Water Injection for Cloud Cavitation Suppression: Analysis of the Effects of Injection Parameters

Abstract

This study investigates cloud cavitation suppression around a model-scale NACA66 hydrofoil using active water injection and explores the effect of multiple injection parameters. Numerical simulations and a mixed-level orthogonal test method are employed to systematically analyze the impact of jet angle α_jet, jet location L_jet, and jet velocity U_jet on cavitation suppression efficiency and hydrofoil energy performance. The study reveals that jet location has the greatest influence on cavitation suppression, while jet angle has the greatest influence on hydrofoil energy performance. The optimal parameter combination (L_jet = 0.30C, α_jet = +60 degrees, U_jet = 3.25 m/s) effectively balances energy performance and cavitation suppression, reducing cavitation volume by 49.34% and improving lift–drag ratio by 8.55%. The study found that the jet’s introduction not only enhances vapor condensation and reduces the intensity of the vapor–liquid exchange process but also disrupts the internal structure of cavitation clouds and elevates pressure on the hydrofoil suction surface, thereby effectively suppressing cavitation. Further analysis shows that positive-going horizontal jet components enhance the lift–drag ratio, while negative-going components have a detrimental effect. Jet arrangements near the trailing edge negatively impact both cavitation suppression and energy performance. These findings provide a valuable reference for selecting optimal injection parameters to achieve a balance between cavitation suppression and energy performance in hydrodynamic systems.

1. Introduction

Cavitation, a prevalent phenomenon in fluid systems engineering [1,2,3], presents multifaceted challenges ranging from material erosion [4,5] and efficiency losses in fluid machinery [6] to increased noise [7] and performance degradation in hydrodynamic systems such as hydrofoils and propellers [1,2,3,4,5,6,7]. Characterized by the formation and collapse of vapor-filled cavities due to local pressures falling below the liquid’s vapor pressure [8], its suppression is crucial for optimizing fluid system performance. To mitigate these adverse effects and optimize fluid system performance, effective cavitation suppression is of utmost importance.

Flow control strategies are dichotomized into passive and active methods. Passive control, leveraging fixed geometries or surface modifications, offers simplicity and cost-effectiveness but often lacks adaptability to varying flow conditions [8]. Conversely, active control methods provide dynamic adjustability, crucial for complex flow scenarios, albeit at the expense of higher complexity and energy requirements [9,10].

Passive flow control methods involve the use of fixed geometries or modifications to the surfaces of the system to influence flow behavior. These modifications can include the use of specially designed surface textures [8], dimples, vortex generators [11,12], or obstacles [13,14] to interfere with flow separation, reduce drag, or suppress cavitation.

Scholars have conducted valuable experimental studies on passive control methods, especially on single hydrofoils. Kadivar et al. [15,16] experimentally investigated the arrangement of bubble generators on the hydrofoil, observing significant reduction in the low-pressure area and effective suppression of cavitation development. Kadivar et al. [17,18] explored cavitation control on circular cylinders using scalloped and sawtooth riblets, analyzing cavitation dynamics and vibration reduction. Lin et al. [19] investigated passive cavitation control on a NACA 0015 hydrofoil with bio-inspired riblets, focusing on cavitation dynamics and noise. Kadivar et al. [20] examined laser-induced cavitation bubble dynamics with V-shaped riblets. Simanto et al. [21] assessed leading-edge protuberances on NACA 0012 hydrofoils for cavitation control and noise suppression. Chen et al. [22] used micro delta-shaped vortex generators on a NACA66 hydrofoil to study cavitation modes and instabilities. Cheng et al. [23,24] introduced overhanging grooves (OHGs) at hydrofoil tips as a novel method for tip-leakage vortex cavitation control, demonstrating significant suppression across various gap sizes with minimal impact on hydrofoil performance through experimental and numerical analysis. Amini et al. [25] found that 10%-bent 90° downward winglets on an elliptical hydrofoil effectively suppress tip vortex cavitation without compromising hydrodynamic efficiency. Yang et al. [26] applied a wavy leading edge on a hydrofoil and focused on cavitation noise characteristics. Gu et al. [27] studied the suppression of tip leakage vortex (TLV) using double-control holes on a hydrofoil, investigating the effects on TLV dynamics and cavitation through the SST-CC and ZGB models under varying Reynolds numbers. Wang et al. [28] studied partial cavitation control on NACA 66 hydrofoils using spanwise obstacles, utilizing Lagrangian coherent structures (LCSs) to analyze vortex dynamics and the impact of obstructions on unsteady cavitating flows. Qiu et al. [29] found that micro-vortex generators angled to the inflow direction can alter flow dynamics around hydrofoils. Liu et al. [30,31] introduced a C groove structure for hydrofoil tips, showing through simulations its benefits in energy efficiency, cavitation reduction, and vortex core minimization.

For rotating machines like propellers, water turbines, and pumps, passive flow control methods have many applications. Zhao et al. [32] studied bionic protuberances on stay-guide vanes, analyzing turbulence and hydraulic loss. Pang et al. [33] investigated J-groove designs for cavitation and vortex rope suppression in pump turbines, focusing on velocity distribution and efficiency. Zhai et al. [34] examined propeller-induced hull vibration suppression using triangular vortex generators. Sun et al. [35] analyzed bionic tubercle leading edges for propellers on cavitating wake dynamics.

Active control methods, characterized by their adaptability, have been explored through various experimental and numerical studies. Timoshevskiy et al. [36,37,38] conducted a study on the impact of tangential liquid injection on the suction surface of a hydrofoil, achieving effective suppression of sheet cavitation through experimental and numerical methods. Chang et al. [39] observed that water injection can delay the inception of tip vortex cavitation, while Lee et al. [40] successfully controlled propeller cavitation using water injection in rotating machinery. De Giorgi et al. [41] employed a single synthetic jet actuator on the hydrofoil, leading to the suppression of cloud cavitation. Maltsev et al. [42] demonstrated that active jet flow can maintain a separation-less flow around the hydrofoil at high attack angles, and Lu et al. [43] achieved efficient suppression of sheet/cloud cavitation by applying water injection to the suction surface of a NACA66 hydrofoil. Liu et al. [44] demonstrated the effectiveness of water injection for TVC (tip vortex cavitation) suppression, with parametric studies showing increased cavitation inhibition rate with higher injection velocity or cross-streamwise velocity component. Gu et al. [45] studied the suppression of cavitation in fluid machinery using a bionic shark gill slit jet structure on a hydrofoil, analyzing the effects on vacuole volume, lift-to-drag ratio, and pressure fluctuations through numerical simulations. Li et al. [46] studied cavitation suppression on a Clark-Y hydrofoil using a periodic jet scheme and large eddy simulation combined with the Zwart-Gerber-Belamri cavitation model, achieving a 47–52% reduction in cavitation volume.

Despite these advancements, a gap remains in the comprehensive understanding of how different control parameters influence cavitation suppression and the hydrodynamic performance. Previous research has predominantly focused on the effectiveness of active control using configurations limited to single-injection parameters. It has restricted the depth of understanding regarding the multifaceted impact of jet variations on cavitation control. Therefore, it inspires us to bridge this gap by providing a detailed investigation into the effects of various injection parameters on cavitation dynamics and the lift–drag ratio of hydrofoils.

