Effect of Tip Rake Distribution on the Hydrodynamic Performance of Non-Planar Kappel Propeller

Abstract

Taking advantage of end-plate effects to enhance propeller efficiency is engaging. This paper applied a 4-order B-spline curve to design the rake distribution of Kappel propellers using five types of Kappel propellers that each possesses different tip rakes, and one type has no constructed end-plate. The RANS method coupled with the γ transition model was utilized to analyze the open-water performance of the six propellers, considering cavitating flow. It was found that the tip rake is conducive to the thrust capacity of the Kappel propellers, mostly improving the propulsion efficiency by 2.5% at a designed advance speed with the appropriate tip rake. The increase in the tip rake will magnify the low-pressure value and area on the suction side blade surface, together with the phenomenon of the stretching tip vortex and the inhibition of wake vortex contraction, which are both beneficial to the elevation of propulsion efficiency. However, the sheet cavitation behavior of the six propellers aggravates as the tip rake rises. Accordingly, the reasonable range of a tip rake for the design of a Kappel propeller in favor of the propulsion performance is suggested in this paper, exhibiting the promising potential of energy savings for the application to marine vehicles.

1. Introduction

Since the first helicoidal screw propeller came into practical application nearly 200 years ago, in 1836, a mature and perfect system of propeller research, design, and manufacturing has been established [1]. However, with the scarcity of power and the appeal of green production, the limitations of conventional propellers have come to the point where they need to be broken. The tip-rake propeller with an end-plate effect that prevents cross-flow in the tip is a specific kind of special propeller that is conducive to improving propulsion efficiency.

The tip-rake propeller can be divided into two categories according to the rake direction of the end-plate, namely, the propeller with an end-plate that bends toward the pressure side, e.g., the CLT (Contracted and Tip-Loaded) propeller, and the propeller with a tip rake that is inclined to the suction side, e.g., the Kappel propeller. The end-plate propeller shares similarities with the winglet that is mounted onto the aerofoil, which is capable of hindering the flow around the blade tip and thus exaggerating the pressure difference of the blade surface, i.e., the positive component of the thrust. Anderson et al. [2,3] were the first to carry out research on Kappel propellers using the potential flow theory and mainly involving the lifting-line theory. Before 2005, Kappel propellers were completely designed based on both the lifting-line theory and the lifting-surface theory, and then a series of basic analyses including open-water performance tests and cavitation tunnel experiments would have been completed on a model scale [4,5]. The sea trial was conducted aboard a product carrier following the manufacturing of the full-scale propeller. It was confirmed that the Kappel propeller could raise the propulsion efficiency by 4% in designed working conditions compared with the conventional propeller during the full-scale operation test [6,7].

The design concept of the Kappel propeller was then taken over by MAN Diesel & Turbo Corporation in 2012, which was simultaneously, highly recommended. When the MAN Alpha Kappel series propeller designed by the corporation was combined with the MAN B&W low-speed diesel engine of the G series, it showed notable energy-saving skills, reducing carbon dioxide emissions by up to 10%. In 2014, the Alpha Kappel propellers authorized by MAN Diesel & Turbo Corporation were installed aboard three 8500CEU pure car and truck carriers (PCTC) built by Xiamen Shipping Industry. In 2019, an 8500 TEU container ship of CMA CGM was replaced with a three-bladed Kappel propeller of 9 m diameter. The exploration of the Kappel propeller application initiated a wave of interest in propeller renewal and upgrading.

A Kappel propeller is a non-planar propeller that possesses a different geometric construction compared to a conventional propeller. Kehr and Wu [8] studied the geometric characteristics of the Kappel propeller and put forward the definition of a three-dimensional coordinate conversion formula considering the tip rake. Due to the special geometry of Kappel propellers, some theories and methods that are applicable to conventional propellers need to be properly transformed in order to be used for the calculation of Kappel propellers. In the field of theoretical analysis, Cai [9] optimized the lifting-line theory, which takes into account the influence of the induced speed in the three-dimensional direction and correctly analyzes the performance of Kappel propellers.

Compared with a conventional propeller, the rake at the blade tip of a Kappel propeller presents magnification; thus, the performance behaves differently. Some scholars were absorbed in the open-water performance and cavitation noise behavior of Kappel propellers. Huang [10] focused on the cavitation performance, varying the geometry parameters of Kappel propellers and carrying out cavitation tests through the cavitation tunnel, and found that the pitch and camber of the blade tip exerted a notable impact on the cavitation performance, showing a guideline of the geometric design that is related to both camber and pitch. He noted that the degree of the tip rake may play an essential role as well.

