Effect of water injection on the cavitation control:experiments on a NACA66 (MOD) hydrofoil

Abstract

The objective of this work is to investigate experimentally controlling cavitating flow over NACA66 (MOD) hydrofoils by means of an active water injection along its suction surface. The continuous water vertically jets out of the chamber inside the hydrofoil through evenly distributed surface holes. Experiments were carried out in cavitation water tunnel, using high-speed visualization technology and the particle image velocimetry (PIV) system to study the sheet/cloud cavity behaviors. We studied the effects of this active control on cavity evolution with four kinds of jet flow at two different jet positions. We analyzed the effect of water injection on the mechanism of the cavitating flow control. The results were all compared with that for the original hydrofoil without jet and show that the active jet can effectively suppress the sheet/cloud cavitation characterized by shrinking the attached cavity size and breaking the large-scaled cloud shedding vortex cavity into small-scaled ones. The optimum effectiveness of cavitation suppression is affected by the jet flow rates and jet positions. The water injection at flow rate coefficient 0.0245 with the jet position of 0.45C reduces the maximum sheet cavity length by 79.4$\%$ and the cavity shedding is diminished completely, which gives the most superior effect of sheet cavitation suppression. The jet blocks the re-entrant jet moving upstream and weakens the power of re-entrant jet and thus restrains the cavitation development effectively and stabilizes the flow field.

Graphic Abstract

1 Introduction

Cavitation is a dynamic phase-change phenomenon and it appears originally when the static pressure drops below the saturated vapor pressure of the liquid at the operating temperature [1]. The cavitation phenomenon exists widely in the hydraulic machinery such as pump, hydraulic turbine, propeller and underwater vehicles [2]. The sheet cavitation has a relatively stable cavity length, while the cloud cavitation has a periodically varying length which is associated with the shedding of large clouds of bubbles [3]. In the pump cavitation evolution process, the cavitation region occupies a large part of the flow passage. Therefore, the pump head and efficiency will sharply drop due to block effect of cavity [4]. The cloud cavitation flow field contains attached cavities and a cluster of cavitation bubbles which interact with each other in a complicated manner [5]. Cutting off the cavity attached to the surface of hydrofoil, rolling up the bubbles from the hydrofoil surface, collapsing in the high pressure region downstream lead to the unsteady cloud cavitation characteristics [6]. The cavitation causes vibration, noise, erosion and degradation of the hydraulic machinery performance, which will seriously affect the safety, quietness and stability of the hydraulic system operation [7,8,9,10,11]. Therefore, the study on cavitation flow control is urgent.

The instability for cavitating flow can be classified principally into two categories: inherent instability and system instability. The former is due to the growth of attached main cavity and the separation of small bubbles during the evolution of cavitation itself; the latter is due to the influence of external equipment and conditions on the stability of cavitation flow, especially the interaction between the inlet and outlet pipes [12]. The instability of cavitating flow is the combined effect of both the two factors in the experiment. Therefore, this paper just studies the observed cavitation phenomenon.

The cavitation local instability develops cavity breakup and shedding, which further results in cloud cavitation. This mostly occurs in the case of large attack angle and low cavitation number. Two main mechanisms of the cavity shedding and breakup have been found. One is the well-known re-entrant flow mechanism. The re-entrant jet is on the hydrofoil suction surface and generated at the closure of the cavity [13]. When the re-entrant jet rushes from the trailing edge to the leading edge of a fully developed cavity, the whole collapse of the attached cavity is triggered [5]. The other one is the shock wave propagation mechanism. When the cavitation number is quite small, the large-scale periodic cloud shedding is associated with the formation and propagation of a bubbly shock within the high void-fraction bubbly mixture in the separated cavity flow [14].

Based on experimental analysis [15,16,17] and numerical simulations [18,19,20], it is found that the re-entrant jet and the lateral jet are responsible for cloud cavitation shedding. Therefore, blocking or weakening the re-entrant jet will control the generation of cloud cavity, and further suppress the cavitation development.

At present, blocking the re-entrant jets can be achieved by the active or passive control strategy. The passive control technology requires no external energy injected into the flow field. It changes the flow field pressure and velocity distribution in order to manipulate the flow. For all the passive methods, cavitation restrain is realized by modifying the wall properties [21]. For instance, the effective cavitation suppression in cloud and sheet cavitation conditions can be achieved by setting obstacles [5] in the suction surface of the hydrofoil, setting through-hole [22], blade perforation [23] or reshaping the surface of the hydrofoil [24, 25]. Differently, the active control technology offers mass and momentum into the flow field to manipulate the cavitation development. For instance, the jet air from the leading edge of the blade can improve the pressure fluctuation in the cavitation zone [26], and decrease the erosion effectively [27]. A small amount of non-condensable gases injected into the shear layer can restrain the bubble formation and the self-vibration [28]. A continuous tangential liquid injection through a spanwise slot along the surface of guide vanes (GV) hydrofoil can suppress the periodic cavity length oscillations and the pressure pulsations in order to achieve a favourable and efficient flow manipulation [29]. The water injection through single-row holes arranged in the surface of modified NACA66 (MOD) hydrofoil changes the pressure distribution on the suction side of hydrofoil and reduces the adverse pressure gradient near the cavity tail, which weakens the power of re-entrant jet. Consequently, the cloud cavitation can be effectively controlled and the lift-drag ratio increases [30].

We have proposed an active flow control method by injecting water into cavitating flow field and confirmed that it showcases superiority on cavitation suppression according to the results of numerical simulation. But the porosity of jet holes has great influence on the ratio of lift and drag coefficient [31]. In order to maintain the hydrodynamic characteristics of hydrofoil, the specific injection porosity should be chosen. At present, there is little attention to this active flow control method by setting single-row holes on suction surface of hydrofoil.

In this paper, we will carry out an experimental investigation in order to further reveal the effectiveness and mechanism of water injection on controlling the cavitating flow. In this experiment, the high-speed imaging visualizations (HIV) system is employed to capture the visual observations of cavity evolution and the particle image velocimetry (PIV) system is used to obtain cavitating flow velocity information under different conditions. The experimental comparison among the original hydrofoil and the modified hydrofoils using jet holes is conducted. It includes (1) the effects of water injection on sheet/cloud cavity evolution, (2) the impacts of jet parameters on cavitation suppression, and (3) the effects of cavitation flow control on velocity distribution.

