## Abstract

The thermodynamic suppression effect of cavitation arising in a NACA0015 single hydrofoil is experimentally investigated in water at mainstream temperatures of T = 20 °C to 140 °C in the present study. The cavity length at T = 140 °C is shorter than that at T = 20 °C at a constant cavitation number for all cavity patterns from inception to supercavitation. On the other hand, the cavity length at T = 80 °C is slightly shorter than that at 20 °C in a certain region in which unsteady sheet-cloud cavitation occurs. This indicates that the thermodynamic suppression effect appears easily in unsteady cavitation. In addition, the temperature reduction inside cavities in water is accurately measured using thermistors, which are inserted from the sidewall directly into the cavity. The temperature measurement is performed at a mainstream temperature of less than 80 °C due to limitation of calibration for the sensor. The temperature reduction at 140 °C is then predicted from the measured cavity length. It is shown that the temperature reduction inside the cavity is approximately ΔT = 0.3 °C at T = 80 °C and ΔT = 0.05 °C at T = 20 °C under supercavitation conditions. The predicted temperature reduction inside the cavity is ΔT = 1.1 K at T = 140 °C under supercavitation conditions. Finally, Fruman's prediction equation for ΔT is examined by fitting to the measured and predicted ΔT values with assuming a volume coefficient of evaporation CQ as a fitting parameter.

## Introduction

The cavity volume is suppressed by an increase in the mainstream temperature for the same cavitation number. This phenomenon is called the thermodynamic suppression effect of cavitation [1]. This effect is favorable for pumps because the suction performance of the pump improves. The suppression mechanism is known, and there are some parameters that express the suppression effect for an attached steady cavity with a clear surface in a steady flow, but we still cannot quantitatively predict the magnitude of the suppression effect in an actual cavitating flow field. Therefore, there is a need to investigate the thermodynamic effect of cavitation in a flow field.

Cavity volume suppression takes place when evaporation occurs in the cavity region, the local temperature decreases around the cavity due to the latent heat of evaporation, and the local saturated vapor pressure decreases. The cavity then shrinks due to a decrease in the evaporation rate and in the inner pressure. The thermodynamic suppression effect is a self-suppression effect because as evaporation increases, evaporation is increasingly suppressed. The thermodynamic suppression effect is intensified based on a change in the thermophysical properties when the mainstream temperature increases. The effect becomes obvious in cryogenic fluids such as liquid hydrogen and liquefied natural gas, as well as in hot water and refrigerant. In past research, some parameters that express the extent of the thermodynamic effect were proposed. Stephanoff proposed the B-factor [1], which is the oldest parameter. Brennen proposed the thermodynamic parameter Σ [2], which was derived from the equation for bubble dynamics. Moreover, Σ was transformed to a nondimensional parameter by Brennen [3] and was termed Σ* by Franc [4]. Note that Σ* takes into account the time of heat transfer. Kato proposed the Y and Z factors [5], which were derived from the evaporation rate on a sheet cavity surface.

The thermodynamic suppression effect of cavitation has been studied for liquid-propellant rocket engines because it is directly connected to an improvement in suction performance of the turbopump in a rocket engine. The thermodynamic suppression effect in the inducer of a rocket engine, which is an axial flow impeller, was studied by Ball et al. using liquid hydrogen [6], by Franc et al. using a refrigerant [4], and by Ito et al. using liquid nitrogen [7], where the suppression effect was estimated based on the cavity length and volume in experiments. The effect in a centrifugal pump was also studied by Tanaka [8] using liquid nitrogen. The thermodynamic suppression effect, especially for cavitation instabilities such as rotating cavitations and cavitation surges, was investigated by the research group of the author [911] in a rocket inducer using liquid nitrogen. As described above, many experimental studies have been reported in the past for liquid-propellant rocket engines. However, unknown factors still remain because of the lack of basic research in simple flow fields.