Relatively speaking, the research about the CLT propeller from the perspective of CFD is more complete, which is worth referencing when discussing Kappel propellers. Daniele et al. [11] compared the test results of the panel method and the CFD method and concluded that the two simulation methods were applicable to analyze the open-water and cavitation performance of CLT propellers, with the errors meeting the engineering requirements. A potential panel method and Star CCM+ software were recommended by Gaggero et al. [12] for the sheet cavitation prediction of CLT propellers after the assessment of the numerical approaches, ranging from the panel code to various RANS solvers, and the experimental results. Thereafter, a strategy of improved delayed detached eddy simulation (IDDES) was tested by Gaggero et al. [13] to observe the mutual interaction of secondary vortices originated by the end-plate of two CLT propellers. From a different perspective, Lungu [14] turned the attention to the scale effect of the propeller P1727, analyzing the open-water performance and vortex structures on a model scale and full scale using the large eddy simulation (LES) method. Wang et al. [15,16] matched the LES methodology with a RANS solver in the research of vortex-induced instability for the wake at different advance speeds. It should be noted that most of the research took no account of the fluid transition phenomenon in a model-scale simulation.

Some scholars have similarly transformed some typical and conventional propellers directly into tip-rake propellers and have since analyzed the change in performance. Maghareh et al. [17] estimated the effect of the end-plate on the efficiency of propeller DTMB4382, finding that the end-plate bending toward the suction side is more beneficial to the improvement in the propulsion efficiency at the design working point than that bending toward the pressure side. For the same propeller, Ghassemi [18] provided a reasonable tip rake that could reduce the noise to a minimum level. On the contrary, Kang et al. [19] adjusted the tip rake of a propeller, which showed better propulsion performance when the end-plate bent toward the pressure side. Gao et al. [20] discovered that a small amplitude of the blade tip inclined to the pressure side helped delay the occurrence of tip-vortex cavitation while hardly affecting the propulsion efficiency, but when the tip rake was excessively large, it caused an even worse cavitation behavior. For some propellers, e.g., DTMB4119 and Kp505, with tip rakes that have a disposition to the pressure side, weaker tip vortices exit in company with the noise lowering down [21,22]. However, Posa [23,24,25,26] detected a greater intensity of the vorticity and acoustic signature near the blade tip in terms of the same tip rake inclination during the simulation for a five-bladed CLT propeller. By contrast, in some cases where the end-plate bends to the suction side, the tip vortex might not be visibly modified or improved due to the tip rake [27,28]. This indicates that the impact of the tip rake on the propeller performance, tip-vortex strength, and acoustic characteristics varies with the geometry of the benchmark propeller, which may not always be the dominant factor compared with other parameters, not counted quantitatively; hence, parametric design will be further applied to the tip-rake’s blade design for an improved end-plate effect in this study.

Cavitation analysis is a vital aspect for Kappel propellers, inasmuch as the generation and collapse of the cavitation bubbles are detrimental to the performance and construction of the propeller. There are two methods for propeller cavitation research: simulation and experimental measurement. Among them, the key points involved in cavitation simulation include the selection of the turbulence model, the cavitation model, and the measurement criteria. The orthodox turbulence models are indirect numerical simulation models; the turbulence model is based on the RANS equation, the large eddy simulation model, and the separated eddy simulation (SES) model [29], in which RANS is widely applied. The cavitation model of the propeller can be divided into two types based on state equation and transport equation, the latter including Singhal, Kunz, Zwart, Schnerr–Sauer, and other cavitation models [30,31,32].

At present, the simulation research on propeller cavitation concentrates on the prediction of a single propeller’s cavitation performance, especially under complex working conditions using variable methodologies. Ge [33] et al. predicted the cavitation performance of a propeller in the wake of a container ship on a model scale compared with the experimental results. Kim et al. Lloyd et al. [34,35] applied an adaptive mesh technology and delayed the separated vortex method to analyze the cavitation of a propeller in uniform flow. Yilmaz et al. [36] proposed a mesh strategy that combined volume control and adaptive mesh refinement, which was integral to the RANS solver together with the Schnerr–Sauer cavitation model. Lee [37] evaluated the application of the full cavitation model, the Zwart–Gerber–Belamri model, and the Schnerr–Sauer model in a propeller simulation, in which the Schnerr–Sauer model presented greater accuracy for the prediction of cavitation and thrust. Ville et al. [38] carried out a cavitation simulation of a propeller in an oblique flow, and the results were relatively consistent with those of the tests.