In the literature, active and passive control methods for cavitation are mainly explored through numerical simulations and some experimental studies. Key techniques include Large Eddy Simulation (LES) and the Reynolds time-averaged method, with LES favored for its detailed turbulence capture despite high computational costs, and the Reynolds method preferred for its efficiency in extensive numerical analyses. Studies by Liu et al. [47], Li et al. [48,49], and Huang et al. [50,51] demonstrate LES’s effectiveness in analyzing cavitating flows. Roohi et al. [52], using LES and K-Omega SST models, studied cavitating flow around a pitching NACA 66 hydrofoil, analyzing vorticity, shear stress, and turbulence, revealing impacts on hydrodynamic loads and flow structures. They also, using LES [53], studied the behavior of a NACA 66 hydrofoil under combined oscillatory motion, examining the effects on cavitation, drag force, and boundary layer transition.

Yu et al. [54], Cheng et al. [55], Long et al. [56], and Han et al. [57] validate the Reynolds method’s accuracy and predictive capabilities in various cavitation scenarios. The Reynolds time-averaged method offers reasonable accuracy and computational efficiency. It is preferable for scenarios involving a large number of multi-group numerical calculations or when conducting parametric studies with numerous simulations.

Current research on active control methods for cavitation suppression primarily examines the effects of injection parameters at fixed positions. However, the comprehensive impact of varying these parameters—including jet position, inclination angle, and velocity—on the cavitating flow field remains underexplored. This gap highlights the need for a more nuanced understanding of how these variables collectively influence cavitation dynamics and hydrofoil performance.

Investigating multiple parameters of active jets can provide insights into how different parameters influence the flow field, leading to a broader understanding of the cavitation suppression mechanism. This research aims to address a gap in the current academic literature where most studies remain at the effectiveness verification level without exploring the detailed impact mechanism of various parameters. The novelty of this work is expanding on existing research by exploring multiple injection parameters, providing a more comprehensive perspective on active control strategies that builds upon the current body of literature.

In our prior work, we demonstrated the potential of active jet methods to significantly suppress cavitation by employing a single row of jet holes on a hydrofoil surface [44,58]. These experiments, however, were constrained by the limited range of jet positions and angles tested. Additionally, our numerical analyses [9] indicated that specific jet placements could adversely affect the hydrofoil’s lift–drag ratio, suggesting a trade-off between cavitation suppression and aerodynamic efficiency. Our previous studies focused on whether the active water injection method is effective, how effective it is, and the mechanisms behind its effectiveness, based on a single jet parameter. In contrast, the current study systematically investigates the influence of three jet parameters—jet position, jet velocity, and jet angle—on cavitation suppression and hydrofoil performance, providing a more detailed understanding of these mechanisms.

By systematically analyzing the roles of these variables, this work proposes an optimal combination of injection parameters, which is capable of enhancing cavitation control while maintaining, or even improving, the lift–drag ratio of hydrofoils. The manuscript is organized as follows: Section 2 describes the methodology, including the computational setup and mesh details. Section 3 elaborates on the numerical methods employed and the setup for the simulations. Section 4 introduces the orthogonal test method used to evaluate the influence of injection parameters on cavitation suppression and hydrofoil performance. Finally, Section 5 discusses the results of these tests, offering insights into the optimal injection parameters for improved flow control.

2. Research Object

2.1. Hydrofoil and Jet Configuration

Figure 1a illustrates the scaled model of the NACA66 hydrofoil, hereinafter referred to as the “original hydrofoil” or “H_ori” for conciseness, which will be discussed in subsequent sections of this paper. The hydrofoil features a chord length (C) of 0.07 m and a span length of 0.021 m, equivalent to 0.3C. Leroux et al. [59,60] performed detailed cavitation experiments on the NACA66 hydrofoil, providing the XY geometric data necessary to define the NACA66 profile’s shape in reference [59]. In our research, we utilize a reduced-scale model of the NACA66 hydrofoil, decreasing the chord length from the 150 mm reported in previous studies [59] to 70 mm, while preserving the proportional relationship between the dimensions.

Figure 1b depicts a hydrofoil equipped with a single row of nine jet holes on its suction surface, designed to inject water into the flow field. Each hole has a diameter (Φ) of 1.4 mm and is spaced 2.35 mm apart from its neighbors. This modified hydrofoil will henceforth be referred to as the “jet hydrofoil” or “H_ori”. In Figure 1e, three critical parameters of the jet configuration are highlighted. The holes are located a distance (L_jet) from the leading edge, and the angle of jet injection (α_jet) is measured in degrees relative to a vertical reference line. Notably, each jet hole releases water at a consistent flow rate, with the jet velocity denoted as U_jet. The mainstream flow direction is parallel to the x-axis, as shown in Figure 1e. Therefore, a positive α_jet implies water injection in the direction of the mainstream flow, while a negative α_jet indicates injection against it. An α_jet of 0° means the water is injected perpendicular to the mainstream flow.

The jet diameter Φ and distance d between each hole are considered fixed parameters. The jet angle α_jet, jet location L_jet, and jet velocity U_jet are regarded as variable parameters, as they are potential factors that may influence cavitating flows.

For ease of reference, these three factors are denoted as A, B, and C, respectively. To investigate their effects on the cavitating flows, a mixed-level orthogonal table of three factors has been constructed, as summarized in

Table 1. Factors and their corresponding levels in the study.

Levels	Jet Angle αjet (°) Factor A	Jet Location Ljet (C) Factor B	Jet Velocity Ujet (m/s) Factor C
Level 1	−60	0.19	2.60
Level 2	−30	0.30	2.74
Level 3	0	0.45	2.89
Level 4	+30	0.60	3.25
Level 5	+60	/	/

. The jet angle α_jet is varied across five different parameters: −60°, −30°, 0°, +30°, and +60°. Similarly, the jet location L_jet is set at four different levels: 0.19C, 0.30C, 0.45C, and 0.60C. Furthermore, the jet velocity U_jet is categorized into four groups: 2.60 m/s, 2.74 m/s, 2.89 m/s, and 3.25 m/s. Note that the pre-design table is just to indicate the factors and levels considered, without implying a direct correspondence between the row data.

By adopting this mixed-level orthogonal table, the numerical test groups are designed systematically to assess the combined and individual effects of these factors on the cavitating flows surrounding the hydrofoil.

2.2. Computational Domain and Mesh Arrangement

In order to establish the computational domain, a cuboid with dimensions of 16.0C × 4.0C × 0.3C is employed, as visually represented in Figure 2. The hydrofoil is positioned within this computational domain, with an attack angle of 8 degrees. The leading edge of the hydrofoil is positioned 4.0C away from the upstream velocity inlet, while the pressure outlet is situated 12.0C downstream from the leading edge to ensure fully developed flow conditions. The combination of velocity-type inlet conditions and pressure-type outlet conditions, which is considered the most robust setup in CFD, is supported by the large body of similar literature [61,62,63,64,65]. No-slip wall boundary conditions are imposed on the hydrofoil surface. Additionally, free-slip wall boundary conditions are implemented on the water tunnel wall, allowing fluid motion along the walls without shear forces. This approach is consistent with the methodology used in similar studies involving hydrofoils with truncated spanwise length [66,67].

Table 1 presents the variable parameters, namely the jet angle and jet position, which influence the structural configuration of the jet hydrofoil, leading to the generation of 20 distinct mesh configurations (4 × 5 = 20). Two representative mesh configurations of the jet hydrofoil are illustrated in Figure 3a,b.

This paper focuses on the operating condition of cloud cavitation. The cavitation number, a dimensionless parameter, is introduced to quantify the state of cavitating flow:

$$\sigma ={\displaystyle \frac{p_{out}-p_v}{0.5{\rho }_l{U}_{\infty }^2}}$$

(1)

where p_out is the outlet pressure and UȂ_∞ is the inlet velocity. p_v is the saturated vapor pressure, which in this paper is set as 1940 Pa (17 °C). ρ_l is the density of water, which is set as 998.2 kg/m³.

