e - e* n

cos

In Eq. (22), Ka is a constant which has to be acquired by experiments. K+ = Kt (J0) and K- = Kt (-J0). Eq. (22) coincides with Eq. (13) at 0 and 180 degree. Hence Eq. (22) contains all of the effects of ambient flow and implies that the total thrust force is composed of thrust force with bollard pull condition, K°, and additional force induced by ambient flow velocity and its incoming angle, / (J0, Q) .

To verify the proposed model, firstly, we operated the thruster under various ambient flow velocities: ±1.2m/ s, ±1.0m/ s, ±0.8m/ s, ±0.6m/ s, ±0.4m/ s, and 0m/ s with a zero degree incoming angle. Then, for 0.4m/s, 0.6m/s and 0.8m/s ambient flow velocities, the thruster was tilted at 5 degree increments from 0 to 180 degree to change incoming angle. For simplicity, we only consider cases where Q>0.

In Fig. 9, the experimental thrust forces are compared with simulation results of the proposed model with an input voltage range from 1.5V to 4.5V, which the whole range is between 0.0V and 5.0V for the positive direction. And the range from 0.0V to 0.8V is dead-

zone. Both results are very similar except at some localized points. The deviation could be caused by the thruster not being located in sufficiently deep water due to the restriction of the experimental environment. Thus, the anti- and vague directional response could have been disturbed by spouting water.

(a) thrust force - experiment (b) thrust force - simulation

Fig. 9. Comparison results of experiment and simulation by the proposed model

To highlight the performance of the proposed model, we compare the results with those of the conventional model described by Eq. (11). The comparison results are shown in Fig. 10. The figure shows that the results of the proposed model are significantly better than the conventional model in the anti- and vague directional regions.

Fluid velocity (rad/s) Fluid velocity (rad/s)

Fig. 10. Thrust force matching error

Fluid velocity (rad/s) Fluid velocity (rad/s)

(a) the proposed model (b) the conventional model

Figures 11(a), 11(c) and 11(e) show the thrust force comparison between experiment and simulation as a function of incoming angle. The errors of matching, as shown in Figs. 11(b), 11(d) and 11(f), are mostly within ±2N. Note that the maximum force of the thrust is up to 50N.

From the matching results with ambient flow velocities and incoming angles, we can say that the initial definition of axial flow is valid, and the proposed model shows good agreement with experimental results under various ambient flow velocities and incoming angles.

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