When cavitation occurs with a certain volume in a cryogenic fluid, the magnitude to which it is suppressed cannot be determined because we do not know the original volume. The only means to determine the magnitude of suppression is to measure the temperature reduction inside the cavity. Several past studies were performed to measure the temperature in a cavitating flow field. For example, for a cryogenic flow, Hord et al. measured the temperature in a cavitation flow in a venturi tube [12] around a hydrofoil [13] and ogives [14] in liquid hydrogen and nitrogen using a thermocouple. Niiyama et al. measured the temperature in a cavitation flow around a NACA16-012 hydrofoil in liquid nitrogen using a diode sensor [15]. For a fluid at room temperature, Fruman et al. measured the temperature during cavitation in a venturi tube using refrigerant R-114 [16], and Watanabe et al. performed measurements using a fluoric solvent [17], both using thermocouples. For hot water, Tagaya et al. measured the temperature during cavitation on a hydrofoil from 100 °C to 140 °C using a thermistor on the hydrofoil surface [18]. Petkovsek et al. measured the unsteady temperature in a venturi tube at 95 °C using infrared thermography [19]. Hosbach measured distributions of fuel temperature in microchannel using also infrared thermography [20]. Thus, temperature measurements have been carried out under various conditions, from cryogenic to high temperatures, and for internal and external flow fields. However, all temperature measurements were not conducted directly in the vapor phase inside a cavity but were conducted on a wall, such as by installing a sensor on a wall, or by measurement from outside a sidewall in the case of using thermography, which often measures the temperature of a liquid phase. In addition, the heat entering from the wall cannot be considered to be negligible during wall sensor measurements because ensuring complete heat insulation is difficult. The temperature decrease inside a cavity is considered to be very small, at less than 1 K. Therefore, a highly accurate sensor is required. In our previous study [21], we developed a high-accuracy measurement technique for the temperature inside a cavity by inserting a thermistor probe directly into the cavity. In the present study, using this technique, the thermodynamic suppression effect of cavitation in a simple flow field in hot water is investigated experimentally.

In water, the magnitude of thermodynamic suppression effect widely varies from room temperature, where it can be neglected, to hot water, where it is similar to that in cryogenic fluids, although the effect is always large in cryogenic fluids. For example, the thermodynamic situation for water at 140 °C corresponds to that for liquid methane at 110 K, liquid nitrogen at 77 K, and liquid hydrogen at 15 K. We, therefore, consider that the magnitude of the effect can be discussed in water by comparison of cavitation between room-temperature water and hot water. In the present study, the thermodynamic suppression effect of cavitation is investigated through the experiment in a high-temperature water cavitation tunnel. Cavitation on a NACA0015 single hydrofoil is investigated in terms of the cavity aspect and length at the mainstream temperature T from room-temperature water at 20 °C to hot water at 140 °C. The aspect is compared with that for the same hydrofoil in liquid nitrogen at T = 76 K, in which the condition of thermodynamic suppression is close to that for water at 140 °C. In addition, the temperature reduction inside the cavities in water is directly measured by the high-accuracy measurement technique developed by the authors [21]. The temperature measurement is performed at mainstream temperatures from 20 °C to 80 °C due to the limitation of temperature in the calibration system. The temperature reduction inside the cavities at T = 140 °C, which is not measured in the present study, is predicted based on the measured cavity length. Finally, an existing quasi-empirical equation for temperature reduction inside a cavity is compared to the measured and predicted temperature reductions, and the results are discussed.

## Experimental Facility

A high-temperature water-cavitation tunnel was built at the Institute of Fluid Science at Tohoku University for research on the thermodynamic suppression effect of cavitation. An overview is shown in Fig. 1. The test section is 30 mm in height, 20 mm in width, and 330 mm in length, which is small for a cavitation tunnel, but control of the mainstream temperature is easier. The flow rate is 700 L/min. The normal operating pressure is 0.51 MPa, and the normal operating temperature is 140 °C. The maximum designed pressure is 1.37 MPa, and the maximum designed temperature is 160 °C. The material is stainless steel. The mainstream is heated by electric heaters, and the temperature is controlled within ±0.1 °C using a microcomputer. To study conditions ranging from no cavitation to supercavitation from room temperature to high temperature, a compressor and a vacuum pump are connected to the pressure tank. The working fluid is tap water passed through activated carbon and degassing filters. In the activated carbon filter, the chloride-ion concentration is reduced to below 1 ppm to prevent stress corrosion cracking under high-temperature operation. In the degassing filter, the dissolved oxygen content is reduced to 30% under room-temperature and atmospheric-pressure conditions.