The numerical cases in this study are conducted under specific working conditions with a Reynolds number Re = 5 × 10⁵ and a cavitation number σ = 0.83, corresponding to the boundary condition of inlet velocity of 7.832 m/s and outlet pressure of 27,325 Pa. The initial field values for velocity and pressure are set according to the initial boundary settings. For the jet hydrofoil, the boundary condition of velocity inlet is set on the surface of the jet hole.

The generation of structured meshes (hexahedral lattice) based on geometric topology is carried out using the ICEM CFD 2021 R1 software. The block around the hydrofoil leading edge is set as topological C-Block in order to generate mesh with fine quality. Although the positions and angles of the jet holes differ among the 20 meshes, their geometric topology remains consistent. Consequently, the span nodes of the jet hydrofoils are uniformly distributed with 92 nodes, while the Y-direction grid of the turbulent core region is refined to 150 nodes. To facilitate a smooth transition from the foil grid to the jet hole grid, the chordwise nodes are also refined, resulting in 270 nodes, as depicted in Figure 3c. In the turbulent core region, the mesh size is approximately 0.35 mm × 0.40 mm × 0.25 mm (x × y × z). For the boundary layer mesh, the height of the first layer mesh on the hydrofoil wall is 0.003 mm, with a growth factor of 1.15 for subsequent layers. Figure 3e illustrates the detailed grid arrangement for the jet holes. In an effort to realistically simulate the shape of the “cylindrical jet” in the experiments [43,44,58], a total of 3875 grids are allocated for each jet hole. The jet hydrofoil is characterized by 92 nodes along its spanwise direction. The final generated mesh has a minimum orthogonal quality of 0.2098, a minimum skewness of 0.336, and a minimum angle of 27.39°.

The mesh independence verification for the hydrofoil is typically assessed by increasing the number of spanwise nodes [47,68]. Lift coefficient C_L and drag coefficient C_D are used as indicators to test the grid independence, and their expressions are as follows:

$$C_L={\displaystyle \frac{F_L}{0.5{\rho }_l{U}_{\infty }^{2}AC}},C_D={\displaystyle \frac{F_D}{0.5{\rho }_l{U}_{\infty }^{2}AC}}$$

(2)

where U_∞ is the inlet velocity and ρ_l is the water density. A and C are the spanwise and chordwise length of the hydrofoil, respectively. F_L and F_D are the lift force and drag force acting on the hydrofoil, respectively.

As shown by Figure 4, the change trend of C_L and C_D stabilizes when the number of spanwise nodes reaches 60. The grid resolution which comprises 60 spanwise nodes has demonstrated sufficient capability to resolve multiphase flow in the previous study [69]. Therefore, this grid configuration is adopted in this work. More information on the grid independence verification and Y plus distribution could be found in our previous work [69].

In comparison, the mesh of jet hydrofoil in

Table 2. Mesh configuration of the original hydrofoil and the jet hydrofoils.

Item	Total Number of Cells	Spanwise × Chordwise Nodes	Cells of Each Jet Hole
Original hydrofoil	6,566,400	60 × 220	/
Jet hydrofoils	13,828,200	92 × 270	3875

boasts superior analytical resolution when compared to the original hydrofoil, attributed to its higher number of spanwise and chord nodes, resulting in a total mesh size of nearly 14 million.

3. Numerical Method and Validation

3.1. Governing Equation and Turbulence Model

The cavitating flow is assumed as homogeneous in this study. The vapor and liquid phases share the same velocity and pressure. The continuity and momentum equations in Cartesian coordinates are expressed as follows:

$${\displaystyle \frac{\partial {\rho }_m}{\partial t}}+{\displaystyle \frac{\partial ({\rho }_m{U}_j)}{\partial x_j}}=0$$

(3)

$${\displaystyle \frac{\partial \left({\rho }_m{U}_i\right)}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_m{U}_i{U}_j\right)}{\partial x_j}}=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\mu }_m+{\mu }_t\right)\left({\displaystyle \frac{\partial U_i}{\partial x_j}}+{\displaystyle \frac{\partial U_j}{\partial x_i}}-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\partial U_k}{\partial x_k}}{\delta }_{ij}\right)\right]$$

(4)

where ρ_m, µ_m, and µ_t signify the density of the mixed phase and the coefficients for the mixed-phase laminar and turbulent viscosities, respectively. i, j and k refer to the coordinate directions, while p and U represent the pressure and velocity of the mixture phase, respectively.

The mass transport equation of the vapor-water two-phase mixture is expressed as follows:

$${\displaystyle \frac{\partial {\rho }_v{\alpha }_v}{\partial t}}+{\displaystyle \frac{\partial \left({\rho }_v{\alpha }_v{u}_j\right)}{\partial x_j}}={\dot{m}}^{+}-{\dot{m}}^{-}$$

(5)

$${\rho }_m={\rho }_l{\alpha }_l+{\rho }_v{\alpha }_v$$

(6)

$${\mu }_m={\mu }_l{\alpha }_l+{\mu }_v{\alpha }_v$$

(7)

where α_v represents the vapor volume fraction, with l and v denoting the liquid and vapor phases, respectively. The two terms on the right hand of Equation (5) are the evaporation term and the condensation term in the phase change process of cavitation, respectively.

The SST k-ω turbulence model, also known as the Shear Stress Transport k-ω turbulence model, is particularly effective in capturing complex boundary layer flow and accurately predicts flow separation, reattachment, and transitional boundary layers. The SST k-ω turbulence model is often used in previous work related to cavitating flows around hydrofoils [30,31,70,71].

In the SST k-ω turbulence model, two transport equations are solved: one for the turbulent kinetic energy (k) and the other for the specific dissipation rate (ω). The turbulent viscosity is computed from these equations, and the model includes a blending function that automatically switches between the k-ε and k-ω formulations based on the local flow conditions. The equations are as follows:

$${\displaystyle \frac{\partial }{\partial t}}({\rho }_{m}k)+\nabla \cdot ({\rho }_{m}kU)=\nabla \cdot \left[\left({\mu }_m+{\sigma }_k{\mu }_t\right)\nabla k\right]+P_k-{\rho }_m\beta *f_{\beta *}\left(\omega k-{\omega }_0{k}_0\right)+S_k$$

(8)

$${\displaystyle \frac{\partial }{\partial t}}({\rho }_m\omega )+\nabla \cdot ({\rho }_m\omega U)=\nabla \cdot \left[\left({\mu }_m+{\sigma }_{\omega }{\mu }_t\right)\nabla \omega \right]+P_{\omega }-{\rho }_m\beta f_{\beta }\left({\omega }^2-{\omega }_0^2\right)+S_{\omega }$$

(9)

where ∇·(ρ_mkU) represents the divergence of the convective term involving density ρ_m, turbulent kinetic energy k, and the velocity U. The terms σ_k and σ_ω are constants. The terms ρ_mβ*f_β* (ωk − ω₀k₀) and ρ_mβf_β(ω² − ω₀²) are the pressure-strain terms, where β* and β are model constants, f_β* and f_β are model functions, ω_k is the specific dissipation rate of turbulent kinetic energy, and ω₀k₀ is a reference value for ω_k. S_k and S_ω represent the turbulent kinetic energy source term and the turbulent specific dissipation rate source term, respectively. They account for any additional energy input to the turbulent flow. P_k and P_ω are the production terms, which represent the generation of turbulent kinetic energy and turbulent specific dissipation rate due to mean velocity gradients, respectively.

