A Study on the Propulsion Performance of Hybrid-Driven Underwater Glider Equipped with a Kappel Tip Rake Propeller

Abstract

In order to solve the problem of the lack of maneuverability of the conventional underwater glider, this paper proposes a hybrid-driven underwater glider equipped with a Kappel tip rake propeller, analyzes the propulsion performance of different types of Kappel tip rake propellers in the wake field of the hybrid-driven underwater glider, optimizes the overall propulsion performance of the hybrid-driven underwater glider, and realizes self-propulsion and gliding with high efficiency and low energy consumption. In the research process, the Schnerr–Sauer cavitation model and the cavitation simulation strategy of VOF two-phase flow were adopted, coupled with the SST k-ω and γ transition turbulence model, and the control calculation error was not more than 3%. Based on the hydrodynamic performance study of the Kappel tip rake propeller, the self-propelled simulation was carried out under the working conditions of 6 kn, 5 kn, 4 kn, and 3 kn, and the gliding simulation was carried out under the working conditions of 1 kn, 0.5 kn, and a glide angle of 12°. The propulsion performance of the hybrid-driven underwater glider with different models of Kappel tip rake propellers was analyzed. It was found that the maximum open water propulsion efficiency of the propeller Kap05 had the largest improvement, which was 3.07% higher than that of the reference base propeller. Under the self-propelled condition, the hybrid-driven underwater glider with the propeller Kap05 had the lowest wake fraction, and the propellers Kap04 and Kap05 had the best propulsion performance in the wake field of the hybrid-driven underwater glider. In the gliding condition, the form of the folding paddle can reduce the gliding resistance generated by the propeller by more than 45% and the gliding negative lift by more than 68%. A moderate tip rake can effectively improve the propulsion efficiency of the Kappel tip rake propeller in the wake field of the hybrid-driven underwater glider, reduce energy loss, and improve the overall performance of the hybrid-driven underwater glider.

1. Introduction

At present, with the rise of global energy prices and the advancement of the dual carbon project, the energy-saving demands of underwater vehicles are becoming increasingly prominent. Among them, underwater gliders have attracted more and more attention in the field of marine engineering by virtue of their outstanding advantages such as low navigation resistance, high energy utilization rate, and large sailing distance. However, the conventional underwater glider realizes the underwater gliding movement of a fixed track by adjusting its own buoyancy and center of gravity, which is difficult to reach a specific position to achieve multi-angle and approach-type detection functions, and lacks maneuverability. At the same time, due to the lack of the large power source of conventional underwater gliders, it is difficult to maintain a stable track in complex ocean current environments. Therefore, a high-efficiency thruster is added to the stern of a conventional underwater glider to form a hybrid-driven underwater glider [1,2]. The Kappel tip rake propeller [3,4] is a special propeller with special blade tip geometry, which can improve the propulsion efficiency to a certain extent, and has high application value and potential market space.

The Kappel tip rake propeller is a type of special propeller that is beneficial in improving propulsion efficiency, with the blade end plate tilted towards the suction side [5]. The end plate of the propeller is similar to the winglet mounted on the wing, and the end plate can block the flow around the blade tip, thereby enlarging the pressure difference between the suction side and the pressure side, that is, becoming a positive component of thrust [6,7,8,9,10]. Andersen et al. [11,12] first studied the Kappel tip rake propeller through the potential flow theory mainly based on the lift line theory. Cai [13] optimized the lift line theory, considered the effect of the geometry characteristics of the blade tip rake on the induced velocity in the three-dimensional direction, and analyzed the hydrodynamic performance of the Kappel propeller. Wu [14] studied the geometry characteristics of the Kappel propeller and proposed a three-dimensional coordinate conversion formula that takes into account the blade tip rake.

Compared to conventional propellers, Kappel tip rake propellers have their own unique advantages. Wang et al. [15,16] showed that the Kappel tip rake propeller improved propulsion efficiency by approximately 4% compared to conventional propellers in real-scale operational tests. Wang Rui et al. [17] tested the propeller performance of the Kappel tip rake propeller in the cavitation state in a cavitation water cylinder, and the efficiency of the Kappel tip rake propeller at the design working point was 1.44% higher than that of the conventional propeller, and its end plate structure could inhibit the propeller tip vortex and block the trend of cavitation along the propeller guide edge. Huang [18] focused on the cavitation performance of the Kappel tip rake propeller and found that the pitch and rake of the blade tip had a significant impact on the cavitation performance of the propeller, which provided guidance for the geometric design of the pitch and rake of the propeller.

The Kappel tip rake propeller geometry has changed, and the performance is also different. Chen et al. [19] obtained the propeller blades with different bending degrees of the 6-type blade tip plate by changing the tip rake and found that the propeller efficiency of the blade tip with a tip rake in the range of 0.08~0.10 was the highest, and it was difficult to improve the propeller propulsion efficiency if the blade tip rake was too large or too small. The effect of increasing the thrust coefficient caused by the increase in the bending degree of the end plate gradually weakened. At the same time, with the increase in the water-soaked area of the propeller, the additional frictional resistance of the end plate part due to the increase in the area increases, resulting in the increase in the torque coefficient of the propeller as a whole, and the benefit of the end plate blocking the backflow of the blade tip is limited, resulting in the reduction in propeller efficiency.

In this paper, a series of Kappel propellers were combined with an underwater glider (UG) to form a hybrid-driven underwater glider, and the application analysis was carried out. Therefore, it is necessary to discuss the research status of traditional underwater gliders and hybrid-driven underwater gliders in this section.