Fig. 1
Fig. 1
Close modal
In this tunnel, the cavitation experiment in water can be performed from room temperature to 140 °C. The thermodynamic parameter Σ [2] for several fluids is shown in Fig. 2. Here, Σ is expressed by the following equation, and is an index of the magnitude of the thermodynamic suppression effect of cavitation
$Σ=ρv2L2ρl2CplT∞al [m/s3/2]$
(1)

where L is the latent heat, Cpl and αl are the isobaric specific heat and thermal diffusivity in the liquid phase, respectively, ρl and ρv are the densities of the liquid and vapor phases, , respectively, and T is the mainstream temperature. Figure 2 shows that Σ for water at 140 °C corresponds to that for liquid methane at 110 K, liquid nitrogen at 77 K, and liquid hydrogen at 15 K. The advantage of using water in the experiment is that Σ can be changed over a wide range using a single fluid. Then, we can compare the cavitation conditions with and without the thermodynamic suppression effect for one type of fluid. The Σ range in the present experiment for water is shown in Fig. 2 by the gray band , which run from 3.9 × 100 to 2.8 × 104 m/s3/2 .

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

Figure 3 shows the distribution of the mainstream velocity in the center of the test section, measured in the vertical direction using a laser Doppler anemometer (FlowLite 2D, Dantec Dynamics) without the test body [21]. The response frequency of the laser Doppler anemometer was 350 Hz and the sensitivity deviation was 5%. As the result, the velocity was confirmed to be uniform. When the mainstream velocity U changed from 8 to 12 m/s, a boundary layer developed on the upper and lower walls, and the thickness reached 15% of the test section height at U = 12 m/s. The vertical distribution of the turbulence intensity was measured using the same laser Doppler anemometer at U = 10 m/s, and the results are shown in Fig. 4. The turbulence intensity was approximately 2% in the mainstream in the present tunnel. The resonance frequency of the tunnel, which was filled with room-temperature water, was determined by a hammering test using an acceleration sensor in the test section at a sampling frequency of 1 kHz. Figure 5 shows that the resonance frequency of the tunnel is approximately 44 Hz.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

In the experiment, the mainstream velocity is measured by an electromagnetic flowmeter, the mainstream pressure is measured in the upstream region in the test section, and the mainstream temperature is measured at the settling tank, which is downstream of the test section.

## Temperature-Measurement Method

In the present study, the temperature inside the cavity is measured at a mainstream temperature of less than 80 °C due to the limitation of temperature in the present calibration system for the probe. Generally, we consider that no obvious thermodynamic effect will appear at a water temperature of 80 °C [1]. In addition, it has been reported that an inverse thermodynamic suppression effect appears in cavity volumes at an approximate water temperature up to 80 °C [22] or up to 55 °C [23]. Therefore, in the present study, a small temperature change is expected to be detected using the high-accuracy measurement technique in such a flow field.

In a conventional measurement technique of temperature in a cavitation flow, a temperature sensor is installed on the body surface or the wall. On the other hand, in the present study, a temperature probe is inserted directly into the flow field from the sidewall of the test section. Then, the temperature inside a cavity can be directly measured. A thermistor (Nikkiso-Thermo Co., Ltd. N317/BR14KA103K/23300/RPS/3/SP, Japan) is used for the measurement. The temperature-sensitive part is a thermistor device in a 0.6-mm diameter polyimide tube filled with epoxy resin. In order to increase the signal intensity, the temperature probe is made by inserting the thermistor in a stainless-steel pipe with an outside diameter of 2 mm and a thickness of 0.5 mm while allowing the temperature-sensitive part to remain outside of the pipe. The temperature probe is inserted from the sidewall into the test section, and the tip of the probe is located at the center of the span, as shown in Fig. 6. Each thermistor is calibrated using a quartz thermometer (DMT-610B, Tokyo Denpa Co. Ltd., Japan). The electrical resistance of the thermistor is measured by a digital multimeter (DMM4040, Tektronix Co., Ltd.,Japan). The extended uncertainty of the thermistor with an inclusion coefficient of 2 is estimated to be 0.032 °C at T = 80 °C, based on the accuracy of the quartz thermometer, the temperature dependence, the long-term stability of the digital multimeter, the difference between the calibration curve and the quartz thermometer temperature, and the variation in repeated measurement. The budget sheet for the uncertainty estimation is listed in Table 1. The error bars in the temperature plots presented later are based on the extended uncertainty.