2 Experimental equipment and procedure

In this chapter, the original test hydrofoil and the modified test hydrofoils with jet holes on the suction surface are introduced at first. Secondly, the HIV system, the PIV system, the experimental equipment and the experimental governing parameters are introduced. Finally, the error analysis is carried out for the experimental cases.

Fig. 1

Schematic and photograph of the modified NACA66 (MOD) hydrofoils. a Original hydrofoil. b H1 hydrofoil (jet at 0.19C). c H2 hydrofoil (jet at 0.45C). d Photograph of modified hydrofoils

2.1 Experimental test hydrofoils

In our previous numerical simulation research about cavitating flow and cavitation control, the NACA66 (MOD) hydrofoils were adopted [31, 32]. Hence, in this experimental investigation, the NACA66 (MOD) hydrofoil with/without jet holes on the suction surface are still selected as test models to analyze the effect of key injection parameters on cavitating flow control and explore the cavitating flow control mechanism.

The chord length of the hydrofoils is C = 70 mm and the span length is a = 67 mm. The maximum hydrofoil camber percentage, which is equal to 2$\%$ of the chord length, occurs at 50$\%$ of the chord, and the maximum thickness, which is equal to 11.74$\%$ of the chord length, occurs at 45$\%$ of the chord. Here, the original NACA66 (MOD) hydrofoil is called H0 hydrofoil. In previous numerical study, the position of 0.19C distance from the leading edge is the apex of the NACA66 (MOD) hydrofoil when the angle of attack of the hydrofoil is $8^{\circ }$, where the injection can obtain better cavitation suppression and hydrodynamic performance [33]. When the cavitation number is 1.44, the position of 0.45C distance from the leading edge is the tail of the cavitation enclosed area around the NACA66 (MOD) hydrofoil. The jet at 0.19C is injected into the inner part of the cavity, and the jet at 0.45C is injected at the tail of the cavity. They are characteristic locations to study the effect of jets on cavitation behaviors.

As illustrated in Fig. 1, the modified hydrofoils have the same structure to the original one. But they additionally have the single-row injection holes on the suction surface, with the jet position of 0.19C (called H1 hydrofoil) and 0.45C (called H2 hydrofoil) distance from the leading edge. The porosity of jet holes has great influence on the cavitation flow development and the ratio of lift and drag coefficients [31]. In order to maintain the hydrodynamic characteristics of hydrofoil, specific jet porosity should be carefully chosen. Hence, the 25 single-row injection holes are evenly distributed on the modified hydrofoils’ suction surface with the diameter of 1.4 mm, thus the corresponding porosity is 47.6$\%$. The modified hydrofoils have a hollow inner part with an area of 15.7 cm$^2$ and a volume of 9.577 cm$^3$. The water will be pumped through auxiliary devices to the hollow hydrofoil and then jetted out of the holes on the suction surface. The jet direction is perpendicular to the local surface of hydrofoil. The hydrofoils are made of stainless steel and the surface machining roughness is about 3.2 µm.

Fig. 2

Schematic of high-speed image visualization

2.2 Experimental visualization techniques and velocity measurement

All the experimental studies of this paper are carried out in a cavitation water tunnel in the Hydrodynamics Lab of Beijing University of Technology [34]. The flow field can be observed from two windows located at the front and bottom side of the test section, respectively. The hydrofoil is mounted horizontally in the tunnel test section at an angle of attack (AOA) $\alpha = 8^{\circ }$, and is fixed to the back wall of the test section with a mechanical locking system. For the convenience of observing the cavity characteristics, the suction side of the hydrofoil is installed towards the bottom of the test section. The flow conditions in the tunnel test section are shown in Table 1. The evolution of the unsteady cavitation patterns can be qualitatively and quantitatively analyzed by the HIV, including a high-speed digital camera, a Hub Sync Unit used to have the frame synchronization within +/-2.5 µs, two dysprosium lamps for enhancing the contrast between the vapor and liquid phases, and a computer with the MotionCentral$^\text {TM}$ software for camera control, image download and viewing. The camera has a sampling frequency of 1500 fps at $752 \times 752$ pixels, with up to 100,000 fps at reduced resolutions. In order to get desirable spatial resolutions, 5000 fps is used in this experiment. Figure 2 depicts some of the components of the HIV system. The technical specifications of the camera and lighting source are indicated in Table 2.

It is known that the PIV is one of the non-intrusive optical flow velocity measurement techniques. The PIV system provided by Dantec is adopted in this experiment. As shown in Fig. 3, the laser has been expanded to a 1 mm thick laser sheet and then hits the suction surface of the hydrofoil from the bottom of the tunnel test section; while the charged-coupled device (CCD) camera is placed on the front of the tunnel in order to capture the instantaneous images. For the online data acquisition, a 3000 Hz trigger rate at $1280 \times 800$ pixels is used. The velocity fields are derived by processing the acquired instantaneous images with the commercial software Dynamic Studio, which has an adaptive multi-pass correlation algorithm. The algorithm performs correlation analysis for each interrogation area (IA) with a minimum of $32 \times 32$ pixels and 50$\%$ overlap in general to obtain the instantaneous velocity distribution of the flow field. Then the time-averaged velocity distribution is obtained through 500 frames of instantaneous velocity data. The algorithm’s procedure of dantec dynamics software is illustrated in Fig. 4 and the selected technical specifications of the PIV are indicated in Table 3.

