V Fig. 1. Propeller race contraction; velocity and pressure changes

If we use the control-volume-horizontal-momentum relation between sections 1 and 2,

Fig. 1. Propeller race contraction; velocity and pressure changes

If we use the control-volume-horizontal-momentum relation between sections 1 and 2,

T = m(v - u). (1) A similar relation for a control volume just before and after the disk gives

Equating these two yields the propeller force

Assuming ideal flow, the pressures can be found by applying the incompressible Bernoulli relation up to the disk

Subtracting these and noting that m = pApUp through the propeller, we can substitute for pb - pa in Eq. (3) to obtain

Finally, the thrust force by the disk can be written in terms of up and u by combining Eqs. (5) and (6) as follows:

Up to this point, the procedures are the same as the previous approaches. Now, we define the axial flow velocity as u = k,u + k„DQ, p 1 2 '

where k1 and k2 are constant. The schematic diagram of the axial flow relation is shown in Fig. 2. Fig. 2. Proposed axial flow model

For quasi-stationary flow, the axial flow only depends on ambient flow and propeller rotational motion. More complex combinations of ambient flow and propeller velocity are possible, but this linear combination is adequate as will be shown later. This somewhat simplified definition gives lots of advantages and physical meanings.

Finally, substituting Eq. (8) to Eq. (7), the proposed thrust model can be derived as follows:

T = 2pAp (k1u + k2DO)(k1u + k2DQ — u), = 2pAp (k'1 u2 + k'2 uDQ + k'3 D 2Q2).

This model will be used in the following non-dimensional analysis.

2.2 Non-dimensional analysis Fig. 3. Thrust coefficient as a function of advance ratio and its linear approximation

Discontinuity points

Fig. 3. Thrust coefficient as a function of advance ratio and its linear approximation

The non-dimensional representation for thrust coefficient has been widely used to express the relation between thrust force, propeller shaft velocity and ambient flow velocity as below:

where J0 = u/DQ is the advance ratio. Figure 3 shows a typical non-dimensional plot found in various references (Manen and Ossanen, 1988; Blanke et al., 2000). In former studies, the non-dimensional relation is only given as an empirical look-up table or simple linear relationship for the whole non-dimensional map as (Fossen & Blanke, 2000).

However, as shown in Fig. 3, Eq. (11) cannot accurately describe the characteristics of the thrust coefficient, especially when J0<0, and, rather than a linear equation, the thrust coefficient seems to be close to a quadratic equation except for the discontinuity points. Even more, Eq. (11) has no physical relationship with thrust force, but is just a linear approximation from the figure.

The proposed axial flow assumption would give a solution for this. The non-dimensionalization of Eq. (9) is expressed as 