Fig. 6
Fig. 6
Close modal
Table 1

Example of budget sheet for uncertainty estimation for temperature probe (T = 80 °C, σ = 1.6, U = 8 m/s)

Reason of uncertaintyValue ±DivisorSensitivity coefficientStandard uncertainty
Dispersion in the iteration of measurements0.0045114.5 × 10−3
Accuracy of the quartz thermometer0.025√311.5 × 10−2
Difference between the calibration curve and quartz thermometer temperature0.006√313.5 × 10−3
Temperature dependence of digital multimeter0.176√32.3 × 10−22.4 × 10−3
Long-period stability of digital multimeter0.26√32.3 × 10−23.5 × 10−3
Reason of uncertaintyValue ±DivisorSensitivity coefficientStandard uncertainty
Dispersion in the iteration of measurements0.0045114.5 × 10−3
Accuracy of the quartz thermometer0.025√311.5 × 10−2
Difference between the calibration curve and quartz thermometer temperature0.006√313.5 × 10−3
Temperature dependence of digital multimeter0.176√32.3 × 10−22.4 × 10−3
Long-period stability of digital multimeter0.26√32.3 × 10−23.5 × 10−3

Combining standard uncertainty = 0.016 °C.

Extended uncertainty (Inclusion coefficient 2) = 0.032 °C.

In order to estimate the influence of heat admission from the tunnel sidewall, a steady heat-transfer analysis was performed using the solidworks 3D cad software [24]. The calculation field is shown in Fig. 7. The temperature probe is in a steady vapor flow at 6 m/s, which means that it is constantly surrounded by a cavity. The mainstream temperature is 80 °C, which corresponds to the tunnel wall temperature. The vapor temperature is assumed to be 79 °C, which corresponds to the temperature inside the cavity. From the calculation results, the measured temperature at the position of the sensor located 1 mm inside the tip of the thermistor is 79.04 °C, which means that the error is 0.04 °C. At this time, the measured temperature reduction is 0.96 °C, and the error is 4%. During unsteady cavitation, where the probe is alternately exposed to water and vapor, the influence of heat admission from the tunnel sidewall decreases. In addition, in the present heat-transfer analysis, the temperature depression is assumed to be as large as 1 °C, so that the actual heat conduction error is smaller. Hence, the heat condition error is estimated to be less than 4% of the measured temperature reduction.

Fig. 7
Fig. 7
Close modal

The sampling frequency during the temperature measurements is 0.5 Hz, which is limited by the digital multimeter. The time constant is 5 s, which is estimated from the transient heat-transfer analysis. This is considered to be large, which means that a 10-Hz order temperature fluctuation in the unsteady cavity flow cannot be measured. Therefore, all temperature data are time averaged over 20 s. Furthermore, the time-averaged temperature in an unsteady cavity flow is not strictly the average value because the measured unsteady temperature does not follow the unsteady flow in which vapor and liquid alternatively cover the probe due to the large time constant. The estimation method of the unsteady temperature in unsteady cavitation is now under development by the authors.

In our past study, it was reported that the temperature distribution inside the cavity in the rear half part is roughly constant [25]. Then the temperature in one position is measured as a reference value of the temperature inside the cavity in this study, although the temperature may be lower around the leading edge of the cavity because of evaporation and higher around the trailing edge of the cavity because of condensation.