Table 1 List of flow conditions in the tunnel test section

Table 2 Technical specifications of the high-speed camera and lighting systems

Fig. 3

Sketch of the PIV configuration and arrangement used in the 2D measurement of the velocity field

Fig. 4

a Schematic diagram of the particle image processing method. b The post-processed velocity field image

In the PIV measurement, the displacement accuracy is 0.1 and the displacement of each particle is within 1/2 of the side length of IA. According to the above technical specifications, the measured velocity vector error $\Delta u_\text {v}$ and the calibration error $\Delta u_\text {c}$ are expressed respectively as follows:

$$\begin{aligned} \Delta {{u}_{\text {v}}}= & {} \sqrt{{{\left( \frac{\Delta {{x}_{\text {v}}}}{\Delta t} \right) }^{2}}}\nonumber \\= & {} \sqrt{{{\left( \frac{0.1\times {32}/{2}\times 50\%\times 20\times {{10}^{-6}}}{{1}/{3000}} \right) }^{2}}}\nonumber \\= & {} 0.048\ \text {m/s}, \end{aligned}$$

(1)

$$\begin{aligned} \Delta {{u}_{\text {c}}}= & {} \frac{\Delta {{x}_\text {c}}}{\Delta t}=\frac{20\times {{10}^{-6}}}{{1}/{3000}}=0.06\ \text {m/s}, \end{aligned}$$

(2)

2.3 Experimental governing parameters

In the experimental study, the definitions of dimensionless injected mass flowrate $C_\text {Q}$, Reynolds number Re, and cavitation number $\sigma$ are given respectively as below:

$$\begin{aligned} {{C}_{\text {Q}}}= & {} \frac{{{m}_{\text {inj}}}}{{{m}_{0}}}=\frac{{{\rho }_{\text {inj}}}{{Q}_{\text {inj}}}}{{{\rho }_{0}}{{U}_{0}}{{S}_{0}}}=\frac{{{Q}_{\text {inj}}}}{{{U}_{0}}{{S}_{0}}}=\frac{{{Q}_{\text {inj}}}}{{{U}_{0}}h\text {a}}, \end{aligned}$$

(3)

$$\begin{aligned} Re= & {} \frac{{{\rho }_{0}}{{U}_{0}}C}{{{\mu }_{0}}}, \end{aligned}$$

(4)

$$\begin{aligned} \sigma= & {} \frac{{{P}_{0}}-{{P}_{\text {V}}}}{0.5{{\rho }_{0}}{{U}_{0}}^{2}}. \end{aligned}$$

(5)

Table 3 Selected technical specifications of the PIV equipment

The dimensionless coefficient $C_\text {Q}$ serves as a quantitative assessment of the flow rate of the controlling flow. $Q_\text {inj}$ is the jet volume flow, and $m_0$ is the equivalent mass flow rate of the main flow passing through the frontal (midsection) area $S_0$ of the hydrofoil in case of its absence. h is the height of the test foil midsection, and it takes the maximum value Hmax at $\alpha = 0^{\circ }$. $U_0$ is the velocity of main flow (reference velocity), $P_0$ is the reference pressure, and $P_\text {V}$ is the saturated vapor pressure. In this experimental study, the dimensionless mass flow rate $C_\text {Q}$ can be simplified to $C_\text {Q} = 0.00048 \times ({Q_\text {inj}}/{U_0})$. The range of volumetric flow rate of the injected water is set to $0\text {-}450\,\text {L/h}$, and therefore the corresponding dimensionless mass flow rate $C_\text {Q}$ ranges from 0 to 0.0356. The value of mass flow coefficient $C_\text {Q}$ is much small, which indicates that the injected mass flow rate $m_\text {inj}$ is smaller than $m_0$. The specific experimental governing parameters are shown in Fig. 5.

2.4 Experimental error assessment for measured quantities

The static pressure $P_0$ was measured by absolute pressure sensors (range $0 {-} 0.6\,\text {MPa}$, accuracy ±0.25$\%$ FS). The end plane of the pressure sensors is flushed with the inner bottom wall of the tunnel, in order to measure the static pressure of the test section. The electromagnetic flowmeter ($\text {m}^3$/h, accuracy ±2$\%$) is fixed at the upstream of the test section to measure the volumetric flow rate of main flow. As for the volumetric flow rate of the injected flow, it is measured by the float flowmeter (range $0\text {-}600\,\text {L/h}$, accuracy ±4$\%$) which is fixed at the downstream of the auxiliary device. Ignoring the measurement error of water tunnel test section and the machining error of jet hole radius, the relative uncertainties of main flow velocity $U_0$ and jet velocity $U_{\mathrm{inj}}$ are ±2$\%$ and ±4$\%$ respectively. Thus, it leads to a ±4.47$\%$ uncertainty in the injected flow coefficient $C_\text {Q}$, a ±4.01$\%$ uncertainty in the cavitation number $\sigma$ and a ±2$\%$ uncertainty in the Reynolds number Re. Table 4 lists the calculation processes and the values of the relative uncertainties of experimental measurements.

For the PIV measurement, two exposure time intervals are controlled by DG535 with a relative error of less than 0.01$\%$. Thus, the error of PIV measurement is mainly determined by the measurement accuracy of the particle displacement. The statistical processing of the velocity measurement data of the 500 frames in the experiment with 95$\%$ confidence interval results in the statistical error $\Delta u_\text {a}\,=\,0.0551\,\text {m/s}$. Combining Eqs. (1) and (2), the total error of the PIV system in the test is:

$$\begin{aligned} \Delta u=\sqrt{{{\left( \Delta {{u}_{\text {v}}} \right) }^{2}}+{{\left( \Delta {{u}_{\text {c}}} \right) }^{2}}+{{\left( \Delta {{u}_{\text {a}}} \right) }^{2}}}=0.0946\ \text {m/s}. \end{aligned}$$

(6)

Besides, the main flow velocity is adjusted from $6.059\,\text {m/s}$ to $9.085\,\text {m/s}$, resulting in the relative error range of velocity measurement as $1.04\% \sim 1.56\%$.

Fig. 5

Governing parameters of the selected main flow of water tunnel and the selected injected flow of hydrofoil Note that five different jet velocities 0, 2.382, 2.598, 2.887 and 3.248 are named as $(\text {\uppercase {i}})$, $(\text {\uppercase {ii}})$, $(\text {\uppercase {iii}})$, $(\text {\uppercase {iv}})$ and $(\text {\uppercase {v}})$ respectively. For instance, H1#1 means the experimental case for the H1 model of $\sigma =1.38$, $U_\text {inj}=0$ and $C_\text {Q}=0$

Table 4 List of error assessments for main measured quantities

3 Results and discussion

In this chapter, we will investigate the influence of the active flow control method on cavitation suppression and the periodic cavity evolution based on HIV technology. We will explore the time-averaged distributions of velocity and turbulence intensity in order to reveal the mechanism of cavitation suppression by means of the PIV system. As the key factors, the impact of jet flow rate and jet positions on manipulating cavitating flow is analysed in detail.