The turbulent viscosity μ_t is introduced to close the turbulence equations in the SST k-ω turbulence model [72,73], of which expression is as follows:

$${\mu }_t={\displaystyle \frac{{\rho }_m{a}_{1}k}{\max \left(a_1\omega ,\Omega F_2\right)}}$$

(10)

For turbulence simulations that take into account phase transition effects, the turbulent viscosity needs to be modified to ensure a more accurate simulation of the vapor phase:

$${\mu }_{t\_DCM}={\mu }_t\cdot {\displaystyle \frac{{\rho }_v+\left({\rho }_l-{\rho }_v\right){({\alpha }_l)}^n}{{\rho }_v+\left({\rho }_l-{\rho }_v\right){\alpha }_l}}$$

(11)

where n is a constant which is set as 10 according to the literature [9,10].

3.2. Cavitation Model

The Zwart-Gerber-Belamri cavitation model (ZGB), as proposed by Zwart et al. [74], has been developed to simulate cavitating flows surrounding a three-dimensional hydrofoil. The ZGB model incorporates meticulous linearization and coupling strategies to ensure favorable convergence behavior. Hence, the ZGB model is adopted as the phase transition model in this research, which is presented as follows:

$$P\le P_v,{\dot{m}}^{+}=C_{vap}{\displaystyle \frac{3{\alpha }_{nuc}\left(1-{\alpha }_v\right){\rho }_v}{R_B}}\sqrt{{\displaystyle \frac{2}{3}}{\displaystyle \frac{P_v-P}{{\rho }_l}}}$$

(12)

$$P\ge P_v,{\dot{m}}^{-}=C_{cond}{\displaystyle \frac{3{\alpha }_v{\rho }_v}{R_B}}\sqrt{{\displaystyle \frac{2}{3}}{\displaystyle \frac{P-P_v}{{\rho }_l}}}$$

(13)

$$P_v=P_{sat}+{\displaystyle \frac{1}{2}}\left(0.39\rho k\right)$$

(14)

where R_B represents the bubble radius, and its reference value is set at 1 μm. Additionally, the reference value for α_nuc, pertaining to the nucleation point, is established as 5 × 10⁻⁴. The parameters C_vap and C_cond, denoting the evaporation and condensation coefficients, respectively, possess reference values of 50 and 0.01.

3.3. Calculation Setup, Uncertainty Analysis and Validation of Numerical Results

The numerical solution of the three-dimensional cavitating flow is carried out using ANSYS Fluent 2021 R1. The simulation process involves multiple steps: initially, a steady simulation is performed with the cavitation model disabled. Then, a steady simulation of cavitating flow is executed, initiated from the results obtained in the previous non-cavitating flow simulation. Finally, the steady-state outcomes from the cavitating flow simulation are employed as the initial condition for the transient simulation.

The coupling of velocity and pressure fields is solved using the SIMPLE algorithm. The employed spatial discretization schemes include “Least Squares Cell Based” for gradient, “PRESTO!” for pressure, “Second-Order Upwind “ for momentum, and “Quick” for volume fraction.

In determining the computational time step for the unsteady simulations, the total flow time is established at 0.3 s, which is sufficient to encompass at least 5 cavitation cycles, thereby ensuring a representative analysis of the cavitation phenomena. The time step is set as Δt = 0.1 ms, and a maximum of 40 iterations per time step are performed to maintain computational accuracy. The under-relaxation factors for pressure and momentum are 0.2 and 0.3, respectively, and for vaporization mass and volume fraction are 0.8 and 0.4, respectively. The residual target of 1 × 10⁻⁵ for continuity is established to ensure convergence.

The uncertainty analysis is utilized as an initial step to assess the reliability of computational results. This study employs three distinct methods: the Correction Factor method (CF), Factor of Safety (FS), and Grid Convergence Index (GCI). Detailed descriptions of these methodologies and their application procedures can be found in references [75,76,77,78]. The monitoring points designated for velocity data extraction are illustrated in Figure 5, with each point positioned 0.1C from the hydrofoil’s suction surface.

Table 3. The uncertainty of the time-averaged velocity.

	V1	V2	V3	V4	V5
UCF	0.022	0.031	0.047	0.114	0.135
UFS	0.017	0.035	0.048	0.092	0.103
UGCI	0.035	0.042	0.056	0.089	0.122

presents the uncertainty associated with the time-averaged velocity measurements. The uncertainties estimated by each method are relatively small for points V₁–V₃ but significantly larger for V₄ and V₅, with some data exceeding the 95% confidence interval. This discrepancy is likely attributed to the periodic shedding of cavitation near the hydrofoil’s trailing edge, causing pronounced periodic fluctuations in key metrics like velocity, pressure, and lift coefficient. Such fluctuations result in a broad uncertainty range for both numerical and experimental data. Consequently, in cavitating flow studies, a qualitative comparison between numerical and experimental results is widely accepted as the standard verification method by most researchers in the field.

Figure 6b depicts the evolution of cavitating flow over one complete cycle. The attached cavitation expands from the leading edge to the trailing edge of the hydrofoil, followed by loss of stability, leading to the formation of a large free cloud that detaches from the hydrofoil’s trailing edge. The numerical simulation successfully captures the observed flow patterns from the experiment. Figure 6a presents the periodic curves of the non-dimensional cavitation area (S_nd = S_cav/S_hydrofoil, where S_hydrofoil represents the cross-section area of the hydrofoil, and S_cav is the cavitation area) obtained from both numerical simulations and experimental measurements. The S_cav can be obtained through digital image processing in experiments. In the numerical simulation, several equidistant cut-planes along the spanwise of hydrofoil are extracted and superimposed to calculate the cavitation area, S_cav. The numerical simulation shows a cavitation cycle duration of approximately 50.27 ms, while the experimental measurement is 53.73 ms, resulting in an error of approximately +6.44%.

Due to the introduction of a new inlet velocity condition for the jet, it is necessary to assess the continued applicability of the numerical method for jet hydrofoil. In our prior experiments, we conducted tests on a jet hydrofoil with specific parameters, featuring L_jet = 0.19C and α_jet = 0 degree. In this context, the experimental data at U_jet = 2.89 m/s serves as the basis for validation.

As shown in Figure 7a, the numerical simulation shows a cavitation cycle duration of approximately 75.14 ms, while the experimental measurement is 71.32 ms, resulting in an error of approximately +5.36%. Hence, the numerical results are capable of satisfactorily predicting the characteristics of the cavitation structure and its periodicity.

As depicted in Figure 7b, the presence of the jet makes significant changes to the attached cavitation structure. The stability of attached cavitation is disrupted, leading to the fragmentation of free clouds that detach from the hydrofoil surface. The dynamic characteristics of cavitation closely align with experimental observations. The cavitation is reduced by approximately 50% compared to the original scenario without the jet. However, it is important to highlight that this injection parameter combination (L_jet = 0.19C and α_jet = 0 degrees) led to a decrease in the lift–drag ratio [9], an issue that this paper aims to address and resolve.

4. Setup of Orthogonal Test Method

The orthogonal test method represents a systematic approach employed to investigate multiple factors and their respective levels. This methodology involves the selection of a well-defined set of representative test cases from a diverse dataset, possessing evenly dispersed and comparable characteristics [79]. This analysis allows for the reduction of the total number of tests conducted, leading to a shorter testing duration [80,81], while effectively identifying the optimal program.