The insertion positions for the temperature probes are shown in Fig. 8. The probes for the temperature inside the cavity Tcav are installed 37 mm downstream from the hydrofoil leading edge. The probe for the mainstream temperature T is 5 mm above the lower tunnel wall. Figure 9 shows the time evolution of the aspect of unsteady sheet-cloud cavitation when the temperature probe is inserted. We found that additional cavitation does not occur around the temperature probe when the sheet cavity length is shorter than the position of the temperature probe, although it somewhat disturbs the flow field downstream from the hydrofoil.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

## Experimental Conditions

In the present experiment, the mainstream temperature is varied from 20 °C to 140 °C. The experimental conditions for the water cavitation tunnel are listed in Table 2. The experiment is performed at a hydrofoil chord length of C =40 mm (blockage ratio 0.33) at a mainstream velocity of U = 8 m/s, an angle of attack of 12 deg, and various values of the mainstream temperature T and pressure p. The definitions of the Reynolds number Re- and the cavitation number σ are as follows:

Table 2

Experimental conditions and corresponding thermodynamic parameters in water cavitation tunnel

Hydrofoil geometryNACA0015
Chord length, C40 mm
Mainstream velocity, U8 m/s
Angle of attack12 deg
Mainstream temperature, T20 °C to 140 °C
Thermodynamic parameter, Σ3.90 m/s3/2 to 2.8 × 104 m/s3/2
Reynolds number, Re3.2 × 105 to 1.5 × 106
Hydrofoil geometryNACA0015
Chord length, C40 mm
Mainstream velocity, U8 m/s
Angle of attack12 deg
Mainstream temperature, T20 °C to 140 °C
Thermodynamic parameter, Σ3.90 m/s3/2 to 2.8 × 104 m/s3/2
Reynolds number, Re3.2 × 105 to 1.5 × 106
Reynolds number
$Re=U∞Cν∞$
(2)
Cavitation number
$σ=p∞−pv1/2ρ∞U∞2$
(3)

where ρ and ν are the density and kinetic viscosity, respectively, of the mainstream, and pv is the saturated vapor pressure at the mainstream temperature T. Please note that Re- also changes in the experiment by changing T∞.

## Results and Discussion

### Aspects of Cavitation.

The characteristics of the cavity aspects are compared for room-temperature water and hot water. The cavity aspects are visualized by a high-speed video camera (FASTCAM Mini AX50, Photron Co., Ltd., Japan) at a frame rate of 2000 fps and shatter-speed of 1/150,000 s with a metal halide lamp (HVC-UL, Photron Co., Ltd., Japan), and a microlens (AF Micro-Nikkor 60 mm f/2.8D, Nikon Co., Ltd., Japan). The special resolution of the images is 9.66 pixel/mm. The typical aspects for the conditions of T = 20, 80, and 140 °C are shown in Fig. 10 for different values of the cavitation number σ. The cavity length cannot be compared because the snapshots are taken at arbitrary times. For each mainstream temperature, with decreasing σ, the cavitation pattern changes from attached sheet cavitation, which is a quasi-steady-state, to unsteady sheet-cloud cavitation with strong oscillation and a large cloud cavity, and finally to supercavitation, which is again a quasi-steady-state. The occurrence condition for each cavitation pattern is approximately the same at each temperature. In addition, the aspect of the cavity in a cryogenic fluid such as liquid nitrogen or refrigerant is creamy or frothy [26,27] because the fluid is composed of many fine bubbles, opposite to the case for icy or glassy inflows in room-temperature water, that were compared also in the same inducer in the same tunnel [28]. The mechanism is generally considered to be thermodynamic suppression of bubble development. However, as shown in Fig. 10, no clear difference can be observed between the aspect of the cavity at T = 20 °C and 140 °C, although the thermodynamic parameter Σ = 2.8 × 104 for 140 °C water is close to that for liquid nitrogen. Moreover, the aspect of the cavity is not creamy, but glassy, at T = 140 °C. For comparison, the aspect of the cavity in liquid nitrogen is visualized for the same hydrofoil configuration (NACA0015) with the same chord length at the same angle of attack in another tunnel for liquid nitrogen [29] using the same visualization setups for the high-speed video camera and lighting as those in water. Snapshots of the cavity for 76 K liquid nitrogen and in 140 °C water are compared in Fig. 11 at roughly the same cavity length. The Reynolds number Re- and the thermodynamic parameter Σ are not the same but are close in these cases. As shown in Fig. 11, the aspect of the cavity in liquid nitrogen is creamy for the same hydrofoil. Consequently, the creamy cavity in cryogenic fluid or refrigerant is not caused by thermodynamic suppression because the cavity in hot water at a similar Σ has a glassy aspect. The cause of the creamy aspect in cryogenic fluid has been reported by a present author to be the influence of surface tension [28].

