Effective parameters in the performance of surface-piercing propellers can be divided into geometrical and physical parameters. Geometrical parameters include diameter (D), pitch (P), number of blades (Z), expanded area ratio (EAR), rake angle (θ_r), skew angle (θ_s), immersion ratio (I_t), chamber profile (f), and thickness of blade (t). Functional conditions have an important and vital role in the hydrodynamic efficiency of surface-piercing propellers along with the geometrical parameters. Advance coefficient (J), Reynolds number (Re), cavitation number (σ), Weber number, shaft inclination angle (γ), yaw angle (ψ), and Froude number (Fr) are known as the physical parameters. Figure 1 shows the different propeller location angles. Some of these parameters, including the number of blades, pitch ratio, expanded area ratio, advance coefficient, Reynolds number, and cavitation number have identical behavior to the submerged propellers. Because of the function of surface-piercing propellers in the water and air, parameters that have lower importance in the design of submerged propellers will have special importance in the design of surface-piercing propellers. Therefore, the thrust and torque can be defined as follows for the surface-piercing propellers (Ghassemi and Ghiasi, 2011):

or

Non-dimensional parameters can be defined as follows:

where υ and κ are kinematic viscosity and dynamic surface tension, h_T is the immersion height of the propeller in water, and n is the rotation of the propeller. Non-dimensional hydrodynamic coefficients are obtained by calculating the mean thrust and torque of the propeller. The propeller's curve including the hydrodynamic coefficients of thrust (K_t) and torque (K_q) along with efficiency are defined as

where T is thrust, Q is torque, ρ is water density, and D is the propeller's diameter.

In this section, required equipment for the testing and calibration of the surface-piercing propellers is introduced. The experimental study for the surface-piercing propellers was conducted in the free surface water tunnel of the Sea-based Energy Research Group of Babol Noshirvani University of Technology. General specifications of the free surface water tunnel are presented in Table 1 (Seyyedi and Shafaghat, 2016).

Table 1General specifications of the free surface water tunnel and the test section (Seyyedi and Shafaghat, 2016).

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Figure 3Dynamometer and the installation details used for measuring of hydrodynamic parameters in the surface-piercing propellers.

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Figure 4Parameters and the locations of the corresponding sensors in the experiment.

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The free surface water tunnel and the test section are shown in Fig. 2.

A dynamometer was used for measuring thrust and torque forces in different shaft inclination angles, yaw angles, and immersion ratios (see Fig. 3). This dynamometer can measure the thrust up to 981 N, torque up to 67 Nm, and rotation up to 3600 rpm.

In order to ensure the accuracy of equipment for measuring different parameters, first, the water tunnel and dynamometer were calibrated. The speed calibration was conducted in the test section using an ultrasonic flow meter (Fluxus ADM 6725, Flexim company) and its connection to a manometer (for measuring the pressure difference). The accuracy of measurements was verified by (based on the different mercury height in manometer) repeating the measurements six times. Table 2 shows the standard deviation and coefficient of variation for speed in the test section. Considering the values in the table, the standard deviation was 0.016 m s⁻¹.

Table 2The standard deviation calculation of speed data in the test section.

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Table 3Calibration data for the shaft inclination angle at 0^∘ and immersion ratio 33 % in two different advance coefficients.

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Since calculation of the thrust and torque of a propeller is one of the major goals during testing of surface-piercing propellers, the dynamometer was used to achieve this goal. Figure 4 shows the measured parameters and the locations of the corresponding sensors in the experiment.

In the next step, the tests were repeated for five states in order to calibrate the thrust and torque sensors of a dynamometer to ensure the accuracy of results for a five-blade propeller (Fig. 5). Table 3 shows a sample of calibration data for the shaft angle at 0^∘ and immersion ratio 33 % in two different advance coefficients. The table shows that the obtained results have an acceptable accuracy (the mean standard deviations for the thrust and torque are less than 0.008 and 0.025, respectively).

Figure 5Front view of a five-blade surface-piercing propeller.