This paper mainly explores the impact of altering the tip rake with a B-spline parametric design on the hydrodynamic performance of a non-planar Kappel propeller. We analyzed and summarized the rule according to the simulation results and found that the tip rake value effectively improves the efficiency of a Kappel propeller. The sheet cavitation performance of Kappel propellers with different tip rakes were also analyzed, which provides a useful reference for the subsequent application of a Kappel propeller.

2. Theory and Model

2.1. Formulas of Blade Geometry for Kappel Propeller

Considering the impact of the rake on the geometry, notably regarding the blade tip, the three-dimensional coordinate conversion of Kappel propeller can be expressed as [8],

$$X=X_m+C\left(S-0.5\right)\sin \varphi \cdot \cos \psi -f\cos \varphi \cos \psi \mp 0.5t\cos \varphi \cdot \cos \psi$$

(1)

$$\theta ={\theta }_m+C\left(S-0.5\right)\frac{\cos \varphi }{r}+f\frac{\sin \varphi }{r}$$

(2)

$$Y=r\cos \theta +C\left(S-0.5\right)\sin \varphi \cdot \sin \psi -f\cos \varphi \cdot \sin \psi \pm 0.5t\cos \varphi \cdot \sin \psi$$

(3)

$$Z=r\sin \theta \pm 0.5t\sin \varphi$$

(4)

where X, Y, and Z are the three-dimensional Cartesian coordinate values of the point on the blade surface, t, θ_m, and r are the thickness, the skew, and the radius of the cylindrical surface at the point, C is the blade section expanded chord length, f is the camber of the propeller foil, S is the dimensionless length distribution of the two-dimensional airfoil point along the chord length direction, X_m is the rake of propeller, and $\varphi$ is the pitch angle, ψ is crucial non-planar parameter, i.e., representing the included angle between the tangent of the rake distribution curve at the blade tip and the X axis shown in Figure 1b. $\pm$corresponds, respectively, to the formulas for the upper and lower blade surfaces.

2.2. Basic Formulas of B-Spline Curve

B-spline curve possesses exceptional local modification capacity with strong convex hull properties, which is why this paper applied it to construct the radial rake distribution curves and geometries of Kappel propellers, which only need great modification near the tip. The equation of B-spline curve can be defined as Formula (5) [39],

$$p\left(u\right)={\displaystyle \sum }_{i=0}^n{d}_i{N}_{i,k}\left(u\right)$$

(5)

where d_i (i = 0,1,..., n) is the control point, N_{i, k} (u) (i = 0,1,..., n) is the basis function of the k-degree normalized B-spline curve, and k is the degree of the B-spline curve. The definition of the basis function is shown in Formula (6),

$$\{\begin{array}{l}{N}_{i,0}\left(u\right)=\{\begin{array}{ll}1& u_i\le u\le u_{i+1}\\ 0& else\end{array}\\ N_{i,k}\left(u\right)=\frac{u-u_i}{u_{i+k}-u_i}{N}_{i,k-1}\left(u\right)+\frac{u_{i+k+1}-u}{u_{i+k+1}-u_{i+1}}{N}_{i+1,k-1}\left(u\right)\\ set\frac{0}{0}=0\end{array}$$

(6)

where subscript i refers to serial number and control node.

2.3. Dimensionless Hydrodynamic Parameters of the Propeller

Assume that the propeller diameter is D, the rotational speed is n, the advance speed is V_A, i.e., the inlet uniform flow of V_∞ in calculation domain, the propeller thrust is T, the torque is Q, the water density is ρ, the ambient pressure is p_∞, the vapor pressure is p_v, and the static pressure on the propeller surface is p. Then, the advance speed coefficient J, thrust coefficient K_T, torque coefficient K_Q, propeller efficiency η_o in open-water condition, the cavitation number σ, and the pressure coefficient C_p can be expressed as follows [40]:

$$J=\frac{V_A}{nD}$$

(7)

$$K_T=\frac{T}{\rho n^2{D}^4}$$

(8)

$$K_Q=\frac{Q}{\rho n^2{D}^5}$$

(9)