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

### Cavity Length.

Next, the cavity lengths at T = 20, 80, and 140 °C are compared. The cavity length is estimated using the images obtained by a high-speed video camera with the same condition in the cavity aspect. In the sheet-cloud cavitation condition, the lengths in five images at the moment of maximum sheet cavitation before cloud shedding are averaged. In the attached sheet cavitation and supercavitation conditions, the lengths in five images at the moment of maximum cavity length during quasi-steady fluctuation are averaged.

The dependence of the measured cavity length on the cavitation number σ is shown in Fig. 12. The error bars show the dispersion of the data. Here, the interval in the data around σ = 2.5 is large at 140 °C because the flow rate strongly oscillates, and, as a result, the data cannot be obtained in this condition. The results show that the cavity length at T = 140 °C is shorter than that at T = 20 °C throughout the Σ region, which is considered to be a result of the thermodynamic suppression effect in hot water. On the other hand, the cavity length does not differ between T = 20 and 80 °C in larger and smaller σ regions. The cavity length at T = 80 °C is slightly shorter than that at 20 °C in the region 2.0 < σ < 3.3, where unsteady sheet-cloud cavitation occurs. This indicates that thermodynamic suppression easily occurs during unsteady cavitation. In the existing other reports, it was reported that cavity length increases according to an increase of water temperature up to 80 °C in the same NACA0015 hydrofoil [22] or increase up to 55 °C and decreases after that in a small-scale venturi tube [23]. Therefore, it indicates that the temperature for the appearance of thermodynamic suppression effect in water depends on the flow pattern in each flow field, which is superposition with a promotion effect by an increase of Reynolds number according to an increase of mainstream temperature.

Fig. 12
Fig. 12
Close modal

### Temperature Reduction Inside the Cavity.

The temperature reduction inside the cavity, which is an index of the amount of suppression of cavitation, is estimated. The temperature measurement is performed at mainstream temperatures less than 80 °C and cavitation number less than σ = 2.35 where the probe is covered by the cavity. Figure 13 shows the temperature reduction inside the cavity, ΔT = TTcav, where Tcav is the temperature inside the cavity, as a function of σ for a constant mainstream temperature T = 80 °C. With decreasing σ, the temperature reduction ΔT increases. This is considered to be caused by the increase in evaporated mass during cavity development. At T = 80 °C, the maximum temperature reduction inside a cavity is roughly ΔT =0.3 °C under supercavitation conditions. In addition, unsteady sheet cavitation with large-cloud-cavity shedding changes to quasi-steady supercavitation at around σ = 1.7. Therefore, ΔT changes continuously at a certain cavitation number, at which the cavitation characteristics drastically change from an unsteady state to a steady-state.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

Figure 14 shows the measured temperature reduction ΔT as a function of the mainstream temperature T from 20 °C to 80 °C under supercavitation conditions at σ = 1.6. Since under these conditions, the probe is covered constantly by the vapor inside a cavity, the data are not affected by the time response of the measurement system or time averaging. Based on the results, ΔT increases with increasing T, where Σ changes from 3.9 to 2.8 × 102 m/s3/2. The maximum temperature reduction is approximately 0.05 °C inside a supercavitation region at T = 20 °C, taking into account the measurement uncertainty and heat admission from the wall. The temperature reduction corresponds to a 72-Pa decrease in saturated vapor pressure. This means that the conventional assumption of constant-temperature conditions is reasonable in a cavitation flow field for room-temperature water.