3.1 Effectiveness of the active control method on cavitation suppression

Based on the above experimental governing parameters in Fig. 5, the HIV experiments measurement for different hydrofoils are conducted and the maximum lengths of the cavitation pattern are recorded as illustrated in Fig. 6. The scatter plot reveals that in comparison to the original hydrofoil, the injected flow can reduce the maximum cavity length of test cases. When the water is injected with different injection velocities, the dimensionless maximum cavity length ${L_{\text {cav}}}^{\text {max}}/C$ fluctuates at a certain cavitation condition. Besides, the highest effectiveness of cavitation control will be achieved when the jet velocity reaches the specific optimum value. It can also be observed that when the jet absolute velocity reaches the same value of $2.887\,\text {m/s}$., the cavitation suppression effects at case 1, case 2 and case 3 are the most significant for both H1 and H2 models. However, the corresponding ${L_{\text {cav}}}^{\text {max}}/C$ values for H2 model are reduced by about 75.5$\%$, 46.2$\%$ and 79.4$\%$, respectively. The reason is that the similar water injection absolute velocities bring similar mass and momentum, but they are supplied to the different cavitating flow field with different Re and $\sigma$. The optimum jet velocity of $2.382\,\text {m/s}$ at case 5 and the value of $3.248\,\text {m/s}$ at case 4 also verify the advantage of cavitation suppression for the modified models. Besides, even for the special cases without injection for modified hydrofoils, the cavitation suppression is more or less achieved, and it works better for the H1 model than the H2 model.

In order to better understand the effect of water injection on cavitating flow under different Re values, the jet velocity coefficient ${U_\text {inj}}/{U_0}$ and the mass coefficient $C_\text {Q}$ are introduced. The parameter $\eta =1-{L_{\text {cav}}}^{\text {max'}}/{L_{\text {cav}}}^{\text {max}}$ is defined as the effectiveness of cavitation suppression. The higher the parameter $\eta$, the better the effectiveness of cavitation suppression is. The variations of $\eta$ with $C_\text {Q}$ are illustrated in Fig. 7. The experiments are conducted under five different cavitation numbers corresponding to cases 1-5, which are divided into three stages: the sheet cavity regime, the transition regime and the cloud cavity regime. The scatter plot reveals that the injected flow can suppress the development of the cavity for both the sheet and cloud cavitation.

However, because the injected mass flow rate $m_\text {inj}$ is smaller than $m_0$, the suppression scheme can achieve better suppression with less energy injection, which shows the superiority of our suppression scheme.

In fact, although the optimum jet mass coefficient varies over different $\sigma$, its rate of change keeps around 0.01 under current test conditions. The cavitation control with the same jet flow $C_\text {Q}$ in a wide range of cavitation numbers still achieves cavitation suppression. For instance, when $C_\text {Q}$ is 0.0276 for the H1 hydrofoil, the values of the effectiveness $\eta$ are 0.545, 0.571, 0.361 for $\sigma$ =1.44, 1.38, 0.99, respectively, which are tolerable to deviate the best values $\eta$ in comparison to the complicated jet flow control.

Fig. 6

The dimensionless maximum sheet cavity length under different cases

Fig. 7

The effectiveness of cavitation suppression under different cases

In addition to the chordwise length ${L_{\text {cav}}}^{\text {max}}$, it is important for conducting the spanwise analysis on the cavity pattern. Fig. 8 shows the instantaneous pattern of the minimum and maximum cavity length (${L_{\text {cav}}}^{\text {min}}$ and ${L_{\text {cav}}}^{\text {max}}$) for three hydrofoils. The injected velocity is selected as the optimum one in a series of jet flow rate. The spanwise cavity pattern shows that the attached cavity on the suction surface is consisted of a transparent cavity region at the leading edge and a two-phase mixture region. With the jet flow, the maximum cavity length is decreased sharply. On the other hand, the jet holes arranged at the vertex position of the H1 hydrofoil can also suppress the cavitation to some extent in the absence of jet flow. This is due to the spanwise jet holes which act as part of the vortex generators, indicating the meaning of passive control in cavitation suppression. By comparing the cavitation patterns in Fig. 8a, b, it can be found that the cavity development depends on $\sigma$ and Re. Under different $\sigma$, the cavity lengths in Fig. 8a, c however are basically similar, which indicates that the cavitation flow field tends to be unsteady with higher Re. Therefore, the influence of turbulence field on the cavity pattern, which is dominated by Re, cannot be ignored. Note that the turbulence characteristic will be discussed in detail in Sect. 3.4.

3.2 Impacts of jet positions and jet flow rate on cavitation suppression

According to the analysis above, the active jet can control the cavitating flow field. The suppression effect is determined by the jet parameters, including the jet diameter, the jet porosity, the jet flow rate and the jet position. This section discusses the detailed influence of two key parameters: the jet flow rate and the jet position on cavitation suppression. The analyzed condition is Case 3 with $\sigma$=1.44, which has better overall cavitation suppression in Fig. 7.

Fig. 8

Top view of the cavitation flow under different cases. a Comparison of the sheet cavity pattern with $\sigma =1.38$ and $Re =3.944\times 10^5$. b Comparison of the sheet cavity pattern with $\sigma =1.32$ and $Re =4.556\times 10^5$. c Comparison of the sheet cavity pattern with $\sigma =1.46$ and $Re =5.916\times 10^5$

Fig. 9

Comparison of the sheet cavity pattern under different flow rate coefficients at case 3 ($\sigma =1.44$ and $Re =5.132\times 10^5$). a H0 (Case:H0#3). b H1 $C_\text {Q}$ = 0 (Case:H1#3 $(\text {\uppercase {i}})$). c H1 $C_\text {Q}$ = 0.0202 (Case:H1#3 $(\text {\uppercase {ii}})$). d H1 $C_\text {Q}$ = 0.0220 (Case:H1#3 $(\text {\uppercase {iii}})$). e H1 $C_\text {Q}$ = 0.0245 (Case:H1#3 $(\text {\uppercase {iv}})$). f H1 $C_\text {Q}$ = 0.0276 (Case:H1#3 $(\text {\uppercase {v}})$). g H1 $C_\text {Q}$ = 0 (Case:H2#3 $(\text {\uppercase {i}})$). h H2 $C_\text {Q}$ = 0.0245 (Case:H2#3 $(\text {\uppercase {iv}})$)

