shows the time duration for which the sheet cavity does not cover the slit, and the lower row shows the duration for which the sheet cavity covers the slit. The sheet cavity covers the slit, although in the condition of higher σ, because sheet/cloud cavitation requires some cavity length, which is known to be more than roughly 75% of the chord length experimentally [30] and theoretically [31], in which the cavitation compliance becomes infinite in the condition of 78% of the chord length.

The jet flow through the slit from the pressure side to the suction side of the hydrofoil is caused by the pressure difference between the two sides. When the sheet cavity does not cover the slit (upper left in Fig. 15), the slit jet caused a shear vortex and additional evaporation in the downstream region of the rear end of the sheet cavity. Subsequently, the temperature drop caused by the additional evaporation resulted in the thermodynamic suppression of cavitation. As shown in the upper right of Fig. 15, when the pressure field around the hydrofoil decreases with a decrease in σ, the flowrate of the slit jet did not change because the pressure difference between the pressure and suction sides of the slit did not change. However, the amount of evaporation increased because the pressure on the suction side decreased. Consequently, in the time duration in which the sheet cavity does not cover the slit, thermodynamic suppression increased under the low σ condition.

The slit jet was caused by the pressure difference between the pressure side and saturated vapor pressure when the sheet cavity covered the slit (lower left in Fig. 15). The slit jet was discharged inside the sheet cavity and the cavity surface region was disturbed by the jet flow. The heat transfer between the mainstream to the cavity through the cavity surface region was promoted, and the temperature inside the cavity decreased once it was recovered by the heat transfer. As shown in the lower right of Fig. 15, when σ decreases in the duration, the pressure on the pressure side decreases; however, the pressure on the suction side does not change because the saturated vapor pressure is constant. Subsequently, the pressure difference in the slit and the flowrate of the slit jet decreases. Heat transfer occurs on weekends, the temperature recovery is delayed, and then the low-temperature region spreads. Consequently, in the time duration, thermodynamic suppression increases under low σ conditions. Therefore, thermodynamic suppression increases through the sheet/cloud cavitation cycle according to the decrease in σ.

In this study, the thermodynamic suppression effect of cavitation was examined in an experiment involving a NACA0009 hydrofoil with and without a slit in hot water as a preliminary step for investigating slit inducers. The cavity occurrence range and the length were investigated by changing the mainstream water temperature T_∞ between 50 and 90 °C and the angle of attack of the hydrofoil from 8 to 14 deg. As a result, the thermodynamic suppression effect on the desinent cavitation was enhanced by a slit on the hydrofoil under a relatively high angle of attack condition (α ≥ 12 deg) in the present hydrofoil. The thermodynamic suppression effect on the cavity length was enhanced by a slit on the hydrofoil under the condition of unsteady sheet/cloud cavitation at a higher angle of attack condition (α = 14 deg) in the present hydrofoil. Further, there was no reduction of the thermodynamic suppression effect by a slit at α = 8 deg and 12 deg in the present hydrofoil.

Additionally, the temperature drop and the distribution inside the cavity were measured at α = 12 deg under T_∞ = 90 °C to investigate the small difference in the thermodynamic suppression effect that does not appear in the averaged maximum cavity length. It was found that the cavity suppression amount in the region downstream of the slit inside the cavity increased under the condition of unsteady cavitation in the hydrofoil with a slit. The enhancement of the thermodynamic suppression of cavitation in the hydrofoil with a slit under unsteady cavitation conditions increased with decreasing σ. This is considered because the evaporation in the shear vortex downstream of the sheet cavity caused by the slit jet increased owing to the decrease in pressure on the suction side, and the heat transfer of the cavity surface region caused by the disturbance of the slit jet was weakened owing to the decrease in the flowrate of the slit jet.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.