Next, the temperature reduction inside the cavity at 140 °C is predicted using the experimental results for the cavity length in Fig. 12. The cavitation number for which the saturated vapor pressure is estimated by the local temperature around the cavity Tcav is denoted as the modified cavitation number σ*. At this time, when the cavity length is the same for two flow fields at mainstream temperatures with and without the thermodynamic suppression effect, the modified cavitation number σ* is the same in the two flow fields
$σL*=σH*$
(4)
where subscripts L and H indicate the flow conditions for low mainstream temperature, where the thermodynamic suppression effect can be ignored, and high mainstream temperature, where the thermodynamic suppression effect cannot be ignored, respectively. In the flow field where the thermodynamic suppression effect can be ignored, the cavitation number σ and modified cavitation number σ* are the same because there is no temperature reduction in the cavity region
$σL=σL*$
(5)
The difference in cavitation number σ between high mainstream temperature and low mainstream temperature when the cavity length lcav is the same is denoted as $|Δσ|lcav$, as shown in Fig. 12, and is defined as follows:
$|Δσ|lcav=σL−σH$
(6)
Substituting Eqs. (5) and (4) into Eq. (6), we have
$|Δσ|lcav=σL*−σH=σH*−σH=|Δσ|H$
(7)
The term $|Δσ|H$ corresponds to the difference between the modified cavitation number and the cavitation number in the flow field of high mainstream temperature. The term $|Δσ|H$ can then be transformed as follows:
$|Δσ|H=σ*−σ=p∞−pv(Tcav)1/2ρ∞U∞2−p∞−pv(T∞)1/2ρ∞U∞2=pv(T∞)−pv(Tcav)1/2ρ∞U∞2=Δpv1/2ρ∞U∞2$
(8)
The Clausius–Clapeyron equation is
$dpvdT=LρvT, Δpv=LρvTΔT$
(9)
where L is the latent heat. The terms $ΔT$ and T, which are comparable to $ΔT=T∞−Tcav$ and T from Eqs. (8) and (9), are then used to obtain the following expressions
$|Δσ|lcav=|Δσ|H=2LU∞2T∞ρvρ∞ΔT, ΔT=U∞2T2Lρ∞ρv|Δσ|lcav$
(10)

Therefore, as indicated by Eq. (10), there is a relationship between the cavity length, which is suppressed by the thermodynamic suppression effect $|Δσ|lcav$, and the temperature reduction inside the cavity generated by the thermodynamic suppression effect ΔT.

Using Eq. (10), the temperature reduction inside the cavity at 140 °C, which was not measured in the present study due to the limitation of calibration can be predicted based on the cavity length at 140 °C, which is easily measured by visualization. The value of $|Δσ|lcav$ is shown at σ = 1.6 in Fig. 12, where for the flow field of water at 20 °C, the thermodynamic suppression effect can be ignored. Here, $ΔT$ at 140 °C is calculated from Eq. (10), which is 1.1 K. The predicted $ΔT$ at 140 °C is plotted as an open circle in Fig. 15.

Fig. 15
Fig. 15
Close modal
Fruman et al. [16] derived a prediction equation for the temperature reduction on a cavity surface as follows:
$ΔT=ρvLCQ0.00695ρlCp(xε)17{1−2.1Rex−0.1(1−Pr)}$
(11)
Equation (11) was derived using the heat-transfer theory for a turbulent boundary layer on a rough plate, in which the cavity surface is assumed to be similar to a rough plate. The equivalent roughness of the cavity surface ε was estimated by fitting the experimental data
$ε=2.2Rex−0.5$
(12)
The volume coefficient of evaporation CQ was obtained from a ventilated cavitation experiment
$CQ=5.2×10−3$
(13)

Fruman et al. Reported that CQ is a universal value. Here, Rex is the local Reynolds number, and Pr is the Prandtl number. Moreover, x is the streamwise distance, and Fruman et al. used the sheet cavity length for x.