$${\eta }_0=\frac{T{V}_A}{2\pi nQ}=\frac{K_T}{K_Q}\cdot \frac{J}{2\pi }$$

(10)

$$\sigma =\frac{p_{\infty }-p_v}{\frac{1}{2}\rho V_{\infty }^2}$$

(11)

$$C_p=\frac{p-p_{\infty }}{\frac{1}{2}\rho V_A^2}$$

(12)

where parameters J, K_T, K_Q, and η_o are required for the assessment of open-water performance, the cavitation number σ indicates the cavitation behavior, and the pressure coefficient C_p is used to describe the pressure distribution of the fluid field. To specify the relationship between the load and propeller performance, the thrust load factor K_T/J² and the power loading coefficient K_Q/J³ should be discussed.

2.4. Turbulence Model and Governing Equation

For incompressible flows, when the body force only counts gravity, the Cartesian tensor form of the NS (Navier–Stokes) equation is as follows:

$$\frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j}=g-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\frac{\mu }{\rho }\frac{{\partial }^2{v}_i}{\partial x_j^2}$$

(13)

where v_i and v_j denote velocity, t stands for time, p is the pressure, μ is the dynamic viscosity of the fluid, and g represents the external force.

In practical applications, in order to obtain the average changes in the flow field induced by turbulence, the Reynolds average method is introduced to solve the time-averaged NS equation. The time-averaged Reynolds equation here is namely, the Reynolds Average Navier–Stokes equation, which is simply referred to as the RANS equation. This paper adopted RANS-based SST k-ω turbulence model that combines the k-ε model for far field with the near-wall k-ω model for wall surface, i.e., blade surface. The model is effectively mixed together and it accurately predicts the pressure gradient changes in the non-equilibrium region of the boundary layer when fluid separation occurs. The governing equations of SST k-ω turbulence model are as follows [41]:

$$\begin{array}{c}\frac{\partial \omega }{\partial t}+U_j\frac{\partial \omega }{\partial x_j}=\alpha S^2-\beta {\omega }^2+\frac{\partial }{\partial x_j}\left[\left(\nu +{\sigma }_{\omega }{\nu }_T\right)\frac{\partial \omega }{\partial x_j}\right]\\ +2\left(1-F_1\right){\sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\partial x_i}\frac{\partial \omega }{\partial x_i}\end{array}$$

(14)

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

(15)

where k is the turbulent kinetic energy, ω is the specific turbulent energy dissipation rate, ν is the kinematic viscosity coefficient, ν_T is the turbulent kinematic viscosity, and τ_ij is the viscous shear stress.

For the open-water rotation of the propeller on model scale, there exists state transition from laminar flow to turbulence on the blade surface at low Reynolds number, and it is necessary to consider the transition phenomenon in simulation [42]. Taking time cost into consideration, the γ transition model [43] was selected for this paper. The intermittent transport equations were set as follow:

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

(16)

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

(17)

where S is the absolute value of strain rate, μ_t is the turbulent vortex viscosity, P_γ is the production item, E_γ is a dissipation term, F_onset controls the location of the transition, and F_length controls the length of transition region.

2.5. Cavitation Model

In this paper, Schnerr–Sauer model based on Rayleigh–Plesset simplified equation was applied, matched with two-phase (liquid phase and gas phase) Volume of Fluid (VOF) methodology, while ignoring the bubble growth acceleration, viscosity effect, and surface tension, which has been verified in the literature [36,37,44], proving that the model is accurate in the cavitation simulation for high-speed motion. The vapor volume fraction α is formulated as follows [32]:

$$\alpha =\frac{V_v}{V_{cell}}=\frac{N_{bubbles}\cdot \frac{4}{3}\pi R^3}{V_v+V_l}=\frac{n_0\cdot \frac{4}{3}\pi R^3}{1+n_0\cdot \frac{4}{3}\pi R^3}$$

(18)

where V_cell is the volume of the computational cell, V_v is the vapor volume, V_l is the volume occupied by liquid, N_bubbles is the number of bubbles in the computational cell, and R is the radius of the bubble nucleus. The setting of relevant parameters will be stated in Section 3.

3. Methods for Simulation

3.1. Modeling of Kappel Propeller with Different Tip Rakes

A 4-order cubic B-spline scheme was applied to vary the tip rake distribution of Kappel propellers while keeping the other parameters unchanged to ensure a smooth transformation from the non-end-plate part of the blade to the end-plate part.