For the limitations brought about by the traditional drive mode, the researchers put forward the concept of the hybrid-driven underwater glider, that is, on the basis of the traditional underwater glider applying other strong propulsion devices. Jenkins et al. [20] proposed to install a propeller in the stern of the Slocum underwater glider to improve the maneuverability of the underwater glider. For the Folaga hybrid-driven underwater glider, propulsion devices were installed in both the bow and stern according to Madureira et al. [21]. Todd et al. [22,23] developed a flat rectangular underwater glider named AUV-Glider, which adopted an open-frame design and installed multiple propulsion devices to realize bidirectional motion.

In the early stage, the researchers did not focus on the design of suitable propellers for the hybrid-driven underwater glider. Clayton [24] installed a propeller with high propulsive efficiency in the stern of the Hybrid Slocum hybrid-driven underwater glider. This propeller can maintain the horizontal speed of 2 kn while ensuring the working efficiency is roughly equivalent to the gliding motion mode, which effectively solves the energy consumption problem of the hybrid-driven underwater glider. In view of this, this paper also tries to combine a series of Kappel propellers with high efficiency into the underwater glider to carry out application research.

At present, there is little research on propellers suitable for hybrid-driven underwater gliders. Chen et al. [25,26] developed and designed a collapsible propeller for the hybrid-driven underwater glider, aiming at the problem of increasing the drag of the propeller when gliding. The propeller is a double-blade propeller; the blade can be shrunk and folded to the center, and the opening and closing angle can be adjusted. Bi et al. [27] combined a hybrid-driven underwater glider with a rotatable folding propeller, and the hydrodynamic simulation results of the hybrid-driven underwater glider showed that, during navigation in gliding mode, if the blade of the foldable propeller is shrunk and folded, the glide resistance of the body will be reduced by more than 45%, and the resistance reduction can even reach 60% under low speed conditions. The above provides a guide to solve the propeller drag increase problem encountered in the application research of the Kappel propeller in this paper.

In this paper, we focus on the tip rake of the blade; adjust, modify, and design the tip rake distribution of the series Kappel propellers; and study the effect of the tip rake on the open water propulsion performance and sheet cavitation performance of the series Kappel propellers [19]. At the same time, the self-propelled propulsion performance of the series Kappel tip rake propeller in the wake field of the underwater glider was further studied from the perspectives of actual propulsion efficiency and companion current fraction, and the optimal tip rake value in different flow field conditions was obtained, which provided guidance for the practical application of Kappel propellers.

2. Introduction to the Theory of Propeller–Ship Interaction and Turbulence Model

2.1. Theory of the Interaction between Propeller and Ship Body

The propeller is generally installed in the stern of the underwater vehicle, and the wake of the underwater vehicle will affect the inflow of the propeller, which is different from the open water condition. In the process, there is an interaction between the propeller and the ship body, and some parameters are needed to evaluate, so as to evaluate the degree of adaptation between the propeller and the underwater vehicle. Due to the difference between propeller advance speed V_A and navigation speed V, wake fraction ω is commonly used to describe the influence of the ship body on propeller advance speed. The formula is as follows [28]:

$$\omega ={\displaystyle \frac{V-V_A}{V}}$$

(1)

where ω is the wake fraction, V_A is the propeller advance speed, V_A = JnD, J is the speed coefficient, n is the propeller speed, D is the propeller diameter, and V is the navigation speed.

At the same time, the operation of the propeller will also accelerate the flow field in the stern of the ship body, resulting in increased pressure differential resistance of the ship body and resulting in thrust reduction. The thrust decrement fraction t_p is commonly used to characterize the propeller’s influence on the ship body, and the formula is as follows [28]:

$$t_p={\displaystyle \frac{T-R}{T}}$$

(2)

where t_p is the thrust decrement fraction, R is the hull resistance without propellers installed at the same speed, and T is the propeller thrust.

Hull efficiency η_H can be used to represent the influence of wake and thrust decrement on the whole system as follows [28]:

$${\eta }_H={\displaystyle \frac{RV}{T{V}_A}}={\displaystyle \frac{1-t_p}{1-\omega }}$$

(3)

where η_H is the hull efficiency.

2.2. Turbulence Model

The SST k-ω+γ transition turbulence model is adopted in this paper. The SST k-ω turbulence model combines the far-field model and the near-wall model, can deal with both low and high Reynolds number regions at the same time, and can predict the pressure gradient changes in the non-equilibrium region of the boundary layer more accurately. The formula is as follows [29]:

$${\displaystyle \frac{\partial k}{\partial t}}+u_j{\displaystyle \frac{\partial k}{\partial x_j}}=P_k-{\beta }^{*}k\omega +{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\nu +{\sigma }_k{\nu }_t\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]$$

(4)

$${\displaystyle \frac{\partial \omega }{\partial t}}+u_j{\displaystyle \frac{\partial \omega }{\partial x_j}}=\alpha S^2-\beta {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\nu +{\sigma }_{\omega 1}{\nu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+2\left(1-F_1\right){\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{\partial k}{\partial x_i}}{\displaystyle \frac{\partial \omega }{\partial x_i}}$$

(5)

where ω is the dissipation rate of specific turbulent kinetic energy; S defines eddy viscosity; P_k stands for production limiter; F₁ stands for first mixing function; ν_t is the viscosity of turbulent motion; and β^*, α, β, σ_k, σ, _ω1, and σ_ω2 are empirical constants.

On this basis, the turbulence transition of layer flow direction exists on the surface of the propeller under low Reynolds number conditions, and it is necessary to add a transition model to the turbulence numerical model to improve the simulation accuracy. The γ transition model is a single equation model; it is a local method to reduce partial definition, which can save computational resources. Therefore, in this paper, the γ transition model is selected with the SST k-ω turbulence model, in which the transfer equation of intermittent factor γ is as follows [29]:

$${\displaystyle \frac{\partial \left(\rho \gamma \right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \gamma u_i\right)}{\partial x_i}}=P_{\gamma }-E_{\gamma }+{\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\gamma }}}\right){\displaystyle \frac{\partial \gamma }{\partial x_i}}\right]$$

(6)

$$P_{\gamma }=F_{length}\rho S\gamma \left(1-\gamma \right)F_{onset}$$

(7)

where S is the absolute value of strain rate; P_γ is the production term of γ; E_γ is the dissipation term of γ; σ_γ is the constant; F_onset controls the location of the transition; and F_length controls the length of the transition area.