The factors considered in this paper are the jet angle (α_jet), jet velocity (U_jet), and jet location (L_jet), each encompassing 5, 4, and 4 levels, respectively. Undertaking all possible parameter combinations would necessitate a total of 5 × 4 × 4 = 80 sets of numerical simulations, which is evidently impractical. Consequently, the orthogonal test method is applied to systematically explore the effects of these three factors and their respective levels on two crucial metrics: the cavitation suppression efficiency (η_cav.) and the energy performance (η_eng) of the hydrofoil. These metrics are expressed in percentiles and defined as follows:

$${\eta }_{cav}[\%]=\left({\overline{S}}_{nd-test}-{\overline{S}}_{nd-ori}\right)/{\overline{S}}_{nd-ori}$$

(15)

$${\eta }_{eng}[\%]=\left({\overline{K}}_{test}-{\overline{K}}_{ori}\right)/{\overline{K}}_{ori}$$

(16)

$$K=C_L/C_D$$

(17)

where the parameters ${\overline{S}}_{nd-test}$ and ${\overline{S}}_{nd-ori}$ denote the time-averaged S_nd of the jet hydrofoil and the original hydrofoil, respectively. Additionally, ${\overline{K}}_{test}$ and ${\overline{K}}_{ori}$ represent the time-averaged lift–drag ratio of the jet hydrofoil and the original hydrofoil, respectively. F_L and F_D are lift force and drag force acting on the hydrofoil, respectively. The negative values in η_cav. indicate a reduction in cavitation volume, implying effective cavitation suppression. On the other hand, positive or negative values in η_eng represent an increase or decrease in lift–drag ratio, respectively, indicating the improvement or deterioration of energy performance.

In this study, the three factors, namely jet angle (α_jet), jet velocity, and jet location, are designed with multiple levels in an orthogonal manner, leading to the generation of 20 different parameter combinations, each identified as H01 to H20. This mixed-level orthogonal table, along with the corresponding values of η_cav. and η_eng, are computed as follows in

Table 4. The mixed-level orthogonal table.

Series	Jet Angle (°)	Jet Location (C)	Jet Velocity (m/s)	ηcav	ηeng
H01	−60	0.19	2.60	−37.71%	−4.62%
H02	−60	0.30	2.74	−38.23%	−4.85%
H03	−60	0.45	2.89	−38.34%	−8.94%
H04	−60	0.60	3.25	−44.46%	−9.47%
H05	−30	0.19	3.25	−47.18%	−4.77%
H06	−30	0.30	2.89	−45.86%	−4.27%
H07	−30	0.45	2.74	−42.94%	−4.78%
H08	−30	0.60	2.60	−39.21%	−18.23%
H09	0	0.19	2.89	−49.09%	−3.69%
H10	0	0.30	3.25	−48.82%	−2.31%
H11	0	0.45	2.60	−42.34%	+1.33%
H12	0	0.60	2.74	−36.74%	−17.41%
H13	+30	0.19	2.74	−47.51%	−3.12%
H14	+30	0.30	2.60	−45.64%	+0.35%
H15	+30	0.45	3.25	−48.38%	+2.28%
H16	+30	0.60	2.89	−34.18%	+1.95%
H17	+60	0.19	2.60	−48.78%	−0.41%
H18	+60	0.30	2.74	−48.54%	+3.27%
H19	+60	0.45	2.89	−46.12%	+5.59%
H20	+60	0.60	3.25	−33.02%	+3.28%

:

In Table 4, a crucial characteristic of the table is the equal occurrence of each factor level across the different combinations. This balanced representation has been scientifically chosen to ensure that the variable is uniformly distributed among the various factor levels. Such a balanced design enhances the reliability and robustness of the test results, as it minimizes potential bias and confounding effects. In the subsequent chapter, a comprehensive analysis will be conducted to examine the effects of the factors and their respective levels on the two metrics η_cav. and η_eng.

5. Results and Discussion

5.1. Analysis of Orthogonal Results

To analyze the data in Table 4, the mean calculation in descriptive statistics is adopted. The average $\overline{i_j}$ with the same level under each factor and the range of levels R_j under the same factor are defined by the following two formulas:

$$\overline{i_j}={\displaystyle \frac{1}{k}}{\displaystyle \sum _{n=1}^k{y}_n}$$

(18)

$$R_j={\overline{i_j}}_{\max }-{\overline{i_j}}_{\min }$$

(19)

where the notation i_j refers to the value of the test result y, corresponding to a specific level i under a particular factor j.

Here, i represents the level within a given factor, and j denotes the column order of the orthogonal Table 4. Specifically, factor A (jet angle) is represented by j = 1, while j = 2 and j = 3 correspond to factor B (jet location) and factor C (jet velocity), respectively. The test result y, in the context of investigating the influence of parameters on cavitation suppression, represents the η_cav. Likewise, when studying the influence of parameters on energy characteristics, y signifies the η_eng. For example, the notation “3₂” would represent column 2 (factor B: jet location) and level 3 (0.45C). Thus, “3₂” would correspond to the test result y for the specific combination of factor B at level 3 in the orthogonal table.

The method aims to determine the average influence of a specific parameter on the indices η_cav. and η_eng. The average $\overline{{\eta }_{cav}}$ and $\overline{{\eta }_{eng}}$ values when a certain parameter in Table 4 remains at a consistent level are calculated. For instance, to obtain the average $\overline{{\eta }_{cav}}$ and $\overline{{\eta }_{eng}}$ at the level of −60 degrees for the parameter jet angle, we perform average processing on the data from series H01 to H04 in Table 4. These series represent different combinations of parameter values, including the other two parameters, jet location, and jet velocity.

The calculated result includes not only the impact of the independent parameter (jet angle) but also the influence of the other two parameters. The orthogonal design characteristics ensure that the frequency of occurrence of the other two parameters is balanced, thus minimizing interference. By considering the influence of all factors under orthogonal design conditions, the final selection of parameter combinations avoids unexpected abnormal results. This approach provides a comprehensive understanding of the combined impact of multiple parameters on the indices $\overline{{\eta }_{cav}}$ and $\overline{{\eta }_{eng}}$, allowing for more informed decisions in the analysis.

It is essential to note that due to the averaging process and the influence of the other parameters under orthogonal design conditions, the combination of parameters eventually selected by this method may not directly correspond to any specific combination in Table 4. Instead, the method may generate completely new combinations that are not originally present in Table 4.

The effect of injection parameters on the cavitation suppression performance is shown as follows in

Table 5. Orthogonal analysis of cavitation suppression efficiency $\overline{{\eta }_{cav}}$.

Levels	Jet Angle αjet Factor A	Jet Location Ljet Factor B	Jet Velocity Ujet Factor C
$\overline{1_j}$	−39.69%	−46.05%	−42.74%
$\overline{2_j}$	−43.80%	−45.42%	−42.79%
$\overline{3_j}$	−44.25%	−43.62%	−42.72%
$\overline{4_j}$	−43.93%	−37.52%	−44.37%
$\overline{5_j}$	−44.12%	/	/
Rj	4.56%	8.53%	1.65%

:

When the jet position is shifted from 0.45C to 0.60C, a significant reduction in $\overline{{\eta }_{cav}}$ is observed, indicating that the cavitation suppression effect is most sensitive to the jet position. Conversely, the jet velocity exhibited relatively minimal impact on $\overline{{\eta }_{cav}}$, with the corresponding R_j only 1.65%. As for the jet angle, its influence on $\overline{{\eta }_{cav}}$ lies between the other two parameters.

In summary, the order of importance of the three parameters concerning cavitation suppression is as follows: jet location (B) > jet angle (A) > jet velocity (C). For optimizing cavitation suppression efficiency alone, the most favorable parameter combination is A3B1C4, signifying a jet configuration with L_jet = 0.19C, α_jet = 0 degrees, and U_jet = 3.25 m/s. This combination is identified as the optimal configuration for achieving the highest cavitation suppression efficiency.

The effect of injection parameters on the hydrofoil energy performance is shown as follows in

Table 6. Orthogonal analysis of cavitation suppression efficiency $\overline{{\eta }_{eng}}$.

