Surprising Behaviour of the Wageningen B-Screw Series Polynomials

Abstract

Undoubtedly, the Wageningen B-screw Series is the most widely used systematic propeller series. It is very popular to preselect propeller dimensions during the preliminary design stage before performing a more thorough optimisation, but in the smaller end of the market it is often used to merely select the final propeller. Over time, the originally measured data sets were faired and scaled to a uniform Reynolds number of 2 · 10⁶ to increase the reliability of the series. With the advent of the computer, polynomials for the thrust and torque values were calculated based on the available data sets. The measured data are typically presented in the well-known open-water curves of thrust and torque coefficients $K_T$ and $K_Q$ versus the advance coefficient $J$. Changing the presentation from these diagrams to efficiency maps reveals some unsuspected and surprising behaviours, such as multiple extrema when optimising for efficiency or even no optimum at all for certain conditions, where an optimum could be expected. These artefacts get more pronounced at higher pitch to diameter ratios and low blade numbers. The present work builds upon the paper presented by the author at the AMT’17 and smp’19 conferences and now includes the extended efficiency maps, as suggested by Danckwardt, for all propellers of the Wageningen B-screw Series.

1. Introduction

1.1. The Wageningen B-Screw Series and Its Polynomial Representation

The Wageningen B-screw Series dates back to 1936 [1], when the first results were published. In the following years, the series was systematically expanded to include more than 120 single propellers. The measured data were presented inter alia in open-water diagrams showing the dimensionless thrust and torque coefficients, $K_T$ and $K_Q$, and the open-water efficiency ${\eta }_o$ as functions of the also dimensionless advance coefficient $J$:

$$J=\frac{v_a}{nD},$$

(1)

$$K_T=\frac{T}{\rho n^2{D}^4},$$

(2)

$$K_Q=\frac{Q}{\rho n^2{D}^5}=\frac{P_P}{2\pi \rho n^3{D}^5},\mathrm{and}$$

(3)

$${\eta }_o=\frac{J}{2\pi }·\frac{K_T}{K_Q},$$

(4)

where T = measured thrust; Q = measured torque; $P_P$ = propeller power; $\rho$ = water density; n = shaft speed (in ${\mathrm{s}}^{-1}$); D = propeller diameter; and $v_a$ = speed of advance. If not stated otherwise, all values are in SI base units.

All tested propeller models had a diameter of 240 $\mathrm{m}$$\mathrm{m}$ and, hence, different section chord lengths due to the variable blade area ratio and number of propeller blades. The propellers were tested in varying model basins of MARIN using a diverse rate of revolutions resulting in considerably different Reynolds numbers for each propeller in the whole series. Oosterveld and van Oossanen engaged in the formidable tasks of scaling all available open-water data sets to a uniform Reynolds number of 2 · 10⁶ (based on chord length and section advance speed) and calculating polynomials for the thrust and torque coefficients by multiple regressions analysis [2]. With the help of these polynomials, it is possible to calculate these coefficients as functions of the advance coefficient $J$, the pitch to diameter ratio $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$, the expanded blade area ratio $\raisebox{1ex}{$A_e$}\!\left/ \!\raisebox{-1ex}{$A_0$}\right.$, and the number of blades Z:

$$K_T=\sum C_{s,t,u,v}·J^s·{\left(\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.\right)}^t·{\left(\raisebox{1ex}{$A_e$}\!\left/ \!\raisebox{-1ex}{$A_0$}\right.\right)}^u·Z^v\mathrm{and}$$

(5)

$$K_Q=\sum C_{s,t,u,v}·J^s·{\left(\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.\right)}^t·{\left(\raisebox{1ex}{$A_e$}\!\left/ \!\raisebox{-1ex}{$A_0$}\right.\right)}^u·Z^v,$$

(6)

where $C_{s,t,u,v}$ = coefficient; and s, t, u, and v = whole-number exponents.

These polynomials are nowadays widely used in either selecting the optimum propeller or as a basis for further refinements. It is therefore of utmost significance that these polynomials are consistent and accurate.

1.2. Time Line of Experimental Results

As shown by Helma for the B4-70 propeller, the open-water characteristics have changed substantially over time [3], see Figure 1. The effect on the widely used $B_p$–$\delta$ diagram was also outlined in the same work, see Figure 2.