3. Setting and Verification of Numerical Simulation Strategy

This section focuses on the calculation domain division and grid division of the self-propulsion simulation and glide simulation after a series of Kappel propellers combined with a diamond wing underwater glider are formed into a hybrid-driven underwater glider.

3.1. Kappel Propeller Modeling

Parametric modeling is the process of controlling the shape of the model using various parameters, and the difference between the Kappel tip rake propeller and conventional propeller is mainly that the rake distribution changes, and it is expected that the energy loss caused by the spire tip flow and tail vortex shrinkage can be effectively reduced by optimizing the blade tip geometry. In this paper, the NACA0066 (a = 0.8, mod) is selected as the basis for the design and modeling of the Kappel tip rake propeller, and the geometric 3D coordinate conversion of the Kappel propeller is changed on the basis of the traditional propeller 3D coordinate conversion [18] as follows:

$$\left[\begin{array}{l}X\\ Y\\ Z\end{array}\right]=\left[\begin{array}{l}{X}_S+C\left(S-0.5\right)\sin \phi \cdot \cos \varphi -f\cos \phi \cos \varphi \mp 0.5t\cos \phi \cos \varphi \\ r\cos \theta +C\left(S-0.5\right)\sin \phi \cdot \sin \varphi -f\cos \phi \sin \varphi \pm 0.5t\cos \phi \sin \varphi \\ r\sin \theta \pm 0.5t\sin \phi \end{array}\right]$$

(8)

where X, Y, and Z are the three-dimensional coordinate values of the points on the blade surface, r is the radius of the propeller at the corresponding section, Xs is the rake, C is the blade chord length, S is the dimensionless length distribution of two-dimensional airfoil in the chord direction, Φ is the pitch, θ is the skew, f and t, respectively, represent the camber and the thickness, and φ is the angle between the tangent of the distribution curve of rake and the axis of abscissa.

In this paper, the radial distribution of rake is adjusted by changing the tip rake X_S/D, taking X_S/D = 0.000, 0.025, 0.050, 0.075, 0.100, and 0.125, and using the B-spline curve to design the distribution of rake along the propeller radius r/R, as shown in Figure 1, corresponding to the Kappel tip rake propellers Kap00, Kap01, Kap02, Kap03, Kap04, and Kap05, respectively. It should be noted here that the radial distribution of rake is consistent with that of the conventional end plate propeller before 0.9R through the fourth-order cubic B-spline control points, and begins to change to varying degrees after 0.9R. Numbers 1–5 in Figure 1 represent the 5 control points used for curve fitting.

The geometry of the Kappel tip rake propeller is shown in Figure 2. It can be seen from the figure that the degree of blade tip bending increases with the increase in rake, and the blade tip bending of Kap05 propeller is the most obvious. Other major geometrical parameters of the Kappel tip rake propeller are shown in

Table 1. Other major geometrical parameters of the Kappel tip rake propeller.

Diameter (mm)	Pitch Ratio at 0.7R	Skew (°)	Projected Area Ratio	Hub Diameter Ratio
250	1.0338	16	0.4788	0.2

.

3.2. Diamond Wing Underwater Glider

As shown in Figure 3, the research object is the diamond wing underwater glider (referred to as diamond wing UG) [30]. Its main geometric parameters [31] are shown in

Table 2. Main geometry parameters of diamond wing underwater glider [31].

Geometric Scale Parameters	Symbols	Numerical Value	Units
Effective maximum length	L	2.00	m
Maximum body diameter	D	0.22	m
Bow length	L1	0.23	m
Length of stern	L2	0.46	m
Long hydrofoil span	B	0.60	m
Wingtip chord length	Ct	0.05	m
Wing root chord length	Cr	0.10	m
Hydrofoil wet surface area	S	0.18	m2

. The performance adaptability of the Kappel propellers in the horizontal propulsion and glide navigation of the hybrid-driven underwater glider is studied.

3.3. Open Water Simulation Settings

As shown in Figure 4, the diameter of the propeller is D, the diameter of the stationary domain is 8D, and the length of the stationary domain is 10D; that is, the distance between the velocity inlet and the pressure outlet is 10D, the distance between the velocity inlet is 4D from the propeller disc surface, the pressure outlet is 6D from the propeller disc surface, the propeller disc surface is located in the center of the rotation domain, and the diameter of the rotation domain is 1.08D and the length is 0.24D.

In this paper, STARCCM+ 2021.3 commercial software is selected for CFD simulation calculation, which adopts the numerical technology of continuum mechanics, which has significant advantages in dealing with complex geometric models and reducing meshing time. In the auto-meshing function of STAR CCM+, the Trimmed Mesher type was selected. The stationary domain and the rotation domain are independently set up with the same mesh base size, and the interface mesh settings of the stationary domain and the rotation domain are completely consistent. The number of prismatic layers is set to 10 and the total thickness of the prismatic layers is set to 2% of the grid base size. At the same time, some areas in the computing domain are encrypted. Taking the meshing of the propeller Kap509 when the grid base size is 0.01 m (0.04D, D is the propeller diameter) as an example, the meshing effect is shown in Figure 5.