Using Eq. (11), we can predict the change in the temperature reduction ΔT with respect to T, as shown in Fig. 15. Although the cavity length at T = 140 °C is suppressed, as shown in Fig. 12, a constant value of 1.5 C is used for x, which is the cavity length at σ = 1.6 and T = 20 to 80 °C. When CQ and ε estimated by Fruman et al. in Eq. (13) are used, the predicted value of ΔT is indicated by the dashed line in Fig. 15. This is an overestimate of the present measurement result for ΔT, although it shows that the order of the measured temperature is appropriate. Since the value of CQ was estimated by the ventilation of air in Fruman's study [16], the value does not exactly correspond to the evaporation rate in the cavity. Therefore, curve fitting is performed for the present measurement from T = 20 °C to 80 °C and the predicted temperature reduction at T = 140 °C assuming that CQ is a variable-model constant. The fitted line is indicated by the solid line in Fig. 15. The value of CQ from the fit is estimated to be CQ = 3.62 × 10−3. In other words, we can predict the volume coefficient of evaporation by measuring the temperature reduction in the cavity.

## Conclusions

In the present study, the thermodynamic suppression effect of cavitation on a NACA0015 single hydrofoil was experimentally investigated in water at mainstream temperatures of T = 20 °C to 140 °C. The results are summarized as follows:

There was no clear difference in the aspect of the cavity between room-temperature water at T = 20 °C and hot water at 140 °C. The cavity surfaces all appeared glassy, although the thermodynamic parameter for water at 140 °C is close to that for liquid nitrogen. Therefore, the creamy cavity, which is generally observed in cryogenic fluid or refrigerant, is not caused by thermodynamic suppression effect, but rather by another factor.

The cavity length at T = 140 °C was shorter than that at T = 20 °C for all cavity patterns from inception to supercavitation. On the other hand, the cavity length at T = 80 °C was slightly shorter than that at 20 °C in the region where unsteady sheet-cloud cavitation occurs, and the cavity lengths did not differ in the other region. This indicates that the thermodynamic suppression effect can easily appear during unsteady cavitation.

The temperature reduction inside the cavity was approximately ΔT = 0.3 °C at T = 80 °C under supercavitation conditions, it was shown that a temperature reduction of approximately 0.3 °C did not suppress the cavity length.

The temperature reduction inside the cavity was approximately ΔT = 0.05 °C at T = 20 °C under supercavitation conditions. This corresponds to a 72 Pa decrease in saturated vapor pressure. Therefore, it was shown that the conventional assumption of constant-temperature conditions is reasonable in the cavitating flow field in room-temperature water.

The temperature reduction inside the cavity at T = 140 °C, which was not measured in the present study was predicted using an equation based on the measured cavity length and was ΔT = 1.1 K at T = 140 °C under supercavitation conditions.

## Acknowledgment

The present research was supported by JSPS KAKENHI Grant 16H04263 and by Komiya Research Grant 2014 from the Turbomachinery Society of Japan. The authors would like to thank Professor Hiroharu Kato for his advice on the experiments.

## Funding Data

• Japan Society for the Promotion of Science (Grant No. 16H04263; Funder ID: 10.13039/501100001691).

• Turbomachinery Society of Japan.

## Nomenclature

C =

chord length

Cp =

isobaric specific heat

CQ =

volume coefficient of evaporation

lcav =

cavity length

L =

latent heat

Pr =

Prandtl number

pv =

saturated vapor pressure

Re =

Reynolds number

Tcav =

measured temperature inside cavity

T =

measured mainstream temperature

ΔT =

temperature reduction inside cavity, TTcav

U =

mainstream velocity

x =

streamwise distance

α =

thermal diffusivity

ρ =

density

σ =

cavitation number

σ* =

modified cavitation number

Σ =

thermodynamic parameter

Σ* =

non-dimensional thermodynamic parameter

ν =

kinetic viscosity

### Subscripts

Subscripts
L =

low mainstream temperature

H =

high mainstream temperature

l =

liquid phase

v =

vapor phase

=

mainstream

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