2. Efficiency Maps

2.1. Basic Efficiency Maps

It is assumed, that the presentation in the form of the standard open-water diagrams for propellers are known to the reader. To recap, these diagrams show three families of curves with the thrust and torque coefficients and the efficiency as functions of the advance coefficient—$K_T\left(J\right)$, $K_Q\left(J\right)$, and ${\eta }_o\left(J\right)$—for a set of constant $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values for each propeller of the Wageningen B-screw Series.

Many authors suggested a different way of representing the nondimensional open-water data (see Appendix A.2); we will call them efficiency maps, as proposed by many of these authors. There are two efficiency maps: the $K_Q$–$J$ and the $K_T$–$J$ map. On the $K_Q$–$J$ efficiency map, we draw the family of $K_Q\left(J\right)$ curves for our set of constant $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values as for the conventional open-water diagram (thin red lines, please refer to Figure 3a). Instead of adding the family of efficiency curves for constant $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values, we add contour lines for the efficiency: ${\eta }_o\left(J,K_Q\left(J\right)\right)=\mathrm{const}$ for a set of selected ${\eta }_o$-values (thin black lines). We can also draw the contour lines for the thrust coefficient: $K_T\left(J,K_Q\left(J\right)\right)=\mathrm{const}$ (dashed red lines). So far, this gives us a diagram with exactly the same information content as for the conventional open-water diagram, just displayed in a different way.

We can now draw two lines of maximum propeller efficiency into this basis efficiency map: “${\eta }_{o,max}$ for $J=\mathrm{const}$” (bold black line) and “${\eta }_{o,max}$ for $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.=\mathrm{const}$” (bold red line). (Appendix A.1 describes their derivation.) In the open-water diagram, the curves corresponding to the “${\eta }_{o,max}$ for $J=\mathrm{const}$” and “${\eta }_{o,max}$ for $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.=\mathrm{const}$” lines would be the envelope to and the line connecting the maxima of all ${\eta }_o\left(J\right)$ curves, respectively.

2.2. Enhanced Efficiency Maps

To enhance the basic efficiency map, these families of curves can be added:

$$\begin{array}{ccc}{T}_D\hfill & =\frac{K_T}{J^2}\hfill & =\mathrm{const},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}{P}_D\hfill & =\frac{K_Q}{J^3}\hfill & =\mathrm{const},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}{T}_n\hfill & =\frac{K_T}{J^4}\hfill & =\mathrm{const},\mathrm{and}\hfill \end{array}$$

(9)

$$\begin{array}{ccc}{P}_n\hfill & =\frac{K_Q}{J^5}\hfill & =\mathrm{const}.\hfill \end{array}$$

(10)

These four curves have a very practical significance, because each of them eliminates one of the two unknowns in propeller optimisation: D or n, the propeller diameter and the shaft speed, respectively. (Examples of how these diagrams can be used for propeller selection are presented in Appendix A.4).

Note that in the $K_Q$–$J$ efficiency map, the lines for $P_D$ and $P_n=\mathrm{const}$ are ordinary curves to the single power of three and five, whereas the curves for $T_D$ and $T_n=\mathrm{const}$ are truly parametric curves. Having established these families of curves, we can now draw the four lines of maximum propeller efficiency for each of them: “${\eta }_{o,max}$ for $T_D=\mathrm{const}$”, “${\eta }_{o,max}$ for $P_D=\mathrm{const}$”, “${\eta }_{o,max}$ for $T_n=\mathrm{const}$”, and “${\eta }_{o,max}$ for $P_n=\mathrm{const}$”, all connecting the points of maximum efficiency (see Appendix A.1.3 for how they are constructed). Figure 3b shows the family of curves for $T_D=\mathrm{const}$ (thin blue lines) and for “${\eta }_{o,max}$ for $T_D=\mathrm{const}$” (bold blue line).

As mentioned earlier, there exists a second diagram: the $K_T$–$J$ efficiency map with $K_T$ as the ordinate. It is composed of the families of $K_T\left(J\right)$ curves for the set of constant $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values and the contour lines $K_Q\left(J,K_T\left(J\right)\right)$ and ${\eta }_o\left(J,K_T\left(J\right)\right)=\mathrm{const}$ for the set of selected $K_Q$- and ${\eta }_o$-values. The lines for maximum efficiency are called “${\eta }_{o,max}$ for $J=\mathrm{const}$” and “${\eta }_{o,max}$ for $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.=\mathrm{const}$” as before. This efficiency map can also be enhanced with the families of curves “$T_D$, $P_D$, $T_n$, and $P_n=\mathrm{const}$”, where the curves for $T_D$ and $T_n=\mathrm{const}$ are curves to the single power of three and five.