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A five-blade aluminum propeller with diameter 130 mm, pitch ratio 1.52, and extended area ratio 0.74 was used in this research. The schematic and specifications of the surface-piercing propeller are shown in Fig. 6 and Table 4. The blade section shape is shown in Fig. 7, which has a sharper leading edge, and the trailing edge is cup-shaped.

Figure 6View of the propeller.

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Figure 7Blade section shape.

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Table 4Model propeller specification.

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For the experimental tests eight advance coefficients (from 0.44 to 0.95), a shaft inclination angle (0 to 15^∘), and five immersion ratios (33 % to 70 %) were studied, and the variation of these parameters was determined using the Shafaghat et al. (2017) research. The test parameters are shown in Table 5.

Table 5Test parameters.

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One of the conditions and limitations of conducting experiments on the surface-piercing propeller is to meet the cavitation, Froude, Weber, and Reynolds non-dimensional numbers. In addition, other effective parameters also have a major role in designing the propellers and their hydrodynamic coefficients (Shafaghat et al., 2017). Similarity rules should hold for testing the propellers in the cavitation tunnel and generalizing results of the model to a real propeller. But similarity rules do not hold for the Weber, Froude, and Reynolds numbers simultaneously in the surface-piercing propellers. Determining the Weber and Froude numbers is also another issue in testing the surface-piercing propellers. In certain conditions, we can ignore the effects of Reynolds, Weber, and Froude numbers. This is very important from a laboratory view because in these conditions, other hydrodynamic characteristics of the propeller can be determined by measuring the thrust and torque in an advance coefficient interval, and there is no need to measure the hydrodynamic characteristics of the propeller by changing the non-dimensional numbers.

Separation of boundary layers does not occur in Reynolds numbers larger than a critical value, and the effect of the Reynolds number on the hydrodynamic characteristic fades in spite of the flow regime. Results of previous research showed that the Froude number has no effect on the results and critical advance coefficients for Froude numbers larger than 4 during the functional phases of surface-piercing propellers. When the Froude number is larger than 4, it can be assumed that the air bubbles have reached their final form and the hydrodynamic specifications approach the final values asymptotically (Shiba, 1953; Olofsson, 1996). According to Shiba (1953), the effect of the Weber number, which is associated with surface tension, on hydrodynamic characteristics of propellers fades away for values larger than 180, and the results of model tests can be generalized to the real prototype. A common method used by Pastocheni et al. (2007) for generalizing the results of a model to prototype in order to design the surface-piercing propellers was determining limits for the Reynolds, Froude, and Weber numbers. They concluded their research by proposing Eq. (5). The surface-piercing propeller function curve is independent of non-dimensional numbers, and the minimum rotation required for testing the propellers, using Eq. (5), is 31.32 rps or 1850 rpm.

The experimental results were validated by considering a four-blade propeller (Olofsson, 1996). The diameter of the propeller in Olofsson (1996) was 250 mm, but due to the dimensional limitation of the test section, the diameter of the propeller has decreased to 125 mm (Fig. 8).

Figure 8Propeller model (D=125 mm).

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Figure 9 shows the comparison of the coefficients of thrust and torque with those of obtained from Olofsson (1996) at Fr = 2 (V=2.21 m s⁻¹). This figure indicates that there is good agreement between the results.

Figure 9Comparison of the coefficient of thrust (a) and torque (b) with those obtained by Olofsson at Fr = 2 (1996).

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In this section, first, the results of a surface-piercing propeller test in different test conditions were compared to the hydrodynamic coefficients predicted by semi-experimental equations of Ferrando et al. (2007) and Montazeri and Ghassemi (2009). Then the effect of advance coefficients, immersion ratio, and shaft inclination angle on the hydrodynamic coefficients and efficiency were evaluated and results were analyzed. The presented data for hydrodynamic coefficients are the mean value of the results in all graphs. Generally, in previous studies, the raw experimental data have been used for comparison with values obtained by numerical or analytical methods, but in this paper, because only an experimental test is done, a continuous curve is used to observe the variations in hydrodynamic coefficients.