The propeller speed is set n = 12 rps, and the advance speed coefficient J = 0.62; that is, the incoming velocity of the set speed inlet is 1.862 m/s. The SST k-ω+γ transition turbulence model was selected with the Schnerr–Sauer cavitation model and the Volume of Fluid (VOF) two-phase flow for CFD simulation. The parameters for the cavitation simulation are set as follows: the density of the liquid phase (water) is set to a constant of 998 kg/m³, and the dynamic viscosity of the water should be set to 0.001008 Pa-s and the saturated vapor pressure should be set to 2338 Pa, assuming a temperature of 20 °C. The gas phase is water vapor, the density at 20 °C is 0.01719 kg/m³, and the dynamic viscosity is 1.26765 × 10⁻⁵ Pa-s. The volume fraction of the gas–liquid phase is α set to 0.1 to capture the water vapor surface. The seed density was set to 1.0 × 10¹²/m³ and the seed diameter was set to 1.0 × 10⁻⁶ m. As shown in Figure 6, the calculation accuracy of the SST k-ω+γ transition turbulence model in the Kappel propeller Kap509 simulation is verified, and the SST k-ω+γ transition turbulence model is used to calculate, which can control the calculation error of the open water propulsion efficiency within 3%, and the calculation error is small.

The calculation results of the propeller Kap509 are shown in Figure 7, and the correspondence between the total number of meshes and the size of the grid foundation is shown in

Table 3. The correspondence between the grid base size and the total number of grids in the mesh independence analysis of propeller Kap509.

Base Size	0.12D	0.1D	0.08D	0.06D	0.05D	0.04D	0.036D	0.032D
Total mesh (×106)	0.908	1.210	1.763	2.394	3.045	4.533	5.422	6.759

. Through the mesh independence verification of the propeller Kap509, the grid foundation size of the open water case simulation is determined to be 0.04D. The y+ distribution of the propeller Kap509, as shown in Figure 8, can be limited to a value of 0 to 3.1 on the propeller surface.

3.4. Self-Propulsion Simulation Settings

In this paper, the calculation domain of the horizontal propulsion and self-propulsion simulation was set as a cylindrical shape region and was divided into stationary domain and rotating domain so as to facilitate the adoption of the sliding grid motion simulation mode. The calculation domain was divided as shown in Figure 9.

The body length of the diamond wing underwater glider is represented by L, and the diameter of the propeller is represented by D. The diameter of the rotating field is 1.08D and the length is 0.24D. The diameter of the stationary domain is 4L and the length of the stationary domain is 7L; that is, the distance between the velocity inlet and the pressure outlet is 7L, the velocity inlet is 2L from the bow apex of the underwater glider, and the pressure outlet is 5L from the bow apex of the body. The propeller disk is the distance from the bow apex L of the underwater glider.

In this paper, from the perspective of computational efficiency and the post-processing effect, the Trimmed Mesher type is selected in the automatic mesh function, the number of prismatic layers is set to 10, and the total thickness of the prismatic layer is set to 5% of the grid base size, as shown in the view in the enlarged box in Figure 10d. In the process of meshing, the line encryption is carried out at the guide edge and accompanying edge of the propeller blade, and at the wing tip and wing root edge of the hydrofoil, as shown in Figure 10c,d. Different surface encryption is adopted for the bow surface, the mid-section surface, the hydrofoil surface, the stern surface, the propeller blade surface, and the interface between the rotation domain and the stationary domain, as shown in Figure 10a–d. The propeller wake region in the rotating domain and the stationary domain were infilled separately to better capture the details of the wake field and the distribution structure of the wake vortex, as shown in Figure 10a,d. Figure 10 shows the meshing of the simulation example of propeller Kap01 and underwater glider when the grid base size is 0.01 m (0.04D, D is the propeller diameter).

Due to the variation of the blade tip rake, the six types of Kappel propellers have significant differences in geometrical structure, which may affect the grid-independent analysis. Therefore, two representative Kappel propellers Kap01 and Kap04 (with smaller tip rake and larger tip rake, respectively) were selected to match with the underwater glider for the grid independence analysis of self-propelled examples.

The Kappel propeller Kap01 and Kap04 were calculated by the SST k-ω+γ transition turbulence model combined with the Schnerr–Sauer cavitation model and VOF gas–liquid two-phase method under J = 0.5 and n = 12 rps conditions. The overall force F, resistance D, propeller thrust T, and propeller torque Q of the propellers are shown in Figure 11 and Figure 12, respectively, from which it can be seen that the total mesh tends to be stable when the total mesh is greater than 4 × 10⁶. In the grid independence calculation example of the self-propulsion simulation for propeller Kap01 and Kap04 with the diamond wing underwater glider, the corresponding relationship between the total number of grids and the grid base size is shown in

Table 4. The corresponding relationship between the mesh base size and the total mesh size in the calculation example with propeller Kap01.

Base Size	0.16D	0.128D	0.112D	0.08D	0.07D	0.06D
Total mesh (×106)	1.553	2.114	2.876	4.671	5.967	8.239

and

Table 5. The corresponding relationship between the mesh base size and the total number of mesh in the calculation example with propeller Kap04.

Base Size	0.16D	0.13D	0.08D	0.07D	0.064D
Total mesh (×106)	1.764	2.565	4.758	6.103	7.288

. When the grid base size is less than or equal to 0.08D, the simulation numerical results tend to be stable.

When the grid base size is 0.08D, the y⁺ distribution of the diamond wing UG combined with the propeller Kap01 and Kap04 self-propulsion simulation is shown in Figure 13. The y⁺ value of the hybrid-driven underwater glider is limited to 0.4~5.5.

3.5. Glide Simulation Setup

The calculation domain division of the gliding simulation example in this paper is shown in Figure 14, which is roughly the same as that of the self-propulsion simulation. Except that the longitudinal axis of the hybrid-driven underwater glider body is at a 12° angle with the direction of the inflow, the other settings were the same as that of the self-propulsion simulation. Similar to the determination of the grid base size of the self-propulsion simulation, propellers Kap01 and Kap04 were selected to match the diamond wing underwater glider, and the grid-independent analysis of the glide simulation was carried out.