The strategy is to maintain the coordinate values of the first two control points invariant, as shown in Figure 2, whereas the control points from 0.5 r/R begin to move, which keeps the turning points of the rake distribution curve basically the same. Xs/D values of control points No. 3 and No. 4 are both 0.008. In this paper, five Kappel propellers with different tip rakes were established, with tip rake values Xs/D of 0.025, 0.05, 0.075, 0.1, and 0.125, respectively, corresponding to propellers Kap01, Kap02, Kap03, Kap04, and Kap05, which are distinguished from the reference propeller Kap00.

The ratio S/R is introduced here as the quantitative index for the tip rake, in which the length of the rake distribution curve is denoted as S. The ratio S/R and immersion area ratio of the six Kappel propellers are shown in

Table 1. Distinguished ratios of the Kappel propellers.

Serial Number	Kap00	Kap01	Kap02	Kap03	Kap04	Kap05
$S/R$	1	1.023	1.059	1.100	1.143	1.187
Immersion area ratio	1	1.0071	1.0202	1.0373	1.0583	1.0796

, of which the 3D models are shown in Figure 3. Immersion area may provide associative reason for the variation of propeller drag, which is adverse to the propulsion performance. The basic geometric parameters of propeller Kap00, as shown in

Table 2. Basic geometric parameters of the reference propeller (Kap00).

Kappel Propeller	Model Scale
Diameter (mm)	250
Pitch ratio at 0.7 r/R	1.0338
Skew (°)	16
Hub diameter ratio	0.2
Projected area ratio	0.4788

, provide templates for the other five Kappel propellers.

3.2. Simulation Setting

The simulation process was conducted in Star CCM+ software. The calculation domain in this paper was uniformly set as a cylinder, as shown in Figure 4a, which was divided into stationary region and rotating region. The propeller diameter is D; the diameter of the stationary region is 8D and the height is 10D; the rotating region diameter is 1.08D and the height is 0.24D; the velocity inlet of the stationary region is 4D away from the propeller disk; and the pressure outlet is 6D downstream.

Turbulence strategy in this paper consists of SST k-ω turbulence model and γ transition model. Schnerr–Sauer method exerts the function of correcting the simulation in cavitating flow when combined with two-phase VOF approach. The density of the liquid phase, namely the density of water, was assumed constant at 998 kg/m³, while the dynamic viscosity and saturation pressure were set to 0.001008 Pa-s and 2338 Pa, given that the temperature was 20 °C. The gas phase is water vapor, of which the density is 0.59531 kg/m³ and dynamic viscosity is 1.26765 × 10⁻⁵ Pa-s at the same temperature. The volume fraction here was set to 0.1 to capture the water vapor surface. According to the literature [20], the appropriate range of the seed density is in excess of 1.0 × 10¹¹/m³, whereas for seed diameter, a value lower than 1.0 × 10⁻⁵ m offers acceptable convergence. In this paper, the seed density was set to 1.0 × 10¹²/m³ and the seed diameter was 1.0 × 10⁻⁶ m for all cases.

To ensure comparability, the cavitation number σ of the cases during open-water analysis listed in Section 4 was equally prescribed as 2.0. The advance speed varied with the inlet velocity, while the rotation speed of the rotating region sustained invariant.

The mesh on the blade surface was bound to be encrypted, as well as the rotating region and the wake area of the stationary region. A circular tube-type wake encryption zone was set behind the propeller to facilitate the capture of the wake velocity field and vortex distribution of the propeller, as shown in Figure 4b.

Due to blade tip rake changes, Kappel propellers have striking differences in terms of their geometric structure, which may affect the mesh independence analysis. Therefore, two representative propellers, Kap01 and Kap04, respectively, with smaller tip rake and larger tip rake, were selected for mesh independence analysis. The mesh independence was explored using Kap01 and Kap04, comparing five cases with the base sizes of 0.005D, 0.006D, 0.008D, 0.01D, and 0.012D under the same mesh strategy.

Here, the base size of mesh is denoted as l_D. Set the rotation speed to 12 rps and the advance speed coefficient J to 0.5. In Figure 5, the simulation results turn stable when the base size is less than or equal to 0.008D. Considering the accuracy and time cost of simulation, this paper chose the base mesh size of 0.008D uniformly in the following research. The value of y⁺ on the blade surface of each propellers was limited between 30 and 90, establishing a suitable thickness of boundary layer with a setting of 10 layers on the propeller surface, and the number of mesh cells was 4.254 × 10⁶.