The comparison between the present experimental results and previous semi-experimental equations proposed by Ferrando et al. (2007) and Montazeri and Ghassemi (2009) is shown in Figs. 10 and 11. Figure 10 compares the hydrodynamic coefficients obtained by the regression equation of Ferrando et al. (2007) and Montazeri and Ghassemi (2009) with the experimental values at immersion ratios of 33 % and 50 % and a shaft inclination angle of 0^∘. As seen, although both semi-experimental equations can correctly predict the increase in hydrodynamic coefficients by an increasing immersion ratio, the difference between the values predicted by these equations with the experimental tests, especially for the thrust, shows that these equations are not accurate enough. The accuracy of the equation presented by Ferrando et al. (2007) for the thrust and torque, especially in the low immersion ratio, is higher than those proposed by Montazeri and Ghassemi (2009).

Figure 10Comparison of the hydrodynamic coefficients predicted by Ferrando et al. (2007) and Montazeri and Ghassemi (2009) semi-experimental equations and the experimental values.

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Figure 11 shows the comparison between the hydrodynamic coefficients obtained by the regression equation of Ferrando et al. (2007), and Montazeri and Ghassemi (2009) with the semi-experimental values in two shaft inclination angles of 0 and 10^∘ for the immersion ratio of 33 %. As seen by changing the shaft inclination angle, the hydrodynamic coefficients predicted by Ferrando and Montazeri's equations remain relatively stable, while in the experimental values, the thrust and torque hydrodynamic coefficients decrease and increase, respectively, by increasing the shaft inclination angle.

Figure 11Comparison of the surface-piercing hydrodynamic coefficients predicted using Ferrando et al. (2007) and Montazeri and Ghassemi (2009) semi-experimental coefficients with the experimental values.

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The comparison indicated that semi-experimental equations do not have suitable accuracy for predicting hydrodynamic coefficients because all of the geometrical and physical parameters effective on the performance of the propellers have not been studied yet. On the other hand, semi-experimental data which are provided using a database of experimental tests are imperfect. One of these defects is the definition of equations based on the submerged area. In a fully ventilated regime, the free surface rises due to the formation of a pressure field such that the immersion cannot be measured correctly. In these equations, effects of immersion ratio and shaft inclination angle were imposed separately to determine the thrust and torque which reduce accuracy of equations. Because the experimental test data for the effect of shaft inclination angle have been used in the semi-experimental equations, the experimental data obtained in this study can be used to improve the accuracy and reduce the semi-experimental equation error.

Immersion ratio is one of the most important parameters in the performance of surface-piercing propellers. In the previous studies, Dyson (2000) and Ferrando et al. (2007) conducted their experiments for a five-blade propeller. Dyson (2000) conducted the experiments for two immersion ratios of 30 % (shaft inclination angle of 4) and 50 % (shaft inclination angle of 8), while Ferrando et al. (2007) investigated a 0.4–0.7 immersion ratio in the shaft inclination angle of 6. In Lorio's study (2011) for a four-blade propeller, the immersion ratio of 33 % to 50 % was studied with a fixed shaft inclination angle.

Figures 12 and 13 show the effect of immersion ratio on the thrust, torque, and efficiency of five-blade propellers in different advance coefficients and two fixed shaft inclination angles 0 and 10. Immersion ratio changes by maintaining a fixed shaft inclination angle in these figures. As seen in Fig. 12a and b, for the shaft inclination angle 0, as the immersion ratio increases the thrust and torque hydrodynamic coefficients increase. Similar results are obtained for the thrust and torque in Dyson (2000) and Ferrando et al. (2007). As the immersion ratio surpasses 50 %, as well as increasing the advance coefficient beyond the critical advance coefficient, the thrust will significantly reduce. Increasing the thrust and torque hydrodynamic coefficients with the increase in immersion ratio is due to the increase in effective blade area in the water and lift and drag forces on the blade which increase the thrust and torque, respectively.

Figure 12Hydrodynamic coefficients and efficiency of a surface-piercing propeller for a 0 shaft degree and different immersion ratios.