Levels	Jet Angle αjet Factor A	Jet Location Ljet Factor B	Jet Velocity Ujet Factor C
$\overline{1_j}$	−6.97%	−3.32%	−4.32%
$\overline{2_j}$	−8.01%	−1.56%	−5.38%
$\overline{3_j}$	−5.52%	−0.90%	−1.87%
$\overline{4_j}$	0.37%	−7.98%	−2.20%
$\overline{5_j}$	2.93%	/	/
Rj	10.95%	7.07%	3.51%

:

The abundant negative values in the table indicate that, regardless of the jet discharge configuration, the lift–drag ratio of the hydrofoil will be reduced. This reduction is likely attributable to the reaction force exerted by the jet flow on the hydrofoil body. Nevertheless, a few promising results stand out: when the jet angle is set to +30 degrees, there is a 0.37% increase in the lift–drag ratio, and at +60 degrees, the lift–drag ratio is even improved by 2.93%. The hydrofoil’s performance is notably influenced by the jet position, and positioning the jet at 0.60C leads to a considerable 7.98% decrease in hydrofoil performance. This analysis, along with the previous findings on $\overline{{\eta }_{cav}}$, suggests that placing the jet near the hydrofoil’s trailing edge is not a reasonable arrangement.

In summary, the order of importance of the three parameters concerning the lift–drag ratio is as follows: jet angle (A) > jet location (B) > jet velocity (C). For optimizing energy performance alone, the most favorable parameter combination is A5B3C3, indicating a jet configuration with L_jet = 0.45C, α_jet = +60 degrees, and U_jet = 2.89 m/s.

However, it is important to note that considering only one index to determine the best jet configuration may lead to a biased evaluation. Additionally, sacrificing energy performance to suppress cavitation cannot be regarded as an ideal flow control approach. Hence, the authors hope to combine both cavitation suppression efficiency and energy performance to identify the best parameter combination with the most balanced capabilities.

In similar studies, scholars have frequently utilized grey relational theory [82,83] to normalize multiple indices and transform multivariate optimization problems into univariate optimization problems represented as grey relational grades. However, it is important to acknowledge that the rankings and results obtained through this method are solely based on the data available within the given dataset and do not guarantee the best combination of parameter values beyond the original dataset.

In the previous content, the final selection of parameter combinations takes into account the combined impact of multiple parameters on the indices $\overline{{\eta }_{cav}}$ and $\overline{{\eta }_{eng}}$. This allows for more comprehensive exploration of the parameter and the potential discovery of novel combinations with excellent performance. Consequently, the graphical representation of the data in Table 5 and Table 6 simplified the process of intuitively selecting the optimal combination, making the decision-making procedure both straightforward and efficient.

Based on the observations from Figure 8a, minimal differences are noted in $\overline{{\eta }_{cav}}$ at jet velocities of 2.60, 2.74, and 2.89 m/s, while the lift–drag ratio exhibits significantly low values at these velocities. In contrast, a jet velocity of 3.25 m/s demonstrates satisfactory performance for both $\overline{{\eta }_{cav}}$ and $\overline{{\eta }_{eng}}$ Consequently, parameter C4 is considered a suitable candidate for further consideration.

Moving to Figure 8b, it becomes apparent that the jet positioned at 0.60C yields poor performance in both $\overline{{\eta }_{cav}}$ and $\overline{{\eta }_{eng}}$. In contrast, jets positioned at 0.19C, 0.30C, and 0.45C exhibit no significant differences in their respective cavitation suppression effects and lift–drag ratios. Thus, parameters B1, B2, and B3 emerge as viable options.

Finally, Figure 8c reveals poor performance of both $\overline{{\eta }_{cav}}$ and $\overline{{\eta }_{eng}}$ at a jet angle of −60 degrees. However, as the angle shifts from −30 to +60 degrees, the lift–drag ratio of the hydrofoil gradually increases, with a notable improvement in $\overline{{\eta }_{eng}}$ when the angle shifts from 0 to 30 degrees. In terms of cavitation suppression, all angles, except for −60 degrees, yield non-significant differences in $\overline{{\eta }_{cav}}$. Consequently, parameters A4 and A5 are identified as promising alternatives.

Considering the analysis above, 1 × 3 × 2 = 6 parameter combinations have the potential to achieve a desirable balance between lift–drag ratio and cavitation suppression. These combinations hold promising potential for optimizing the performance of the hydrofoil in terms of both energy efficiency and cavitation mitigation.

To further screen out parameter combinations of better performance, two constraints are imposed: (1) The lift–drag ratio must be improved, i.e., η_eng > 0, and (2) The cavitation suppression efficiency must surpass 44%, i.e., $\overline{{\eta }_{cav}}$ < −44% (based on the evidence of the mean $\overline{{\eta }_{cav}}$ of the 20 combinations in Table 4, which is −43.44%). Following these constraints, two combinations, A5B1C4 and A5B2C4, remain as potential candidates.

However, it is evident that for B1 (L_jet = 0.19C), $\overline{{\eta }_{eng}}$ is −4.62%, whereas for B2 (L_jet = 0.30C), $\overline{{\eta }_{eng}}$ is −1.16%, indicating a noticeable difference. On the other hand, for $\overline{{\eta }_{cav}}$, B1 is −46.05%, and B2 is −45.42%, with no significant difference between the two. As a result, the combination A5B2C4 (L_jet = 0.30C, α_jet = 60°, and U_jet = 3.25 m/s) is chosen to minimize any adverse impact on the energy characteristics, even though it sacrifices approximately 1% of $\overline{{\eta }_{cav}}$. This final combination does not belong to any cases in Table 4.

Subsequently, numerical simulations are conducted for the final selected combination, A5B2C4 jet hydrofoil. The next two sections present the findings related to its cavitation suppression efficiency, energy performance, and flow control mechanisms. Through these comprehensive analyses, a deeper understanding of the optimal jet configuration’s capabilities and effects will be achieved.

In our previous study, the single jet configuration (L_jet = 0.19C, α_jet = 0°, U_jet = 3.25 m/s) was observed to significantly reduce the cavitation volume in the experiment [44]. However, our numerical simulations [9,10] showed that this configuration damaged the lift–drag ratio, with the lift force being reduced by 23.71%. The newly optimized A5B2C4 jet configuration reduces the cavitation volume by 49.34% while increasing the lift–drag ratio by 8.55% and reducing the lift by only 1.43%. Compared to previous studies, these latest results ensure significant cavitation suppression while improving the lift–drag ratio, which is an exciting outcome.

5.2. Cavitation Suppression Performance (σ = 0.83)

The A5B2C4 jet hydrofoil is referred to as H_opt for short, and the original hydrofoil with no jet is referred to as H_ori. They are conducted in numerical simulation with the same boundary condition (U_∞ = 7.832 m/s, P_out = 27,325 Pa, σ = 0.83). The monitored data of non-dimensional cavitation area for these two hydrofoils are depicted in Figure 9. The temporal evolution of cavitation structures, combined with absolute pressure contours on the hydrofoil, in one cycle are displayed in Figure 10. The structure of the cavitation cloud is plotted by an iso-surface with the vapor phase volume fraction of 10%.

After implementing the optimal jet scheme (A5B2C4), a noticeable reduction in cavitation volume is observed. The S_nd of the original hydrofoil (H_ori.) predominantly varies between 0.2 and 1.4, while the S_nd of the jet hydrofoil (H_opt) varies mainly between 0.1 and 0.8. Through calculations, the cavitation volume under the A5B2C4 scheme is reduced by an impressive 49.34%. Comparing cavitation periods, T_ori (the cavitation period of H_ori.) is measured at 51.78 ms, whereas T_opt (the cavitation period of H_opt) is 56.84 ms. The jet’s effect extends the cavitation period by approximately 9.77%.

