406 



where t= max. thickness of propeller section 

 c= chord length of propeller section 



The drag is calculated using the characteristics 

 of the equivalent profiles of the NSMB B-series 

 propellers. 



Lifting Surface Calculations 



The lifting surface theory calculates the induced 

 velocities over the propeller blades, in chordwise 

 and radial direction, thus including the effects 

 of finite aspect ratio of the blades. The draw- 

 back is that the theory is linearized, which 

 restricts the validity to lightly loaded propellers. 



Van Gent (1977) has shown in his thesis how 

 heavily loaded propellers can be treated with a 

 linearized theory since the vorticity in the wake 

 induces an additional axial velocity component in 

 the propeller plane, keeping the angles of attack 

 of the propeller sections small. 



The boundary conditions on the propeller blades 

 are fulfilled at a number of chordwise and spanwise 

 points. In our calculations four chordwise and 

 ten radial points per blade were chosen. The pitch 

 of the vortex sheet in the wake was taken rather 

 arbitrarily as the pitch at 0.7D. 



A very approximate description of the viscous 

 effects is used. The drag force of the propeller 

 sections is split into two parts : a drag force 

 as a result of losses in the suction peak at the 

 leading edge and a drag force due to friction. The 

 latter is calculated using a friction coefficient 

 of 0.0080, irrespective of the Reynolds number. 

 The first drag force is taken as half the theoretical 

 suction force. The same correction is also applied 

 to the sectional lift, which is obtained from chord- 

 wise integration of the lift distribution. In the 

 calculation of the induced velocities the geometrical 

 pitch angle is reduced by 3/4 degree to simulate 

 viscous effects on the zero lift angle. 



The open-water diagrams as calculated with the 

 lifting surface theory as described by Van Gent 

 (1977) are shown in Figure 7 together with experi- 

 mental results and lifting line calculations. The 

 general agreement with measurements is as good as 

 the lifting line calculations. This makes clear 

 that the linearized lifting surface theory can 

 indeed produce reliable open-water characteristics 

 up to high propeller loadings. At very low advance 

 ratio's the calculations deviate from the measure- 

 ments but this might well be caused by an erroneous 

 estimate of the viscous effects. 



Calculation of the Pressure Distribution 



Lifting line as well as lifting surface calculations 

 give the radial distribution of the lift coefficient, 

 of the angle of attack, and of the induced camber 

 (or camber distribution) which can be translated 

 into a zero lift angle. In Figure '8 these results 

 are compared for propeller A at A0% slip. The 

 lifting line calculation gives a higher loading at 

 the tip and a lower loading at inner radii, compared 

 with the lifting surface calculation. This is 

 characteristic for all four propellers in all 

 conditions. The total thrust does not differ very 

 much. Large differences, however, are foiind for 

 the angle of attack and for the zero lift angle. 



04 



LIFT SURFACE 

 LIFT LINE 



■'TOT- '-^INCIDENCE 



ttn = ZERO LIFT ANGLE 



FIGURE 8. Radial distribution of lift coefficient and 

 angle of attack on propeller A at 40% slip. 



Since these values will be used in the calculation 

 of the pressure distribution this discrepancy needs 

 further attention. 



The source of the discrepancy is the choice of 

 Eqs. 8 and 9, used in the lifting line calculation. 

 The reduction of the slope of the lift curve with 

 the lifting surface correction factor for the camber, 

 K(, (Eq. 8), is an empirical one, first suggested 

 by Lerbs (1951) when he analyzed the lift slopes 

 of his "equivalent profiles". The physical meaning 

 of this correction is not clear, but it still can 

 lead to correct results for thrust and torque, 

 since the lift slope for the equivalent profiles 

 was derived using a lifting line theory and experi- 

 mental values of thrust and torque. Therefore, this 

 correction for the lift slope, used in combination 

 with the same lifting line theory, should give 

 results for thrust and torque not too far from the 

 experimental results. The definition of the three 

 dimensional zero lift angle (Eq. 9) is another 

 empirical relation, bringing the calculated open 

 water characteristics in line with experiments. 

 However, this does not necessarily mean that the 

 three dimensional angle of incidence and zero lift 

 angle have a physical meaning and can be used for 

 the calculation of the pressure distribution. 

 Therefore, the results of the lifting surface cal- 

 culations are used in the following to calculate 

 the pressure distribution. 



To calculate the pressure distribution on the 

 blades, the effect of propeller thickness has to 

 be calculated and the leading edge singularity of 

 the lift distribution has to be dealt with. Tsakonas 

 et al. (1976) calculated the pressure distribution 

 on the propeller blades using a singularity distri- 

 bution for the thickness, in combination with a 

 linearized lifting surface theory. These calcula- 

 tions, however, remain linearized, producing an 

 infinite velocity at the leading edge, which was 

 removed by the Lighthill correction for thin air- 

 foils [Lighthill (1951)]. In our study, three- 

 dimensional effects on the pressure distribution 

 are neglected. Interaction effects between thickness 