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Figure 12 shows the transition region for a 33 %–50 % immersion ratio at J_a=0.67. As seen by an increase in the immersion ratio, the critical advance coefficient reduces. Figure 12c shows that in low advance coefficients, the efficiency increases by the increase in the immersion ratio, while in a high advance coefficient the efficiency increases for the immersion ratio up to 50 % and then decreases. As seen at 0^∘ shaft inclination, the maximum efficiency in all advance coefficients occurs at a 50 % immersion ratio. The maximum efficiency changes from 65 % to 33 % when the immersion ratio changes from 78 % to 50 %. The highest efficiency in all immersion ratios was achieved for shaft inclination 0^∘. This result is not consistent with the results of Ferrando et al. (2007) and Dyson (2000) in which the immersion ratio had no significant effect on changing the efficiency of the propeller, and its increase only increased the thrust and torque.

In Misra's study (2012) on four-blade propellers with different geometries, the maximum efficiency occurred at a 50 % immersion ratio. The experimental results for the four-blade propellers also showed that with increasing the immersion ratio for the propellers, the maximum efficiency reduces from 61 % in the 30 % immersion ratio to 27 % in the 70 % immersion ratio. Therefore, the results indicate that for each propeller with different geometrical characteristics and profiles, a different result can be observed for the effect of the immersion ratio on efficiency. Figure 13 shows the effect of immersion ratio on the hydrodynamic coefficients and propeller efficiency at 10^∘ shaft inclination. As seen by increasing the immersion ratio, the thrust and torque increase. With increasing the immersion ratio from 40 % to 60 %, and in high advance coefficients, the thrust will slightly increase in a certain advance coefficient. Figure 13c shows that the maximum efficiency ranges from 30 % at a 33 % immersion ratio to 41 % at a 40 % immersion ratio, which is the highest efficiency in all immersion ratios and advance coefficients (for 10^∘ shaft inclination). This means that despite the 0^∘ shaft inclination, the maximum efficiency occurs at a 40 % immersion ratio. Therefore, the propeller's location angle has an impact on maximum efficiency at a different immersion ratio. Figures 12 and 13 show that by increasing the immersion ratio, K_T and K_Q will significantly increase. Propeller efficiency is maximum at a 40 %–50 % immersion ratio. When higher thrust is needed without exceeding the allowable torque, increase in the immersion ratio can be useful or vice versa; when thrust condition is met but torque exceeded the limit, reducing the immersion can help to reach an optimum condition.

Figure 13Hydrodynamic coefficients and efficiency for the surface-piercing propeller at 10^∘ shaft inclination and different immersion ratios.

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Figure 14Effect of the shaft inclination angle on the hydrodynamic coefficients and efficiency of the surface-piercing propeller at 33 % immersion ratio.

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This section examines the effect of shaft inclination angle on the hydrodynamic coefficients and efficiency of the propeller at a fixed immersion ratio. By keeping the immersion ratio constant, and changing the shaft inclination ratio, significant patterns occur in the thrust, torque, and efficiency of the propellers which are shown in Fig. 14 for the 33 % immersion ratio. By changing the shaft inclination angle from 0 to 5, the thrust will significantly reduce and, with further increase in the inclination angle, although the thrust decreases, these coefficients are close to each other (see Fig. 14a). By increasing the shaft inclination angle, the critical advance coefficient reduces. The effect of the free surface also decreases and changes from a partially ventilated phase to a total fully ventilated phase. As seen in Fig. 14b, the variation of torque with changing the shaft inclination angle is contrary to the thrust variation. The torque decreases for the shaft inclination angle from 0 to 5^∘ and then increases beyond with further increase in the shaft inclination. A similar observation was reported by Lorio (2011) in which with the increase in the shaft inclination angle, the thrust remained constant and torque increased. Oloffson (1996) concluded that at 33 % immersion ratio, by increasing the shaft inclination angle from 0 to 5, both thrust and torque increase. As seen in Fig. 14c, the efficiency reduces by increasing the shaft inclination angle such that the maximum efficiency reduces from 65 % in 0^∘ to 25 % at 15^∘ (reduces by 60 %). In other words, by increasing the shaft inclination angle from 0 to 15, the maximum efficiency reduces by 15 %. The reason for this significant decrease, according to Eq. (4), is the decrease in thrust and increase in torque. Changing the shaft inclination angle will change the force imposed on the propeller. This reduces the thrust and influences the efficiency and performance of the propeller.