For the original hydrofoil H_ori, the physical process corresponding to the monitoring curve in the rising stage involves the continuous expansion of attached cavitation until it reaches the trailing edge of the hydrofoil. The wave peak of the curve signifies the formation of highly unstable three-dimensional cloud structures, followed by the descending stage, which corresponds to the shedding of the unstable cloud. Similarly, under the influence of the jet, the cavitation motion pattern for H_opt is similar. It also undergoes the growth of attached cavitation (from 1/6 to 3/6 of T_opt) and the subsequent shedding of free clouds (from 4/6 to 5/6 of T_opt).

It is observed that the attached cavitation passes through the jet holes during 1/6 to 2/6 of T_opt. However, with a value of σ = 0.83, representing a relatively severe cloud cavitation state, the jet cannot completely block the expansion of attached cavitation. Nevertheless, unlike the stable strip structure observed in H_ori, the attached cavitation of H_opt exhibits a distinctive comb-like structure under the influence of the jet. Although the attached cavitation passes through the jet hole, it becomes destabilized and fractured shortly after (at 3/6 of T_opt). This indicates that the jet destabilizes the attached cavitation before it reaches its most intense level. By 4/6 of T_opt, the unstable cloud exhibits a group of scattered clouds, likely due to the jet flow disrupting the internal structure of the cloud. These scattered clouds then undergo incomplete fusion or collapse, forming smaller free clouds that subsequently shed away from the hydrofoil (at 5/6 of T_opt).

The pressure contour distribution on the surface of the hydrofoil provides additional insight into the cavitation dynamics. Cavitation results from a drop in local fluid pressure. During the most intense stage of cavitation development, between 3/6 T_ori/T_opt and 4/6 T_ori/T_opt, the pressure on the suction surface of the original hydrofoil is almost entirely below 2000 Pa, whereas high-pressure areas are still visible on the jet hydrofoil. At 5/6 T_ori/T_opt, the low-pressure distribution at the trailing edge of the hydrofoil explains the difference in the size of the shed cloud. Additionally, when the cavitation volume is at its minimum, corresponding to 1/6 T_ori/T_opt and 6/6 T_ori/T_opt, the pressure on the pressure surface of the jet hydrofoil is globally higher than that of the original hydrofoil. Overall, the surface pressure on the jet hydrofoil is generally higher than on the original hydrofoil, indicating an improved pressure environment in the flow field. The pressure environment created by the jet hinders the development of cavitation.

In Figure 10, t₁ and t₂ represent the moments when the cavitation volume of the original hydrofoil (H_ori) and the jet hydrofoil (H_opt) reach their maximum, respectively. At these two specific moments, ten y-o-z sections are taken along the chordwise direction of both hydrofoils. The obtained data for vapor phase volume fraction is presented in Figure 11.

Examining the vapor phase distribution at 0.1C and 0.2C, it is observed that for the original hydrofoil (H_ori), the red regions indicate high vapor phase concentration in the attached cavitation. Conversely, in the case of the jet hydrofoil (H_opt), the front exhibits an incomplete attached cavitation structure with a relatively lower vapor phase concentration. This suggests that the jet, situated at 0.3C, exerts a controlling effect on the cavitation in front of it. This is attributed to the low pressure near the leading edge and the high pressure near the jet hole, which attracts the jet flow towards the front, acting on the attached cavitation and leading to the fragmented vapor phase structure. At 0.3C, 0.4C, and 0.5C, a significant number of vapor phase distributions are observed above the suction surface of H_ori, with their thickness increasing. In contrast, for H_opt, there is almost no vapor phase, confirming that the jet causes the attached cavitation to break.

From 0.6C to 1.0C, it is evident that the area of vapor phase expands significantly for both hydrofoils. However, the height of the vapor phase region in the jet hydrofoil (H_opt) is notably lower than that in the original hydrofoil (H_ori). Particularly, at 0.7C to 0.9C, for H_ori, the distribution of α_v > 90% is continuous and complete, while for H_opt, the distribution of α_v > 90% appears scattered. The red contour regions are interspersed with green contour regions (approximately α_v = 50%). This observation provides supporting evidence that the jet disrupts the internal structure of the clouds, leading to the formation of fragmented and scattered vapor phase regions.

Derived from Equations (3) and (4), the mass transfer equation for the conservation of the vapor volume fraction is shown as follows:

$${\displaystyle \frac{D{\alpha }_v}{Dt}}={\displaystyle \frac{\partial {\alpha }_v}{\partial t}}+\left(u\cdot \nabla \right){\alpha }_v={\displaystyle \frac{{\rho }_m}{{\rho }_l-{\rho }_v}}\nabla \cdot u$$

(20)

Combining the above equation with the transport equation of vapor volume fraction Equations (5), (12), and (13), the evaporation and condensation are derived to the following equation [84]:

$$\dot{m}={\displaystyle \frac{{\rho }_l{\rho }_v}{{\rho }_l-{\rho }_v}}\nabla \cdot u$$

(21)

This expression shows that the velocity divergence is directly associated with the rate of mass transfer between the vapor and liquid.

The contours of $\dot{m}$ on the hydrofoil’s mid-plane are captured at three critical moments: during the development of attached cavitation, when the cavity volume reaches its maximum, and during the shedding of unstable scattered clouds, as illustrated in Figure 12.

A negative value of $\dot{m}$ signifies a local tendency for fluid condensation, while a positive value indicates evaporation. During the development of attached cavitation, it is evident that the evaporation area is intercepted by the jet. At the point of maximum cavitation volume, as depicted in Figure 12b, the original hydrofoil H_ori predominantly exhibits areas prone to evaporation. In contrast, the jet hydrofoil H_opt shows a markedly reduced evaporation area, with a noticeable increase in negative regions, particularly near the jet hole—highlighted within the circular box. This suggests that the jet fosters local fluid condensation, thereby suppressing cavitation. When the cloud becomes unstable and detaches, the original hydrofoil displays large, alternately distributed positive and negative regions of $\dot{m}$, indicating drastic vapor–liquid exchange. However, the pattern on H_opt is significantly more subdued. The introduction of the jet promotes vapor condensation and moderates the intensity of the vapor–liquid exchange process, effectively suppressing cavitation. This observation corroborates the findings in Table 4, affirming that cavitation can be mitigated with various jet parameter combinations, thus supporting the general rule that jet intervention facilitates cavitation control.

Figure 13 presents the chordwise distribution of the vapor phase volume fraction (α_v) near the wall of the two hydrofoils. The data are collected from sampling points located at the mid-section of the hydrofoil’s near-wall region, representing time-averaged values. From 0 to 0.2C, the time-averaged α_v of both hydrofoils appears to be nearly identical. However, at 0.29C, 0.30C, and 0.31C, α_v shows zero, suggesting the absence of cavitation in these regions. Interestingly, unusually high values of α_v are observed at 0.25C to 0.28C and 0.32C to 0.33C, indicating the presence of cavitation in the vicinity of the jet hole due to the interaction of the jet with the hole edge. After 0.34C, the time-averaged α_v of H_opt is consistently lower than that of H_ori.

5.3. Energy Performance and Pressure Distribution

Figure 14 presents the time-domain information of the lift coefficient, drag coefficient, and lift–drag ratio for both hydrofoils. The dotted line represents the time-averaged value. The time fluctuations of the three energy performance indexes for the jet hydrofoil exhibit similarities with those of the original hydrofoil. Their periodicity corresponds to the respective cavitation periods. The jet’s influence is observed to reduce the lift coefficient by 1.43%. Moreover, Figure 14b illustrates a remarkable 9.19% reduction in the drag coefficient, indicating the jet’s significant drag reduction effect on the hydrofoil. Most notably, the lift–drag ratio experiences an exciting 8.55% increase in Figure 14c.