3.3. Verification of the Methods for Open-Water Simulation

This paper further verified the accuracy of the open-water performance simulation. Due to the lack of available experimental references for Kappel propellers, here we have presented the calculation errors of tip-rake propeller P1727 without considering transition, as shown in Figure 6a, and with γ transition, as shown in Figure 6b, for which the model and the test results were provided in the literature [45,46]. The mesh strategy was parallel to that suggested in Section 3.2, while the rotation speed was set at 12 rps as well. 3.42 × 10⁶ was value of the total cell number.

It can be seen that the results considering γ transition are quite consistent with the EFD (Experimental Fluid Dynamics) data, which possess narrower gap in value within 5% deviation than those ignoring transition phenomenon. The calculation error of K_T notably dwindled by 16%~51%, in contrast to the error magnification of 10K_Q at medium and high advance speeds, which resulted in the calculation error of η_o sharply narrowing by a general range between 51% and 86%, with the calculation efficiency nearly identical to EFD data when J = 0.3. The comparison result shows the necessity of counting transition phenomenon, and the accuracy of the numerical models adopted in this paper was convincingly proved from the perspective of open-water performance.

3.4. Streamline Distribution of Kappel Propellers in Simulation

The way to directly perceive the transition phenomenon on the Kappel propeller’s surface is to extract the streamline distribution, as shown in Figure 7, for propellers Kap00, Kap02, and Kap04 with the advance speed coefficient J of 0.8 and rotation speed of 12 rps, together with the y⁺ distribution on the suction side blade surface, as shown in Figure 8.

In this working condition, the Reynolds number was 4.723 × 10⁵ according to the prescription from ITTC78 as the Formula (19).

$$\mathrm{Re}=\frac{b_{0.75R}\sqrt{V_A^2+{\left(0.75\pi nD\right)}^2}}{\nu }$$

(19)

where b_0.75R stands for the chord length of the blade section foil at the 0.75R radius.

Based on the conclusions summarized in the literature [47], it can be seen that the stream originating from the blade leading edge experiences a transition from laminar flow state to turbulence, and then, to a separation state, which is represented as the streamlines turning from straight lines to curved lines, and then converging near the trailing edge, with a vacuum zone of streamlines left. In terms of the advanced speed coefficient ranging from 0.3 to 1.0, the Reynolds number varies between 4.508 × 10⁵ and 4.858 × 10⁵, which will involve the transition phenomenon for all cases in this paper.

3.5. Convergence Analysis of Cavitation Simulation

For the observation of sheet cavitation, the visual stability is bound to be guaranteed. To quantify the convergence of the calculation, the volume of cavitation, i.e., volume of vapor bubbles, was monitored as the calculated time passed. Extraction occurred for propeller Kap01 when the advance speed coefficient J was 0.5, the rotation speed was 30 rps, and the cavitation number σ was 2.0, as shown in Figure 9. This demonstrated that the calculated volume of cavitation tends to be stable within the time range of 0.05 s.

4. Discussions of Simulation Results

4.1. Effect of Kappel Propeller Tip-Rake Change on Propeller Performance

The thrust and torque of the Kappel propellers with different tip rakes were obtained under the working conditions with an advance speed coefficient J of 0.3~1.0 and a rotation speed of 12 rps. The open-water simulation results are shown in Figure 10.

The thrust coefficient and the torque coefficient of the Kappel propellers were generally uplifted with the enlargement of tip rake, despite the case of propeller Kap01, where the performance is worst. When at a low advance speed (J = 0.3), a decline in the thrust coefficients of propellers Kap04 and Kap05 occurred, which implies that excessive tip rake is adverse to the thrust output of the Kappel propeller at a low advance speed. As for the propulsion efficiency, propellers Kap02 and Kap03 behaved excellently at both low and medium advance speeds (J = 0.3 and 05) among the propellers, while at medium and high advance speeds (J = 0.5~1.0), the propellers Kap03 and Kap04 were outstanding. At a low advance speed, the tip vortex of the propeller stayed weak, and the effect of the end-plate was inapparent. On the contrary, the addition of the tip rake increased the submerged area of the propeller blade, resulting in a resistance growth that weakened the gain brought by the end-plate effect, especially for propeller Kap01.