Figure 15Effect of shaft inclination angle on the hydrodynamic coefficients and efficiency of the surface-piercing propeller at different advance coefficients and 40 % immersion ratio.

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The effect of shaft inclination angle on the thrust hydrodynamic coefficient at 40 % immersion ratio and different advance coefficients is shown in Fig. 15. According to this figure, the thrust reduces by increasing the shaft inclination angle. As seen at $\mathit{\gamma }={\mathrm{0}}^{\circ }$ and $\mathit{\gamma }={\mathrm{5}}^{\circ }$, the maximum thrust occurs at J=0.67, and by increasing the shaft inclination ratio, the critical advance coefficient reduces to J=0.59. The maximum and minimum thrusts in all advance coefficients occur at $\mathit{\gamma }={\mathrm{0}}^{\circ }$ and $\mathit{\gamma }={\mathrm{15}}^{\circ }$. In Fig. 15b, by increasing the shaft inclination angle, initially, the torque reduces at $\mathit{\gamma }={\mathrm{5}}^{\circ }$ and, then, it increases except for the fully ventilated area in which the propeller is under a heavy load. Based on Fig. 15c, by increasing the shaft inclination angle, the efficiency reduces in all advance coefficients due to a simultaneous reduction of the thrust and increase in the torque. The maximum efficiency also occurs at 0^∘ shaft inclination angle. According to Fig. 15, it can be concluded that at 30 % immersion ratio, the increase in the shaft inclination angle reduces the thrust and efficiency and increases the torque.

Similarly to Fig. 15, Fig. 16 shows the effect of shaft inclination angle on the hydrodynamic coefficients and efficiency at a 60 % immersion ratio. According to this figure, by increasing the shaft inclination angle, the thrust reduces. On the other hand, the torque increases by an increase in the shaft inclination angle, except at 15^∘. Comparing the torque in Figs. 15 and 16 shows that the torque variation depends on the immersion ratio, in addition to the shaft inclination angle. According to Fig. 16c, the maximum efficiency was obtained at a 0^∘ shaft inclination angle (low advance coefficients) and the minimum efficiency at 15^∘ shaft inclination. Therefore, by increasing the shaft inclination angle, the efficiency decreases.

Figure 16Effect of shaft inclination angle on the hydrodynamic coefficients and efficiency of the surface-piercing propeller at different advance coefficients and 60 % immersion ratio.

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Figure 17Variation of maximum efficiency at different immersion ratios and shaft inclination angles.

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Figure 17 shows the maximum efficiency variations of propellers for all immersion ratios and shaft inclination angles. As seen by increasing the immersion ratio at $\mathit{\gamma }=\mathrm{0}{}^{\circ }$, the maximum efficiency increases from 64 % at 33 % immersion ratio to 78 % at 50 % immersion ratio, and after that decreases to 66 % at 70 % immersion ratio. It is also clear that by increasing the shaft inclination angle, except at 70 % immersion ratio, the maximum efficiency reduces. In all shaft inclination angles, except 0^∘ shaft inclination, the maximum efficiency is at 50 % immersion ratio. The best position for installing of the propeller in this research is at 50 % immersion ratio and $\mathit{\gamma }={\mathrm{0}}^{\circ }$, because a propeller has the highest efficiency.

All the data used in this paper can be obtained by request from the corresponding author.

SMS and MS conducted experiments, analyzed data and wrote the manuscript with support from RS. RS verified the results and supervised the whole project.

The authors declare that they have no conflict of interest.

This paper was edited by Amin Barari and reviewed by three anonymous referees.