In this example, the inlet velocity is 0.5 kn (0.2572 m/s), which was calculated by the SST k-ω+γ transition turbulence model, Schnerr–Sauer cavitation model, and VOF gas–liquid two-phase method. The calculated values of the overall resistance D, overall lift L, propeller resistance D_propeller, and propeller lift L_propeller of the hybrid-driven underwater glider are shown in Figure 15 and Figure 16, respectively, for the changes in the total number of grids in the example, from which it can be seen that the total number of grids tends to be stable when greater than 4 × 10⁶ in the grid independence verification of the hybrid-driven underwater glider. The corresponding relationship between the total number of grids and the size of the grid foundation is shown in

Table 6. Correspondence between the mesh base size and the total mesh size of the diamond wing UG+ propeller Kap01 gliding simulation example.

Base Size	0.3D	0.18D	0.12D	0.1D	0.08D
Total mesh (×106)	0.932	1.691	3.251	4.228	5.234

and

Table 7. Correspondence between the mesh base size and the total mesh size of the diamond wing UG+ propeller Kap04 gliding simulation example.

Base Size	0.3D	0.18D	0.12D	0.096D	0.08D
Total mesh (×106)	0.947	1.784	3.344	4.237	5.280

, respectively. When the size of the grid foundation is less than 0.1D, the numerical results of the simulation calculation tend to be stable.

The y⁺ distribution of the glide simulation example when the grid base size is 0.08D is shown in Figure 17, and the y⁺ values are limited to 0~3.2 and 0~1.5, respectively.

Considering the accuracy and time cost of the simulation calculation, the base mesh size of 0.08D was selected uniformly in the gliding simulation in this study.

4. Discussion of Simulation Results

In this paper, before studying the self-propulsion performance and glide performance of the Kappel propeller combined with the diamond wing underwater glider, the open water performance of the Kappel propeller was analyzed under the working conditions σ = 2.0 and n = 12 rps according to reference [19]. The open water propulsion efficiency is shown in Figure 18.

From the analysis of Figure 18, it can be seen that, in the range of the advance coefficient J ≤ 0.5, with the increase in the bending degree of the end plate of the Kappel tip rake propeller, the open water propulsion efficiency of the propeller gradually decreases, and the propulsion efficiency of the propeller Kap01 is the best.

In the range of the advance coefficient J ≥ 0.8, the moderate end plate bending is conducive to improving the propulsion efficiency of the Kappel tip rake propeller, the open water propulsion efficiency of the propeller Kap05 is the highest, the open water propulsion efficiency of the propellers Kap01 and Kap02 is the lowest, and the open water propulsion efficiency of the propellers Kap01 and Kap02 shows a sharp decrease. When the advance coefficient J = 0.9, the open water propulsion efficiency of the propeller Kap05 is increased by 2.21% compared with that of the propeller Kap00. When the advance coefficient J = 1.0, the open water propulsion efficiency of the propeller Kap05 is greatly improved compared with that of the propeller Kap00, with an increase of 3.07%.

In general, in the range of the low advance coefficient, the tip plate of the Kappel tip rake propeller cannot play a role in improving the efficiency of the propeller, but will increase the propeller resistance due to the increase in water immersion area. In the range of the high advance velocity coefficient, with the increase in the bending degree of the end plate of the Kappel tip rake propeller, the open water propulsion efficiency of the propeller gradually increases, and the moderate and large end plate bending is conducive to improving the propulsion efficiency of the propeller.

4.1. Self-Propulsion Simulation

This section aims to find the self-propulsion points of the hybrid-driven underwater glider under four speed conditions of 6 kn (3.0867 m/s), 5 kn (2.5722 m/s), 4 kn (2.0578 m/s), and 3 kn (1.5433 m/s), and then evaluate the propeller performance parameters. The interaction and adaptation between the propeller and UG body were evaluated by wake fraction and ship body efficiency.

4.1.1. Self-Propulsion Point Calculation and Determination

The self-propulsion point calculation of the series Kappel propeller combined with the diamond wing underwater glider under four working conditions of speeds of 6 kn, 5 kn, 4 kn, and 3 kn is shown in Figure 19. The black dotted line in Figure 19 represents F_total = 0 N.

At different speeds, the propeller is balanced with the net force of the hybrid-driven underwater glider, i.e., F_total = 0, and the propeller speed is called the self-propelled point. From the analysis of Figure 19, it can be seen that the F_total of the mixed-drive underwater glider gradually decreases with the increase in propeller speed at different speeds. The self-driving point speed of the mixed-drive underwater glider is Kap05, Kap04, Kap03, Kap02, and Kap01 corresponding to the Kappel propeller model from small to large, and the self-driving speed of the mixed-drive underwater glider corresponding to Kappel05 can be reduced by about 7.5% compared with the self-driving speed of the mixed-drive underwater glider corresponding to Kappel01. Its thrust gradually increases, and it can reach the self-propelled state at low speed, which is conducive to the hybrid-driven underwater glider to achieve the energy-saving goal in the self-propelled process.

As shown in Figure 20, the relationship between the self-driving point speed and the series Kappel propeller model can be further analyzed, and the self-driving point speed of the mixed-drive underwater glider shows a similar trend of gradually decreasing with the increase in the bending degree of the end plate of the series Kappel propeller at different speeds, and the combination of propeller Kap05 and diamond wing underwater glider has the lowest self-driving point speed, which once again confirms that, within a certain range, the more bent the Kappel blade tip pitch propeller with the underwater glider, the greater the thrust and the better the propulsion performance.