There are certain advantages of efficiency maps over open-water diagrams for finding the optimum propeller. These are described in Appendix A.3 and Appendix A.4. For the purpose of this article, we will concentrate on the shape of the lines for maximum efficiencies. To recap, these lines are the solutions of the optimisation problems under different constraints. We will use the $K_Q$–$J$ efficiency map enhanced by the addition of the $T_D$ and $T_n=\mathrm{const}$ curves and the $K_T$–$J$ map enhanced with the $P_D$ and $P_n=\mathrm{const}$ curves, as proposed by Danckwardt [6]. We will call them the T–J and P–J Danckwardt diagrams (see Appendix A.2 for a short historical overview).

2.3. Remake

The author presented efficiency maps for the Wageningen B-screw Series during the smp’19 conference [7]. For the purpose of this paper, the two Danckwardt diagrams for all propellers of the Wageningen B-screw Series, as outlined in

Table 1. Summary of the propeller models of the Wageningen B-screw Series.

Z	${\displaystyle \raisebox{1ex}{${\mathit{A}}_{\mathit{e}}$}\!\left/ \!\raisebox{-1ex}{${\mathit{A}}_0$}\right.}$
2	0.30	0.38
3		0.35			0.50			0.65			0.80
4			0.40			0.55			0.70			0.85			1.00
5				0.45			0.60			0.75			0.90			1.05
6					0.50			0.65			0.80			0.95
7						0.55			0.70			0.85

, were recreated with the help of a purpose-made computer program. This program employs the polynomials (5) and (6), as described in Section 1 and published by Oosterveld and van Oossanen in 1975 [2]. For these newly generated diagrams, the symbols were updated to the ITTC nomenclature [8] (see also

Table A1. Differences between the nomenclature used by Danckwardt and ITTC.

Name	Danckwardt	ITTC
Pitch	H	P
Blade area ratio	$F_a/F$	$A_e/A_0$
Speed of advance	$v_e$	$v_a$
Thrust	S	T
Torque	M	Q
Delivered power	$N_W$	$P_D$
Advance coefficient	$\mathsf{\Lambda }$	$J$
Thrust coefficient	$k_s$	$K_T$
Torque coefficient	$k_m$	$K_Q$
Open-water efficiency	${\eta }_p$	${\eta }_o$
Slip	$s_n$	$S_R$

for the differences to the nomenclature used by Danckwardt). In addition to the curves presented in the original diagrams, the line for “${\eta }_{o,max}$ for $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.=\mathrm{const}$” and the contour lines for $K_T(J,K_Q)$ or $K_Q(J,K_T)=\mathrm{const}$ were added. All these recalculated Danckwardt diagrams are presented in Appendix B, and

Table A4. Composition of the Danckwardt P–J ($K_T$–$J$) and T–J ($K_Q$–$J$) efficiency maps. Additionally shown is the significations of line colour and type. $K_T$ = thrust coefficient; $K_Q$ = torque coefficient; ${\eta }_o$ = open-water efficiency; $J$= advance coefficient; T = thrust; Q = torque; $P_P$ = propeller power; D = propeller diameter; P = propeller pitch; n = shaft speed (in ${\mathrm{s}}^{-1}$); $v_a$ = speed of advance; and $\rho$ = water density. If not stated otherwise, all values are in SI base units.

shows an overview of the composition of these diagrams. Examples of how these diagrams are used are explained in Appendix A.4.

The computer program calculates the lines of maximum efficiency by finding all points, where the tangents to the $T_D$, $T_n$, $P_D$, and $P_n$ curves are tangential to the efficiency contour lines. As a free variable, the pitch to diameter ratio $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$ was used. This choice was found to be necessary to be able to remove possible multiple solutions for the ${\eta }_{o,max}$ lines. It must be mentioned that, as a possible consequence, the calculation of the ${\eta }_{o,max}$ lines for $T_D$, $T_n$, $P_D$, and $P_n$$=\mathrm{const}$ sometimes did not succeed at one or the other boundary due to numerical difficulties. The artefacts on the left boundary at low $J$- and $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values were manually deleted and extrapolated by hand whenever possible. The right border, where the $J$- and $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values are high, posed a different numerical challenge. In all cases however, it was possible to reconstruct the valid line manually, but sometimes not right up to the maximum value of $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$. It should be mentioned that these difficulties never arose at high $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values when the ${\eta }_{o,max}$ line doubles back, as discussed in the following section.