For the sheet cavitation, two instantaneous images (top view) record the cavity patterns at the time of cavitation inception and cavitation full development, respectively. The left graphs display the instantaneous evolution of the leading position and trailing position of the cavity over a period of time. The distance between the tail and leading position of partial cavities represents the length of attached cavity. The sequence images in Fig. 9 a–f show the influence of jet flow rate on cavitation suppression. It is observed that the cavitating flow is suppressed obviously. Besides, with the increase of jet flow rate, the time average length of the attached cavity decreases first and then increases, which shows that there is an optimal suppression at the optimum jet rate. For instance, when $C_\text {Q}$ = 0.0245, the maximum cavity length ${L_{\text {cav}\triangle }}^{\text {max}}$ of the H1 hydrofoil is equal to 0.177C, which decreases by 64.1$\%$ in comparison to ${L_{\text {cav}}}^{\text {max}}$= 0.496C of the H0 hydrofoil. Furthermore, the fluctuation amplitude of the attached cavity at the tail position becomes smaller, and thus the cavity becomes more stable after water injection.

Next, we investigate the impact of jet positions on the cavitation suppression. As shown in Fig. 9g, h, when $C_\text {Q}$ = 0.0245, the maximum cavity length ${L_{\text {cav}\square }}^{\text {max}}$ of H2 hydrofoil is 0.102C, which decreases by 79.4$\%$ in comparison to that of the H0 hydrofoil. The cavity evolution is controlled obviously when the jet position is set at 0.19C or 0.45C. We also extract the experimental cavity areas of the H0 hydrofoil and other modified hydrofoils with the optimum injection rate ${C_\text {Q}}^\text {opt}$ in Fig. 10. The instantaneous cavity area of 1800 frames corresponding to 0.06 s is captured by the image binarization method based on MATLAB. The graph clearly indicates that the time-averaged cavity areas and the related fluctuation magnitude around the modified hydrofoils decrease sharply. Particularly, the H2 hydrofoil shows the most superior effect on sheet cavitation suppression, whose cavity length can also be summarized by the quantitative index as indicated in Fig. 7.

3.3 Effects of water injection on sheet cavity evolution

Cavitation is a periodic hydrodynamic phenomenon. This section focuses on the influence of active jet control on the periodic evolution of cavity pattern. The instantaneous images of sheet cavity of the original hydrofoil and the modified hydrofoil at an attack angle of $\alpha = 8^{\circ }$ can be captured by the HIV system. The cases are selected when the suppression effect is the most obvious in Fig. 10 for a clear comparison.

Fig. 10

The area of cavity form evolution along time at $\sigma$ = 1.44 corresponding to the sheet cavitation regime

Fig. 11

Instantaneous images of partial cavities (side view and top view) of the original hydrofoil with $\sigma$ = 1.44 corresponding to sheet cavitation (case: H0$\#$3)

Figure 11 illustrates the sequence images of cavitating flow around the original hydrofoil, wherein $t_0$ is the initial time corresponding to the minimum cavity length. The cavity on the suction surface of the hydrofoil is composed of an attached cavity at the leading edge and a small-scaled shedding cavity at the tail of attached cavity. During the typical period of sheet cavitation evolution, the length and the position of attached cavity remain steady (${L_{\text {cav}}}/C$ $\approx$ 0.45 ± 0.05), while the thickness fluctuates greatly. During the time period $t_0$-$t_4$, with the evolution of thickness, the attached cavity becomes plumper spanwise, which can be clearly observed from the top view. This indicates that the volume of attached cavity and its corresponding low pressure region increase. Thereafter, it results in the generation and development of re-entrant jet, which are caused by the adverse pressure gradient. In fact, at the instantaneous time $t_3$, the re-entrant jet has already made a strong shear effect on the attached cavity, causing the appearance of shedding cavity at the tail of attached cavity. The shedding cavity continues to develop and eventually collapse during the time period $t_{3}$-$t_{6}$. It is observed from the side view that the collapse of shedding cavity is not far from the hydrofoil wall, indicating that the boundary layer may play an important role. Another interesting phenomenon is that the attached cavity and the shedding cavity show significant difference in the gray level, which is more obvious from the top view. This reveals that the vapor phase volume fraction of these two parts is quite different from each other.

Fig. 12

Instantaneous images of partial cavities (side view and top view) of the H1 hydrofoil with $\sigma$ = 1.44,$C_\text {Q}$ = 0.0245 (Case:H1#3 $(\text {\uppercase {iv}})$)

Figure 12 shows that the evolution of cavity shape during a time period is not obvious when the jet rate coefficient is $C_\text {Q}$ = 0.0245 with the jet position of 0.19C. For the sheet cavitation, the fully developed attached cavity around the original hydrofoil is about 45$\%$ of the chord length. Therefore, when the jet is carried out at a distance of 0.19C, the energy carried by the jet is imported into the cavity directly, thus significantly increasing the pressure inside the cavity, and reducing the adverse pressure gradient which generally triggers the re-entrant jet. As a result, the thickness and volume of the attached cavity are greatly decreased. In an evolution period, the attached cavity is steady, with a length of about 0.17C, and the shedding cavity is diminished completely.

The jet flow rate of this case is the optimum in a series of jet flow rates, which has been discussed in detail in Sect. 3.2. We now focus on the relative position of the injected flow and the cavity. As discussed above, the attached cavity maintains a steady regime at the position of 0.17C under the effect of injected flow at the position of 0.19C. The jet is already behind the trailing edge of the cavity. When the jet flow comes out of the jet holes, firstly, it will be mixed with the low-velocity main flow near the hydrofoil wall in order to improve the local velocity distribution. Therefore, the influence of the velocity boundary layer in this process cannot be ignored, and will be discussed in detail in Sect. 3.4. On the other hand, the additional tiny bubbles are found near the jet holes, which may be caused by the direct mixing effect between the jet flow and main flow. But this phenomenon seems not to be critical as the jet has already shown a good suppression effect under this working condition.