4.1.2. Evaluation of Propeller Performance in Self-Propulsion Point

Because the propeller was installed in the stern of the underwater glider, the actual flow field under the self-propulsion condition is different from that under the open water condition. Without considering the non-uniformity of the flow field, the average inflow velocity in front of the propeller disk under the self-propulsion condition is different from that of the hybrid-driven underwater glider, so it is necessary to obtain the actual inflow velocity of the propeller. In the self-propulsion simulation example, the velocity distribution in a circular section with an outer diameter of 0.25 m, an inner diameter of 0.05 m, and an area of 0.04712 m² at 0.16D upstream of the propeller disk is intercepted, as shown in Figure 21. The axial flow through this section is monitored so as to obtain the average inflow velocity through this section. The average inflow velocity of this section is defined as the actual advance velocity V_A of the propeller [32], and the actual advance velocity V_A, the advance coefficient J, the thrust coefficient K_T, the torque coefficient K_Q, and the propulsion efficiency η of the propeller in the self-homing point state of the hybrid-driven underwater glider at four working conditions of 6 kn, 5 kn, 4 kn, and 3 kn are calculated, as shown in Figure 22.

As can be seen from Figure 22, the actual advance speed V_A, advance speed coefficient J, thrust coefficient K_T, and propulsive efficiency η of the propellers of each hybrid-driven underwater glider show a trend of first decreasing and then increasing with the increase in the bending degree of the end plate of the Kappel propellers. However, the propeller torque coefficient K_Q of the hybrid-driven underwater glider shows a monotonically increasing trend with the increase in the bending degree of the propeller end plate. From the analysis of Figure 22a,b, it can be seen that, for the same type of Kappel propeller, the actual advance speed V_A and the advance coefficient J of the propeller change in the same direction as the speed of the hybrid-driven glider, which is the same as the actual trend of the project. At the same speed, except for the conventional propeller Kap00, the actual advance velocity V_A and the advance coefficient J of the propeller gradually increase with the increase in the bending degree of the end plate of the series Kappel propeller. The actual advance velocity V_A and the advance coefficient J corresponding to the propeller Kap05 are the largest, indicating that the increase in the bending degree of the end plate of the Kappel propeller can effectively improve the utilization of the kinetic energy of the convective field of the hybrid-driven underwater glider, and at the same time can reduce the resistance of the hybrid-driven underwater glider in the process of self-propulsion to improve energy efficiency. From the analysis of Figure 22c, it can be seen that, at the same speed, except for the conventional propeller Kap00, the thrust coefficient K_T gradually increases with the increase in the bending degree of the end plate of the series Kappel propeller, indicating that the increase in the bending degree of the end plate of the Kappel propeller can effectively improve the output thrust. For Kappel propellers of the same model, the thrust coefficient K_T is reversed from the speed of the hybrid-driven glider. From the analysis of Figure 22d, it can be seen that, when the diamond wing underwater glider is operated at 6 kn, 5 kn, 4 kn, and 3 kn speed self-propelled point conditions, in addition to the conventional propeller Kap00, the propulsion efficiency η gradually increases with the increase in the bending degree of the end plate of the series of Kappel propellers. Among these, the propellers Kap04 and Kap05 perform the best, which once again proves that the bending of the propeller end plate can effectively improve the efficiency of the propeller. It can improve the overall performance of hybrid-driven underwater gliders.

4.1.3. Calculation of Wake Fraction in Self-Propulsion Point

On the basis of calculating the average flow velocity of the inflow section, the wake fraction ω of each self-propulsion point of the hybrid-driven underwater glider was further calculated. It should be noted that the inflow velocity defined in the wake fraction formula is the average axial velocity at the propeller disk surface. However, since the section area at the propeller disk surface cannot be effectively calculated, the average axial velocity at Section 1 at 0.16D upstream of the propeller disk was used to calculate the wake fraction. The wake fraction ω of each hybrid-driven underwater glider in the self-propulsion point state is shown in Figure 23, and the value ranges from 0.11 to 0.13, in line with the value range of submarine wake fraction 0.10 to 0.25. From the analysis of Figure 23, except for the conventional propeller Kap00, under the same velocity, the current retention fraction of the hybrid-driven underwater glider under the self-propelled condition shows a trend of increasing first and then decreasing with the increase in the bending degree of the end plate of the series Kappel propeller. Among these, the hybrid-driven underwater glider with the propeller Kap02 has the highest current retention fraction under the self-propelled condition, and the hybrid-driven underwater glider with the propeller Kap05 has the lowest current holding fraction under the self-propelled working condition, which indicates that it is within a certain range. Appropriately increasing the bending degree of the end plate of the series of Kappel propellers can effectively reduce the accompanying phenomenon of the hybrid-driven underwater glider in the process of self-propulsion, reduce the interference of the propeller in the wake field of the hull, and improve the overall self-propelled performance of the hybrid-driven underwater glider. For the same type of Kappel propeller, the mate current fraction of the mixed-drive underwater glider under self-propelled conditions changes inversely with the speed of the mixed-drive glider. With the increase in the bending degree of the end plate of the series of Kappel propellers, the gap between the trail fraction at 3 kn and the trail fraction at 6 kn is 5.25%, and the gap between the trail fraction at 3 kn and the trail fraction at 6 kn decreases by 4.87%, indicating that the difference between the trail fraction at 3 kn and the trail fraction at 6 kn decreases by 4.87%. This indicates that, within a certain range, the speed of the hybrid-driven underwater glider can be reduced by appropriately increasing the bending degree of the end plate of the Kappel propeller so as to achieve the same self-propelled effect.

The hull efficiency of the self-propulsion point is shown in

Table 8. Hull efficiency η_H of self-propulsion point of hybrid-driven underwater glider.