Evidence of the observed increase in lift–drag ratio can be observed in Figure 15. Notably, the X velocity distribution near the jet hole significantly exceeds values at other positions, resembling the effect of installing a power jet device on the wing. The velocity distribution downstream at 0.35C from the jet hydrofoil shows a resemblance to that of the original hydrofoil, indicating similar re-entrant jet strengths in both scenarios. Despite the water barrier emitted by the jet holes preventing the re-entrant jet from advancing toward the hydrofoil’s leading edge, it does not appear to weaken its strength. Unlike traditional studies on cavitating flow control, which generally explain the mechanism as weakening the re-entrant jet to suppress cavitation, the mechanism of cavitation suppression by the jet method in this study may differ from the conventional approach.

Next, the pressure distribution of H_opt is investigated, and the pressure coefficient C_p is defined as follows:

$$C_p={\displaystyle \frac{p-p_{out}}{0.5\rho U_{\infty }^2}}$$

(22)

The adverse pressure gradient promotes the propagation of the re-entrant jet, resulting in serious flow separation, which is the main cause of cavitation instability. In this paper, the gradient of pressure coefficient to space is defined, as shown in the following equation:

$$grad{C}_p={\displaystyle \frac{\partial C_p}{\partial \left(x/C\right)}}$$

(23)

Negative values of gradC_p indicate an adverse pressure gradient from the trailing edge to the leading edge, while positive values indicate a favorable forward pressure gradient from the leading edge to the trailing edge. Figure 16 depicts the time evolution of gradC_p over two cavitation cycles as a function of both space and time.

In Figure 16a, enclosed by dotted lines in the circular box, the adverse pressure gradient on H_ori is observed to develop steadily and continuously over time. In contrast, Figure 16b shows scattered red regions interspersed with blue areas. The jet creates an interweaving pattern of forward and inverse pressure gradients on the hydrofoil’s surface, disrupting the continuous space-time development of gradC_p. This observation indicates that the jet has a favorable influence in decelerating flow separation. Notably, a blocky area of inverse pressure gradient (marked by the rectangular box) is observed near the jet location. This leads to the attraction of the jet towards the leading edge and consequently disrupts the attached cavitation structure.

Figure 17 shows the chordwise distribution of time-averaged pressure coefficient from the near-wall region of the two hydrofoils. Before the jet position at 0.3C, the pressure coefficient (C_p) distributions of the two hydrofoils exhibit close similarities. However, beyond 0.3C, the C_p of the jet hydrofoil demonstrates a notable overall increase. Compared with the original hydrofoil, the C_p after the jet position exhibits an average increase of 9.07%. As widely acknowledged, the lift generated by hydrofoil results from the pressure difference between its suction surface and pressure surface. Hence, the decrease in lift coefficient of the jet hydrofoil can be attributed to this rise in C_p. Nonetheless, this phenomenon may not necessarily be disadvantageous. The augmented pressure coefficient on the suction surface creates an unfavorable environment for the survival of the vapor phase, which is favorable from the perspective of cavitation suppression. Consequently, this reaches the issue of achieving a trade-off between energy performance and cavitation suppression, which returns to the primary objective of this research.

5.4. Influence Mechanism of Injection Parameters on Flow Performance

Based on the conclusions and analyses presented in the preceding sections, this section aims to explore and summarize the mechanisms underlying the influence of injection parameters on hydrofoil energy performance and cavitation suppression. To facilitate comprehension, the interaction between the jet and cavitation is conceptualized in a graph, where the jet is divided into vertical and horizontal components, as depicted in Figure 18.

The jet plays a vital role in impeding the expansion of attached cavitation, attributed to its vertical component. In Figure 18a, when the jet angle (α_jet) is positive, a more inclined jet direction relative to the X-axis results in a larger horizontal component. As illustrated in Figure 8c, the average $\overline{{\eta }_{eng}}$ increases with an escalating jet angle, ranging from 0 degrees to +60 degrees. Thus, it is inferred that the positive horizontal component of the jet contributes to an increase in the hydrofoil’s lift–drag ratio. Conversely, with a negative jet angle, the horizontal component conflicts with the main stream and even strengthens the incoming re-entrant jet, as depicted in Figure 18b, leading to a decline in the hydrofoil’s lift–drag ratio. This inference is further corroborated by the observed low levels of $\overline{{\eta }_{eng}}$ at negative angles in Figure 8c.

Notably, when the jet angle is −60 degrees, the jet plays a role of an additional re-entrant jet stream. This exacerbates cavitation instability. The strengthened re-entrant jet and its counteracting impact on cavitation suppression ultimately result in a very low cavitation suppression efficiency.

The stability of attached cavitation diminishes as it extends towards the trailing edge, becoming extremely unstable when it nears the hydrofoil trailing edge. As depicted in Figure 18c, placing the jet in close proximity to the trailing edge yields unfavorable intervention effects. The cavitation instability on the suction surface disrupts the hydrofoil’s energy performance, rendering the jet at this position insufficient in improving the lift–drag ratio. Consequently, the jet at 0.60C in Figure 8b exhibits lower levels for both η_cav. and η_eng.

These analyses shed light on the intricate interplay between injection parameters and their impacts on cavitation suppression and hydrofoil energy performance. The findings underscore the importance of considering jet angle and position for achieving an optimal balance between lift enhancement and cavitation mitigation, contributing to a deeper understanding of the jet method’s potential for flow control applications.

6. Conclusions

This study explored the impact of multiple injection parameters in active water injection, including jet angle, jet location, and jet velocity, on cloud cavitation (σ = 0.83) suppression around a model scale NACA66 hydrofoil through numerical simulation. The mixed-level orthogonal test method is employed to systematically investigate the influence of the three factors and their levels on cavitation suppression efficiency and energy performance of the hydrofoil. The detailed conclusions are as follows:

(1) The jet location (L_jet) emerged as the paramount factor in cavitation suppression, with the most effective suppression observed at L_jet = 0.19C, α_jet = 0 degrees, and U_jet = 3.25 m/s. In contrast, the jet angle (α_jet) was identified as the primary influencer on the hydrofoil’s energy performance, achieving optimal results at L_jet = 0.45C, α_jet = +60 degrees, and U_jet = 2.89 m/s.

(2) Striking a balance between cavitation suppression and energy efficiency, an optimal jet configuration was found at L_jet = 0.30C, α_jet = +60 degrees, and U_jet = 3.25 m/s. This setup notably reduced the cavitation volume by 49.34% and improved the lift–drag ratio by 8.55%, albeit with a slight decrease in the lift coefficient by 1.43%.

(3) Detailed analysis of the cavitation suppression mechanism revealed that the active jet significantly alters the internal structure of cavitation clouds. The jet significantly promotes vapor condensation, diminishes the intensity of the vapor–liquid exchange. It induces fragmentation in the vapor phase regions, disrupts the formation of intense attached cavitation, and raises the pressure on the hydrofoil’s suction surface. This counteracts the adverse pressure gradient, curbing flow separation and fostering flow field stabilization.

(4) Analyzing the A5B2C4 case and 20 other cases revealed how jets affect cavitation and lift–drag ratios, with positive jets enhancing and negative angles diminishing lift–drag ratios. Jet placements near the hydrofoil’s end were found to harm cavitation control and energy efficiency, offering valuable reference for injection parameter optimization to balance cavitation suppression with energy performance.