3.4 Effects of cavitation flow control on velocity distribution

In order to investigate the mechanism of cavitation suppression, we use the PIV technology to measure the instantaneous velocity around the cavitating flow field. The chordwise mean flow velocity, the turbulence intensity distributions and the sheet cavity patterns of three kinds of hydrofoil are processed and compared respectively. The experimental cases are also selected as the same with that in Fig. 10.

Figure 13 illustrates the mean velocity streamwise distribution, the boundary layer (BL) and the sheet cavity pattern. The suction sides of the hydrofoils in every cross-section are divided into 9 equal chordwise segments from the leading edge to the trailing edge. The origin of the coordinates (x/C = 0, y/C = 0) represents the leading edge point of the hydrofoil, and the ending of coordinates (x/C = 0.99, y/C = 0) represents the trailing edge point. These equidistant points are regarded as the monitoring starting points. The straight lines from the monitoring points upwards, normal to the main flow, are the monitoring lines (line x/C = 0, line x/C = 0.11, ..., line x/C = 0.99). In order to facilitate the comparison study, all the analysis results are graphed together for three different hydrofoils.

3.4.1 Influence of the jet flow on the boundary layer

The BL thickness ${\delta }_{\text {BL}}$ is defined as the y-axis distance from the foil surface to the position where the mean velocity is no larger than 0.99 of that at infinity. There are different BL trends for three kinds of hydrofoil and they all become distorted before the position of x/C = 0.3 where the cavity happens. For instance, the ${\delta }_{{\text {BL}}\circ }$ is equal to 0.15 at x/C = 0.11 and ${\delta }_{\text {BL}\circ }$ is equal to 0.075 at x/C = 0.33 for the H0 hydrofoil. It may be caused by the following two reasons. First, cavitation occurs with the generation of the low pressure region in the flow field. The differential pressure between the hydrofoil leading edge point and the interior of the cavity accelerates the flow and then causes converging BL. Second, since the hydrofoil is mounted at the attack angle $\alpha = 8^{\circ }$, the converged and accelerated flow is also generated between the leading edge point and the vertex position of the hydrofoil with the limited experimental test section size.

Fig. 13

Chordwise evolution of the mean velocity streamwise distribution corresponded to the sheet cavity regime ($\sigma =1.44$ and $Re =5.132\times 10^5$) Note that the circle, triangular and square symbols in lines denote the mean velocity distribution for H0 model (without jet), H1 model with $C_\text {Q}$ = 0.0245 and H2 model with $C_\text {Q}$ = 0.0245, respectively. The bottom axis x/C is the dimensionless chordwise position, the short axis $(u/{{{U}_{0}}-1})$ is the dimensionless velocity, and the left axis y/C is the dimensionless length orthogonal to the main flow. The vertical dash lines ($u/U_0-$1=0) represent the criterion lines where the local mean velocity u is equal to the main flow velocity $U_0$. The dark dash curves give the boundary layer outline of three cases. The solid and broken curves attached to the suction sides represent the time-averaged cavity pattern. The flow direction is from left to right

Likewise, the BL trends of H1 and H2 hydrofoils are illustrated as dash curves in Fig. 13. When the water is injected into the main flow field around the H1 hydrofoil, the pressure distribution inside the cavity will be improved and the velocity gradient ${\partial {(u/{U}_{0}-1)}}/{\partial {(y/C)}}$ inside the BL will also be enhanced. Affected by the jet at the position of 0.19C, the BL trend upstream of H1 hydrofoil is much thinner than that of H0 hydrofoil. When the water is injected into the main flow field around the H2 hydrofoil, there is little influence on the BL upstream of the jet holes as the water injection position is far away from the center of the cavity, and there is not remarkable pressure improved inside the cavity. Thus, the BL upstream trend is finally similar to that of the original hydrofoil.

After the position of x/C = 0.22, the BL expands the downstream because of the friction resistance loss. In particular, the BL near the trailing edge point of the hydrofoils grows significantly because the backflow develops and the turbulence worsens there. For example, we have ${\delta }_{\text {BL}\circ }$=0.2 at x/C = 0.88, ${\delta }_{\text {BL}\circ }$=0.225 at x/C = 0.99 for the H0 hydrofoil. The results reveal that the water injection reduces ${\delta }_{\text {BL}}$ significantly, thus improving the velocity distribution of the flow field over the suction surface of H1 and H2 hydrofoils. The flow field around the H2 hydrofoil indicates the smallest ${\delta }_{\text {BL}}$, e.g., the ${\delta }_{\text {BL}}$ ratio of the H2 hydrofoil to the H0 hydrofoil is ${{\delta }_{\text {BL}\square }}/{{\delta }_{\text {BL}\circ }}$=25$\%$ at x/C = 0.88 and ${{\delta }_{\text {BL}\square }}/{{\delta }_{\text {BL}\circ }}$=53.33$\%$ at x/C = 0.99. This means that the enhanced velocity gradient and the relatively stable flow near the wall will weaken the strength of the re-entrant jet.

3.4.2 Influence of the jet flow on the cavity region

The time-averaged cavity length of H0, H1 and H2 hydrofoil (${L_{\text {cav}\circ }}^{\text {mean}}$ , ${L_{\text {cav}\triangle }}^{\text {mean}}$, ${L_{\text {cav}\square }}^{\text {mean}}$ ) are 0.34C ,0.22C and 0.13C, respectively, as observed in Fig. 13. The smallest cavity (solid black squares) happens in the flow field around the H2 hydrofoil, which reveals the most effective way on suppressing the sheet cavitation. Even though the water injection position of the H2 hydrofoil is located in the exterior of time-averaged cavity, as shown in Fig. 13, this injection position (0.45C) in fact is just at the trailing edge of instantaneous fully developed attached cavity of H0 hydrofoil. We have known that the re-entrant jet starts rushing towards the upstream direction from the termination line of the fully developed sheet cavity according to Ref. [5]. Therefore, there is more complicated interaction among the jet, the re-entrant flow and the mainstream flow when the jet works. It will be possible to block the re-entrant jet travelling upstream or weaken the power of re-entrant jet. Besides, part of the jet with extra mass and momentum can be carried into the cavity by the re-entrant jet, and the local pressure inside the cavity is increased. Consequently, the cavity area is shrunk sharply and cavity shedding is reduced.