Propeller Paddle Type	Kap00	Kap01	Kap02	Kap03	Kap04	Kap05
6 kn speed	1.1431	1.1416	1.1444	1.1437	1.1390	1.1341
5 kn speed	1.1415	1.1417	1.1459	1.1458	1.1436	1.1345
4 kn speed	1.1474	1.1469	1.1491	1.1482	1.1441	1.1384
3 kn speed	1.1496	1.1600	1.1629	1.1548	1.1467	1.1501

. Since the total resistance of the hybrid-driven underwater glider is close to the same as the propeller thrust under the condition of the self-propulsion point, the thrust derating fraction is close to zero numerically.

4.1.4. Distribution of Self-Propulsion Point Flow Field

Through simulation analysis, it is found that the velocity distribution of the flow field at each self-propulsion point is roughly the same under the four operating conditions of speed 6 kn, 5 kn, 4 kn, and 3 kn. Therefore, taking the operating condition of speed 6 kn as an example, the velocity distribution cloud diagram of the flow field at each self-propulsion point is shown in Figure 24. In the state of self-propulsion point, the flow field velocity distribution of each hybrid-driven underwater glider is basically the same. There is a high-speed zone on both sides of the bow turning point, both sides of the stern turning point, and the large radial position of the propeller wake area of the underwater glider, as well as a low-speed zone on the front side of the bow, near the stern, and near the center axial part of the propeller wake area.

4.2. Glide Simulation Research

4.2.1. Gliding Research with Blade Extension

In the blade extension state, the Kappel propellers were combined with the underwater glider in order to carry out the simulation under the gliding state with the speed of 1 kn and 0.5 kn and the glide angle of 12°, and the simulation physical time was at least 25 s. In the gliding mode with a speed of 1 kn, the total drag D_total, propeller drag D, total lift L_total, and the lift L generated by the propeller of the hybrid-driven underwater glider are shown in

Table 9. Glide calculation results of hybrid-driven underwater glider with propeller blade extension (speed of 1 kn, gliding angle of 12°).

	Force (N)
Combo	Total Resistance D_Total	Propeller Drag D	Total Lift L_Total	Propeller Lift L
UG + Kap00	7.3082	3.4759	14.0652	0.6066
UG + Kap01	7.3385	3.4869	13.3335	0.6186
UG + Kap02	7.4030	3.5385	13.5168	0.6413
UG + Kap03	7.4079	3.5439	13.2918	0.6571
UG + Kap04	7.4513	3.6008	13.0136	0.6532
UG + Kap05	7.3936	3.4745	13.4936	0.6491

.

As can be seen from the table, propeller resistance accounts for a large proportion of the overall resistance, about 47~49%. However, the negative impact of the propeller on the overall lift is small, and its value is basically 4.3~5.1% of the overall lift. The existence of the propeller not only increases the overall drag of the hybrid-driven underwater glider, but also weakens the overall lift of the hybrid-driven underwater glider, which is not conducive to the gliding performance of the hybrid-driven underwater glider in the gliding mode.

The axial velocity distribution diagram of the flow field of the hybrid-driven underwater glider gliding at 1 kn speed is shown in Figure 25. In this gliding state, the flow field velocity distribution of each hybrid-driven underwater glider is basically the same, and there are high-speed areas in the upper turning point of the bow curve, the lower turning point of the stern curve, and the blade clearance of the underwater glider. And, there are low-speed zones in the front of the bow of the underwater glider, near the surface of the stern, and behind the blade, in which the low-speed zone at the back of the blade has the smallest axial speed, and even shows the low-speed flow in the direction of the reverse flow. According to Figure 25, as the bending degree of the Kappel propeller end plate increases, the distribution range of the low-speed region (blue area) in the wake of the Kappel propeller gradually decreases. The distribution range of the wake high-speed area (yellow and red areas) gradually increases and shows a trend of expanding towards larger radius positions. This indicates that, as the bending degree of the Kappel propeller end plate increases, the axial kinetic energy of the Kappel propeller wake gradually increases, and the axial kinetic energy of the wake is the main contributing component to achieving propeller propulsion, which has a positive significance for the thrust output of the propeller. This further indicates that the combination of the Kappel series propellers can effectively improve the propulsion performance of hybrid-driven underwater gliders, especially when paired with the Kapp5 propeller with a higher degree of end plate bending.

In gliding mode at a speed of 0.5 kn, the total drag D_total, propeller drag D, total lift L_total, and the lift L generated by the propeller are mixed to drive the underwater glider, as shown in

Table 10. Gliding calculation results of hybrid-driven underwater glider with propeller blade extension state (speed of 0.5 kn, gliding angle of 12°).

	Force (N)
Combo	Total Resistance D_Total	Propeller Drag D	Total Lift L_Total	Propeller Lift L
UG + Kap00	1.8497	0.8470	3.2301	0.1293
UG + Kap01	1.8574	0.8553	3.3110	0.1280
UG + Kap02	1.8678	0.8669	3.3084	0.1438
UG + Kap03	1.8548	0.8505	3.3321	0.1402
UG + Kap04	1.8566	0.8446	3.2140	0.1381
UG + Kap05	1.8748	0.8533	3.3529	0.1369

. Among them, the proportion of propeller resistance in the whole resistance is basically in the level of 43~46%, and the propeller negative lift is basically 2.7~4.1% of the overall lift. The proportion of drag force and negative lift force caused by the propeller decreases in the glide condition with a speed of 0.5 kn compared with the glide condition with a speed of 1 kn.

4.2.2. Gliding Research When Blade Folded

Through the simulation analysis, it is found that, after the hybrid-driven underwater glider is equipped with Kappel propellers, the propeller significantly increases the overall resistance of the hybrid-driven underwater glider and weakens the overall lift of the hybrid-driven underwater glider. In order to deal with this problem, this paper puts forward the strategy of folding the blade in order to reduce the adverse effect of the propeller on the gliding performance of the hybrid-driven underwater glider.