When J was 0.8 as the designed advance speed, with the rise of the tip rake, the efficiency first increased and then slipped down, reaching the maximum value near X_S/D = 0.075, which was 2.53% higher than the value of 0.712 of the propeller Kap00, as shown in Figure 11. The rake value X_S/D of the perfect mode of the Kappel propeller was between 0.075 and 0.1, and the ratio S/R was between 1.1 and 1.143, correspondingly. There was a sharp decline in propulsion efficiency for the propeller Kap05, which had a large tip rake, which was due to the inferior amplitude of the thrust increase compared with the amplitude of the torque rising. In Section 4.2, this paper will explain the decreased thrust gain from the perspective of the pressure difference between the blade surfaces.

The open-water efficiency of different Kappel propellers should be compared under the same thrust load factor K_T/J², which is shown in Figure 12, including the relationship of the power loading coefficient K_Q/J³ and propulsion efficiency.

It can be seen from Figure 12 that the efficiency of the propellers Kap01, Kap02, and Kap05 under a high load are numerically close to those of propeller Kap00 without an end-plate. At the lower load factors, the efficiency of the propeller Kap01 is the lowest. It is worth noting that the propellers Kap03 and Kap04 with moderate tip rakes behaved better than propeller Kap00 at a medium or high load. When the thrust load factor is 0.2 or the power loading coefficient is 0.05, the propulsion efficiency of propeller Kap04 is about 2% higher than that of propeller Kap00. The proper tip rake will greatly improve the propulsion performance of the Kappel propeller, helping either to reduce fuel consumption or achieve a higher navigation speed at a given engine rotation rate.

4.2. Effect of Tip-Rake Change on Pressure Distribution of Kappel Propellers

When the advance speed coefficient is 0.8, the pressure coefficient distribution on the suction side of the Kappel propeller blade is shown in Figure 13. There exists an obvious pressure gradient distribution on the blade surface. With the enlargement of the tip rake, the concentration area of the negative pressure expands from the center of the blade toward the blade tip, which registers as the dominant region latter, and the maximum value of the negative pressure also gradually rises on the blade tip, indicating that the cavitation is prone to generate in this region.

The load of the Kappel propellers concentrates on the blade between 0.75 r/R and 0.85 r/R, which is the part where the tip rake begins to grow. To evaluate the pressure distribution in more detail, from the expanded view of the blade section at 0.8 r/R in Figure 14, it can be seen that the pressure distribution of the different Kappel propellers at this section is basically the same. The negative-pressure area is concentrated on the suction side, showing a trend of scaling up as the tip rake enlarges, which is beneficial for the expansion of pressure differences between the suction side and the pressure side and the improvement in thrust output.

In Figure 14, the maximum pressure appears near the leading edge, within a positive-pressure region that is shrinking with the tip rake. There is a small part of the low-pressure area that is prone to broaden near the leading edge on the pressure side, as shown in Figure 14d–f, which may account for the performance decline of propeller Kap05.

4.3. Effect of Tip-Rake Change on Wake Vortex Distribution of Kappel Propellers

In this paper, the Q criterion is applied to treat propellers with different tip rakes. The definition of the Q criterion is shown in Formula (20) [48].

$$Q=0.5({\Vert \mathrm{S}\Vert }^2+{\Vert \Omega \Vert }^2)$$

(20)

where ||S|| = [tr(SS^T)]^0.5, ||Ω|| = [tr(ΩΩ^T)]^0.5, S is the vorticity tensor, and Ω is the strain rate tensor.

The same Q value is prescribed as 500 s⁻² to visualize the vortex structures of Kappel propellers with different tip rakes when J is 0.8. The tip-vortex structures are shown in Figure 15, together with the axial velocity distribution of the wake field as shown in Figure 16, in which the dimensionless parameter U stands for the ratio of the axial velocity to the advance speed.

It shows that the length of the tip vortex tends to approach the downstream to a greater extent as the tip-rake value magnifies, following the build-up of the diameter of the vortex, which suggests a smaller energy loss in the wake according to the momentum theory. Moreover, the diameters of the tip vortices of propellers Kap03, Kap04, and Kap05 are even larger, which is partly due to the swelling of the vortex tube diameter.