As for the H1 hydrofoil, the jet with extra momentum accesses to the interior of cavity directly and improves the low pressure distribution inside the cavity, causing diminished cavity area. Meanwhile, the re-entrant jet gets more momentum when it moves to the jet position of 0.19C from the trailing point of attached cavity at 0.45C. So the injected water is less capable to block the re-entrant jet compared to the case of H2 hydrofoil. Moreover, the velocity gradient near the trailing edge point is smaller than that of the H2 hydrofoil, resulting in a larger strength of the backflow generally due to the adverse pressure gradient. As can be deduced from the analyses, the effect of active jet on suppressing the cavitation is weaker than that of the H2 hydrofoil under the sheet cavitation condition.

The water injection improves the low velocity region near the hydrofoil wall, and the velocity of the main flow region is slightly decreased compared with that of the original hydrofoil. It illustrates that the velocity distribution tends to be relatively uniform in the whole flow field. As the cavity evolution is an unsteady process, further research on the turbulence characteristics of the cavitation flow field is required to study whether the velocity fluctuations have been decreased in an unsteady periodic time requires or not.

3.4.3 Influence of the jet flow on the cavity region

As discussed in Sect. 3.1, the turbulence plays an important role in unsteady flow field and determines the cavity regime. The cavitation suppression by active jet may have changed some turbulence characteristics. We employ the turbulence intensity, a statistical value, to reveal the influence of jet on the turbulence field. The definition of turbulence intensitye $I$ is given as below:

$$\begin{aligned} I=\frac{\mathrm{RMS}(u^{*2})}{u}=\frac{\sqrt{\sum ^N_{i=1}u^{*2}}}{u}. \end{aligned}$$

(7)

As shown in Fig. 14, the high turbulent intensity (HTI) thickness $\delta _{\text {HTI}}$ is defined as the y-axis distance from the hydrofoil surface to the position where the turbulent intensity is no more than 10%. The symbols and axes are the same as that in Fig. 13 except that the top axis $I$ is the turbulence intensity. The dash lines ($I$=0.1) in the graph are the criterion lines which measure whether the turbulence intensity is high or not. The PIV result of the original hydrofoil reveals that the turbulence intensity near the wall is higher. Particularly, the leading edge and the trailing edge of the hydrofoil are always in the HTI region ($\delta _{\text {HTI}\circ }$ =0.20 at x/C = 0.11 and ($\delta _{\text {HTI}\circ }$ 0.15 at x/C = 0.66$\sim$0.99). When the jet is implemented at the position of 0.45C, the $\delta _{\text {HTI}\square }$ is the lowest among three kinds of hydrofoil after the position of x=0.3C, especially near the trailing edge, such as $\delta _{\text {HTI}\square }$/$\delta _{\text {HTI}\circ }$=13.33$\%$ at x/C = 0.88 and $\delta _{\text {HTI}\square }$/$\delta _{\text {HTI}\circ }$=25$\%$ at x/C = 0.99. It illustrates that the flow field around the H2 hydrofoil becomes more stable and the cavity pattern is the smallest among the three cases.

Fig. 14

Chordwise evolution of the turbulence intensity distribution corresponded to the sheet cavitation regime ($\sigma =1.44$ and $Re =5.132\times 10^5$)

Considering the effect of jet positions on the cavitation suppression, it can be seen that the jet of H1 hydrofoil is effective at improving the velocity gradient, reducing the $\delta _{\text {BL}}$ and $\delta _{\text {HTI}}$ in the vicinity of the leading edge. But such effect near the trailing edge is not obvious. On the contrary, the jet of H2 hydrofoil is helpful to improve the velocity gradient and reduce the $\delta _{\text {BL}}$ and $\delta _{\text {HTI}}$ near the trailing edge. It can be further analyzed that the BL of the leading edge is affected by the differential pressure, the converged channel and the installed attack angle; while the BL near the trailing edge is affected by the complex interaction of the water injection, the re-entrant jet and the turbulence. The reduction of time-averaged cavity size shows that the H2 hydrofoil is more advantageous to suppress cavitation under the sheet cavitation condition.

3.5 Effects of water injection on cloud cavity evolution

In order to better understand the effectiveness and mechanism of water injection on the cavitation suppression, we further investigate the impact of water injection on the cloud cavitation around the H1 hydrofoil. Fig. 15 shows the evolution of the cloud cavity pattern for the H0 hydrofoil. The cloud cavity over the suction surface can be divided into three parts as illustrated in Fig. 15a: $(\text {\uppercase {i}})$ the transparent cavity at the leading edge of the hydrofoil; $(\text {\uppercase {ii}})$ the attached cavity in the middle of the hydrofoil; and $(\text {\uppercase {iii}})$ the shedding cavity at the trailing edge of the hydrofoil. The figures show that the shedding cavity leaves the surface of the hydrofoil with rolling and collapsing downstream according to the re-entrant flow mechanism. However, when the jet is injected into the flow field ($C_\text {Q}$ = 0.0278), as shown in Fig. 16, the strength of re-entrant is reduced obviously. The length and thickness of the attached cavity are smaller than those of the original one. Besides, the cavity remains steady throughout the time period and the large-scaled vortex shedding has disappeared, but small eddies still remain.

Fig. 15

Instantaneous images of partial cavities (side view and top view) of the H0 hydrofoil with $\sigma$ = 0.99 corresponding to cloud cavitation (Case:H0#4)

Fig. 16

Instantaneous images of cloud cavities (side view and top view) of the H1 hydrofoil with $\sigma$ = 0.99, $C_\text {Q}$ = 0.0278 (Case:H1#4 $(\text {\uppercase {v}})$)

In fact, the flow rate in this case is the optimum one in a series of flow rates. As illustrated in Fig. 7, the ${L_{\text {cav}}}^{\text {max}}$ of H1 hydrofoil is decreased by 36.1$\%$ compared with that of the H0 hydrofoil. The suppression effect seems not good compared with that of the sheet cavitation. The reason is analyzed as follows. When the cloud cavitation develops dramatically, the areas of the cavity region and the corresponding low pressure region increase rapidly, causing the increase of the momentum of re-entrant jet. Thus, more jet flow rate is required to weaken the intensity of the re-entrant jet. However, it is impossible to increase the jet flow rate infinitely because of the limit of the jet structure, the bubble occurrence around jet holes and the serious reduction of dynamic characteristic. In fact, too much jet flow will aggravate the disturbance in the flow field and generate more serious cavitation as the cloud cavity is unsteady itself.