Taking the combination of propeller Kap05 and lozenge wing underwater glider as an example, the propeller blade in the geometry model is set to rotate and fold 60° in the direction of the stern, and the simulation is carried out under the gliding conditions of 1 kn and 0.5 kn and gliding angle of 12° by using the SST k-ω+γ transition turbulence model. The calculation results of total drag D_total, propeller drag D, total lift L_total, and lift L generated by propeller of the hybrid-driven underwater glider are shown in

Table 11. Gliding calculation results of hybrid-driven underwater glider with propeller blade folding in working condition.

Gliding Speed (kn)	D_Total (N)	D (N)	L_Total (N)	L (N)
1.0	5.7351	1.7166	13.7148	0.0944
0.5	1.4997	0.3872	3.7141	0.0291

. Where the value of L is negative, the direction is vertical downward, opposite to the direction of the total lift L_total.

Comparing the calculation results in Table 11 and Table 9, it can be seen that, if the propeller Kap05 blade folds by 60°, the total drag D_total of the hybrid-driven underwater glider decreases by 22.43% and the total lift L_total increases by 1.64% under the glide condition of 1 kn speed. The propeller drag D decreased by 50.59%, and the negative lift L amplitude generated by the propeller decreased by 85.46%. In the gliding condition with a speed of 0.5 kn, compared with Table 9 and Table 7, it can be seen that the total drag D_total of the hybrid-driven underwater glider decreased by 20.01%, the total lift L_total increased by 10.77%, the propeller drag D decreased by 54.62%, and the negative lift L amplitude generated by the propeller decreased by 78.74%.

Thus, it can be seen that the measures of folding blades can indeed weaken the adverse effects of propellers on the gliding performance of the body, and effectively help reduce the gliding resistance and negative lift caused by propeller blades, with a significant overall effect.

When the propeller Kap05 is combined with the underwater glider and folds the blades, the axial speed distribution cloud map under the glide conditions of 1 kn and 0.5 kn speed is shown in Figure 26. Compared with the unfolded state, the low-speed zone of the stern when the blade is folded is no longer dispersed in the rear of each blade and shaft, but presents centralized; that is, there is only one low-speed zone in the stern.

5. Conclusions

In this paper, the propulsion performance of a series of Kappel tip rake propellers combined with an underwater glider at a range of self-navigation speeds (6 kn, 5 kn, 4 kn, and 3 kn) and at certain gliding states (gliding speeds of 1 kn and 0.5 kn, gliding angle of 12°) were studied. The following conclusions can be drawn from the simulation study.

• The simulation strategy determined in this article can ensure that the calculation error of propeller propulsion efficiency does not exceed 3%. This article takes Kappel propeller Kap509 as the verification object, adopts the Schnerr–Sauer cavitation model and VOF two-phase flow cavitation simulation strategy, couples SST k-ω and γ transition models, considers the influence of the transition phenomenon on simulation accuracy, and verifies the accuracy and precision of simulation with experimental data from the literature. Both self-propulsion simulation settings and gliding simulation settings are based on this template.
• The Kapp05 propeller with blade tip pitch value Xs/D = 0.125 is most beneficial for improving the open water propulsion performance of Kappel-type propellers. A greater degree of blade tip pitch helps to improve the thrust output, torque output, and open water propulsion efficiency of Kappel-type propellers. This gain mainly occurs in the medium- to high-speed state, and the greater the tip pitch, the more significant the effect. Among them, the improvement in the extreme efficiency of propeller Kapo5 is the largest, with a 3.07% increase compared to the reference propeller.
• The propulsion performance of propellers Kapo4 and Kapo5 is optimal in the wake field of a hybrid-driven underwater glider. The hybrid-driven underwater glider equipped with a series of Kappel tip pitch propellers gradually reduces the wake fraction at the waypoint as the bending degree of the propeller end plate increases. The hybrid-driven underwater glider equipped with the Kappel propeller 05 has the lowest wake fraction and performs the best under autopilot conditions, which helps to improve the actual propulsion performance of Kappel-type propellers in the wake field. Meanwhile, the speed also affects the actual propulsion efficiency of Kappel-type propellers. The lower the cruising speed at the waypoint, the higher the actual propulsion efficiency of Kappel-type propellers.
• According to the gliding simulation results, the drag caused by the propeller accounted for a large proportion of the overall drag of the hybrid-driven underwater glider, about 44~48%. However, the negative lift generated by the propeller has a relatively limited effect on the overall lift of the hybrid-driven underwater glider, and its value is about 2.6~5.0% of the overall lift. The propeller not only increases the overall drag of the hybrid-driven underwater glider, but also weakens its overall lift, which is not conducive to the gliding movement of the hybrid-driven underwater glider. However, by folding the propeller blades, the adverse effects can be greatly reduced, and the glide resistance caused by the propeller can be reduced by about 45~55%, while the negative lift caused by the propeller can be reduced by about 68~85%.

Although this article provides a detailed and comprehensive hydrodynamic study on the series of Kappel tip pitch propellers and their hybrid-driven underwater gliders, due to time constraints, there are still some aspects that have not been fully addressed and require further in-depth research, such as evaluating the simulation effects of different aerodynamic numerical models and combining them with noise models in STARCCM+ 2021.3 commercial software to complete a noise analysis, and comparing and applying different types of cavitation models and couple noise monitoring models in cavitation simulation to evaluate the cavitation noise during the operation of Kappel propellers. By analyzing the streamline distribution and y + distribution on the surface of the propeller, the transition phenomenon during the operation of the Kappel propeller is presented, and the key role of the gamma transition model in propeller simulation is demonstrated. When using different simulation strategies, there are often significant differences in the calculation results of high-speed operating conditions. Exploring the transition phenomenon can help reveal the underlying reasons and provide a direction for optimizing numerical simulation strategies.