The tip vortex of propeller Kap01 stretches longer than that of propeller Kap00, as shown in Figure 17, and there is a smaller secondary vortex structure near the blade tip that is not found in other propellers. It is speculated that the reason for this is that when the B-spline curve is used to design the Kappel propeller, the tangent of the rake distribution curve at 1.0 r/R is perpendicular to the abscissa axis, thus the curvature changes greatly, resulting in a large geometric variation at the propeller tip.

4.4. Effect of Tip-Rake Change on Sheet Cavitation Distribution of Kappel Propellers

In order to evaluate the sheet cavitation behavior of the Kappel propellers related to various tip rakes, the surfaces of sheet cavitation based on the gas volume fraction of 0.1 were captured when J was 0.5, with a rotation speed of 30 rps, and a cavitation number σ of 2.0, as shown in Figure 18. Heretofore, the working conditions with the rotation speed ranges of 12 rps, 14 rps, 16 rps, 18 rps, 20 rps, and 25 rps were tested, in which no visual cavitation phenomenon could be discovered.

With the growth of the tip rake, the projected area of the sheet cavitation on the disk area experienced a process of increasing, which was confirmed by the monitored data of cavitation volume, as shown in Figure 19. The sheet cavitation adhered to the surface and centralized on the blade tip near the leading edge, with a movement expanding toward the trailing edge, which may not be so unfavorable for the thrust output and propulsion efficiency as the cavitation is generated in the tip vortex. For propeller Kap00, there was nearly no cavitation that could be detected, whereas propeller Kap01 slightly generated sheet cavitation on the blade tip.

To further check the cavitation performance of the Kappel propellers, we added a group of cases with the advance speed coefficient raised to 0.7 and the cavitation number changed to 0.75, forming the relationship sketch between the cavitation number and the thrust load factor, as shown in Figure 20. The sheet cavitation in these six cases was so minor that it was tough to observe visually; thus, the sheet cavitation distribution of the six cases is not presented as Figure 18. It can be seen in Figure 20 that when the thrust load factor is equal, the cavitation number comes down with the enlarging tip rake, indicating that the Kappel propeller with a larger tip rake possesses heavier cavitation properties. The cavitation performance when J = 0.8 shows the same relationship between the Kappel propellers as the cases in Figure 18 and Figure 19, which is not displayed here either.

5. Conclusions

In this paper, five Kappel propellers with different tip rakes were designed by the 4-order cubic B-spline curve, and a reference propeller was offered. The open-water performance and cavitation performance of the six Kappel propellers were analyzed by the CFD simulation coupled with the γ transition model and the Schnerr–Sauer cavitation model, after the verification of the numerical methodologies. The main conclusions are listed as follows:

(1)

The transition of the fluid state does exist on the blade surface of a rotating Kappel propeller on a model scale. The application of the γ transition could actually reduce the calculation error of the open-water efficiency by 51%~86%, notably reducing the error to a considerably small range.

(2)

The addition of end-plates would raise the thrust and torque outputs of Kappel propellers to an extent. This can result in the propulsion efficiency of the propeller Kap04 exceeding that of the reference propeller by 2.5% at the designed advance speed or when K_T/J² = 0.2 and K_Q/J³ = 0.05. The rake value X_S/D of the perfect mode of the Kappel propeller shall be between 0.075 and 0.1, and the ratio S/R shall be between 1.1 and 1.143, correspondingly. The advancement of propeller propulsion efficiency can help the application vehicle save fuel and lower the EEDI index, which is more environmentally friendly and compliant with green production.

(3)

The rise of the tip rake will contribute to the boost of the low-pressure value and area on the suction surface of the Kappel propeller, which makes the greatest contribution to the propulsion exaltation, which is weakened by the frictional resistance of the blade surface and low-pressure occurrence on the pressure side near the leading edge.

(4)

Benefiting from the effective end-plate effect, with the increase in the Kappel propeller’s tip rake, the length of the tip vortex tends to stretch and the vortex contraction is restrained, indicating a reduction in the energy loss of the wake, which is conducive to the elevation of the propulsion performance.

(5)

The addition of a tip rake will aggravate the sheet cavitation behavior in terms of either the distribution area or the volume of cavitation.

Sheet cavitation and the accrescent frictional resistance induced by the expansion of submerged blade area following the rise of the Kappel propeller’s tip rake will exert a negative impact, which can be covered by the gain brought by the increase in the pressure difference between the suction side and the pressure side when the tip rake is in a reasonable zone of value. Due to the lack of time in our study, we did not carry out a cavitation noise assessment, which may be conducted in future research.