Although the influence of jet flow on ${L_{\text {cav}}}^{\text {max}}$ is not obvious for the cloud cavitation field, the influence on the vortex dynamics behavior changes the evolution of cloud cavity pattern significantly. As can be seen from Fig. 15, the trailing edge cavity falls off, rolls up and lifts away from the hydrofoil wall and collapses finally. Figure 17 shows the collapse process of the shedding cavity at the trailing edge of the H0 hydrofoil. It is well known that the attached cavity is cut off from the surface due to the re-entrant jet. As shown in Figs. 15d, 17a, the main flow moves towards the trailing edge of the hydrofoil; while the re-entrant jet mixed with the shedding cavity moves towards the leading edge of the hydrofoil along the opposite direction. Because of the density difference, the shedding cavity gets the buoyancy $F_{\text {b}}$. Thus, those combined factors cause the shedding cavity to roll up and fall off along the clockwise direction. On the other hand, due to the velocity circulation $\varGamma _{\text {cav}}$ in the rolling cavity, Joukowski’s Theorem implies that the shedding cavity will get lift $F_{\text {cav}}$ along the direction which is perpendicular to the main flow. The velocity circulation around the hydrofoil $\varGamma _{\text {C}}$ will decrease due to the separation of the BL. Consequently, the H0 hydrofoil’s lift coefficient will decrease correspondingly. When the shedding cavity further collapses, the attached cavity shrinks and the re-entrant jet gradually disappears, as shown in Figs. 15f and 17b. During the rolling process, the upper surface of the shedding cavity is accelerated along the same direction as the main flow, while the lower surface of the shedding cavity is slowed down as it moves against the main flow. According to the Magnus effect, the shedding cavity still receives upward lift $F_{\text {cav}}$. Correspondingly, the lift coefficient of the hydrofoil continues to be decreased.

Fig. 17

Schematic diagram of the shedding cavity’s collapse process of the H0 hydrofoil with $\sigma$ = 0.99 corresponding to cloud cavitation (Case:H0#4)

Fig. 18

Schematic diagram of the shedding cavity’s collapse process of the H1 hydrofoil with $\sigma$ = 0.99 corresponding to cloud cavitation (Case:H1#4 $(\text {\uppercase {v}})$)

As shown in Fig. 18, contrary to the H0 hydrofoil, the thickness and length of cavity for the H1 hydrofoil decrease. It can be observed from the experiment that the weak vortex cavity constantly occurs in the attached cavity which tries to leave the hydrofoil suction surface but eventually dissipates in the BL. The weak shedding vortex formed by mixing effect among the jet, the re-entrant jet and the main flow increases the energy in the BL and improves the velocity distribution. The decrease of re-entrant jet strength leads to the increase of velocity circulation $\varGamma _{\text {C}}$ around the hydrofoil. Thus, the lift coefficient of the modified hydrofoils increases. On the other hand, the smaller shedding cavity region means that the periodic oscillations of lift and drag on the hydrofoil are no longer severe in the unsteady cavity pattern evolution compared with that of the original hydrofoil. Therefore, the jet flow can also mitigate the fatigue damage caused by the unsteady force on the hydrofoil to some extent.

4 Conclusions

This paper conducts an experimental study of the water injection effects on the cavitation suppression by concerning the cavity behaviours and the mechanism of cavitation flow control. The experiments are carried out in the cavitation water tunnel for different injection rates and two jet positions using the HIV and the PIV system. The main conclusions are as follows:

• (1) The water injection can restrain the cavitation development effectively for the sheet cavitation and cloud cavitation. At the water injection rate of $C_\text {Q}$=0.0245 for H1 hydrofoil, the length of maximum cavity was decreased by 64.1$\%$ and 34$\%$ for $\sigma$=1.44 and 0.99, respectively. Besides, the large-scaled cavity cloud shedding disappears and is replaced by the small-scaled cavity shedding. The study of the instantaneous sheet cavity pattern evolution reveals that there is complicated interaction among the jet, the re-entrant flow and the mainstream flow when the jet works.

• (2) The jet rate and jet position play important roles in cavitation suppression. At $C_\text {Q}$=0.0245 with the jet position of x=0.45C (H2 model), the value of the sheet cavitation suppression effectiveness reaches up to 79.4$\%$ and its fluctuation magnitude of the tail position becomes smaller for $\sigma$=1.44. At $C_\text {Q}$=0.0276 with x=0.19C (H1 model), the value of the cloud cavitation suppression effectiveness reaches up to 36.1$\%$ for $\sigma$=0.99. It indicates that H2 modified hydrofoil structure gives the most superior effect on the sheet cavitation suppression.

• (3) When the water is injected into the flow field just from the end of the fully developed sheet cavity, it will be possible to block re-entrant jet moving upstream or weaken the power of re-entrant jet. Part of the jet with extra mass and momentum can be carried into the cavity by the re-entrant jet, and the local pressure inside the cavity is increased. Consequently, the cavity area is shrunk sharply and the cavity shedding is reduced.

• (4) For the flow mechanism of sheet cavitation around the modified hydrofoil, which is characterized by the boundary layer and turbulence intensity, the velocity distribution tends to be relatively uniform in the space, and the velocity fluctuation tends to be gentle in the periods of time. The extra momentum input with injected water to the cavity shrinks the size of sheet cavity.

• (5) From the instantaneous images of cloud cavitation, we find that the water injection affects the form of cavity shedding. When the main flow direction is from left to right, the shedding cavity always rolls back and falls off clockwise. The water injection makes the shedding vortex occur in the form of small eddy dissipation, which may supplements the energy of the boundary layer.