Dq

And, the quadratic thrust coefficient relation is obtained as following:

Hence, contrary to other models, the axial flow definition of Eq. (8) gives an appropriate relationship between the thrust force equation and non-dimensional plot since the derivation was done by physical laws. Also, Eq. (13) can explain the characteristics of the quadratic equation of the thrust coefficient. From this phenomenon, we can perceive that the axial flow definition in Eq. (8) is reasonable. The coefficients of quadratic equations could be changed depending on hardware characteristics. However, there is still a question of the discontinuities of thrust coefficient in Fig. 3 which has not been answered yet by existing models. This problem will be addressed in the following section.

3. Thrust force with ambient flow model

If ambient flow varies, thrust force changes even with the same propeller shaft velocity, which means that the ambient flow disturbs the flow state under the bollard pull condition. Flow state is determined by a complex relation between propeller shaft velocity, ambient flow velocity and its incoming angle. This will be shown in the following subsections.

3.1 Flow state classification using CAR

In this subsection, we define three different flow states according to the value of advance ratio and the condition of axial flow. To distinguish them, we introduce Critical Advance Ratio (CAR), J*.

The three states are as below: • Equi-directional state

Anti-directional state

Figure 4 shows the flow states schematically.

Figure 4 shows the flow states schematically.

(a) equi-directionai state (b) anti-directional state (c) vague directional state

Fig. 4. Three flow states

Equi-directional state: The equi-directional state occurs when the ambient flow direction and axial flow direction coincide. In this state, if the ambient flow velocity increases, the pressure difference decreases. Hence the thrust force reduces, and the streamline evolves as a general form. (Fig. 4(a))

Anti-directional state: The anti-directional state happens when the ambient flow and axial flow direction are opposite. However, the axial flow can thrust out the ambient flow, hence the streamline can be built as sink and source. The Bernoulli equation can be applied and the thrust equation is still valid but the coefficients are different from those of the equi-directional state. Also, the thrust force rises as the ambient flow velocity increases, because the pressure difference increases. (Fig. 4(b))

Vague directional state: In the vague directional state, the axial flow cannot be well defined. The axial flow velocity cannot thrust out the ambient flow; hence the direction of axial flow is not obvious. This ambiguous motion disturbs the flow, so the thrust force reduces. In this case, we cannot guess the form of the streamline, so the thrust relation cannot be applied. However, the experimental results show the proposed thrust relation is still valid in this state. (Fig. 4(c))

Fig. 5. Thrust coefficient as a function of advance ratio and Critical Advance Ratio (CAR)

Former studies did not consider the anti- and vague directional states, however they can be observed frequently when a vehicle tries to stop or reverse direction. The CAR divides between the anti- and vague directional states as shown in Fig. 5. It would be one of the important characteristics of a thruster. At this CAR point, the ambient flow and propeller rotational motion are kept in equilibrium. Hence, to increase the efficiency of the thruster in the reverse thrust mode, an advance ratio value larger than the CAR is preferable, as shown in Fig. 6.

Fig. 5. Thrust coefficient as a function of advance ratio and Critical Advance Ratio (CAR)

Force

Ambient flow velocity

Fig. 6. Thrust force as a function of ambient flow velocity

Ambient flow velocity

Fig. 6. Thrust force as a function of ambient flow velocity

3.2 Effects of incoming angle on thrust force

Fig. 7. Incoming angle of ambient flow

In this subsection, the incoming angle effect on thrust force is analyzed. Figure 7 shows the definition of incoming angle. Naturally, if the angle between ambient flow and thrust force is non-parallel, the thrust force varies with the incoming angle. Basically, by multiplying ambient flow velocity by the cosine of the incoming angle, the thrust force can be derived from Eq. (13). In that case, however, the calculated thrust force will not coincide with experimental results except at 0 and 180 degrees, which shows that the incoming angle and ambient flow velocity have another relationship. Hence, we develop the relationship based on experiments, and Fig. 8 shows the result.

Fig. 8. Thrust force as a function of incoming angle

In Fig. 8, the whole range of angles is also divided into three state regions as denoted in the previous subsection. And, we define the borders of the regions as Critical Incoming Angles (CIA) which have the following mathematical relationship.

where aj, a2, b2, and c2 are all positive constants. And, 0* (u) and 02 (Q, u) are the first and second CIA, respectively. Theoretical reasons have not been developed to explain the CIA equations, but empirical results give a physical insight and the above equations can be correlated to experiments. The equi-directional region and anti-directional region are differentiated with the first CIA. The first CIA only depends on the ambient flow velocity. At the first CIA, the thrust coefficient is the same as the thrust coefficient with no ambient flow velocity. The second CIA separates the anti-directional region and the vague directional region. The second CIA depends not only on ambient flow angle but also on propeller shaft velocity. From Eqs. (20) and (21), the three regions shift to the left as the ambient flow velocity increases.

Now, we derive the incoming angle effect on the thrust force as following:

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