3. Ambiguity of the ${\displaystyle {\mathit{\eta }}_{\mathit{o},\mathit{max}}}$ Lines

3.1. Introductory Example

We have seen that the Danckwardt diagrams lend themselves to finding the optimum propeller under certain constraints. For the sake of argument, let us assume that we have already decided on the blade number ($Z=5$) and blade area ratio ($\raisebox{1ex}{$A_e$}\!\left/ \!\raisebox{-1ex}{$A_0$}\right.=0.9$), and we know the torque Q (from the given available power $P_P$), the inflow velocity $v_a$, and propeller diameter D but not the shaft speed n, which should be optimised together with the pitch to diameter ratio $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$. Using these known values, we can calculate P_D from Equation (A4), which we assume to be 0.15. On the Danckwardt P–J diagram for the Wageningen B5-90 propeller (see Appendix B), we find the intersection of the curve for $P_D=0.15$ with the “${\eta }_{o,max}$ for $P_D=\mathrm{const}$” line (blue lines). We can read off the values for $J$, $K_T$, $K_Q$, $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$, and ${\eta }_o$ (approximately 0.65, 0.24, 0.041, 1.04, and 0.60, respectively).

Let us now assume that we want to investigate a three-bladed propeller with the blade area ratio $\raisebox{1ex}{$A_e$}\!\left/ \!\raisebox{-1ex}{$A_0$}\right.$ of 0.80 for the same operating condition. In the diagram for the Wageningen B3-80 propeller, we find the intersection for our assumed value of $P_D=0.15$ with the “${\eta }_{o,max}$ for $P_D=\mathrm{const}$” line (blue lines) at $J$, $K_T$, $K_Q$, $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$, and ${\eta }_o$ equal to 0.62, 0.20, 0.035, 1.00, and 0.57, respectively. However, there also exists another intersection between the $P_D=0.15$ and the “${\eta }_{o,max}$ for $P_D=\mathrm{const}$” lines at a higher $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-value: 0.74, 0.295, 0.062, 1.30, and 0.56. It looks to be that there are two optimum propellers for this condition, since the ${\eta }_{o,max}$ lines are solutions to the optimisation problem under the given constraints!

The reason for this somewhat puzzling behaviour is obviously the doubling back of the ${\eta }_{o,max}$ lines. Two solutions to the optimisation problem exist in the region of this overlap, whereas no solution to the optimisation problem exists right of this region of overlap.

3.2. Classification

To classify this overlap, the value of ${\left.T_D\right|}_{{\left.\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.\right|}_{\max }}$, where the “${\eta }_{o,max}$ for $T_D=\mathrm{const}$” line intersects the maximum ${\left.\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.\right|}_{\max }$ curve, was calculated for every propeller in the Wageningen B-screw Series (please refer to Figure 4) as proposed in [7]. The intersection of this $T_D=\mathrm{const}$ curve with the “${\eta }_{o,max}$ for $T_D=\mathrm{const}$” line was found. The pitch to diameter ratio at this intersection is denoted as $\widehat{\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.}{|}_{T_D}$. The minimum value of ${\left.T_D\right|}_{\min }$ is determined, where there is also no optimum propeller right of this curve! The difference between these two values for $T_D$ is denoted as $\mathsf{\Delta }{T}_D$. For the $T_n$, $P_D$, and $P_n$ curves, these values are calculated accordingly.

Table 2. Main characteristics of all overlaps found in the recreated Danckwardt diagrams. (For symbols and definitions, see Section 3.2 and Figure 4).

Propeller	${\mathit{T}}_{\mathit{D}}$			${\mathit{T}}_{\mathit{n}}$			${\mathit{P}}_{\mathit{D}}$			${\mathit{P}}_{\mathit{n}}$
	${\displaystyle \widehat{\raisebox{1ex}{$\mathit{P}$}\!\left/ \!\raisebox{-1ex}{$\mathit{D}$}\right.}}$	${\left.{\mathit{T}}_{\mathit{D}}\right|}_{\min }$	$\mathsf{\Delta }{\mathit{T}}_{\mathit{D}}$	${\displaystyle \widehat{\raisebox{1ex}{$\mathit{P}$}\!\left/ \!\raisebox{-1ex}{$\mathit{D}$}\right.}}$	${\left.{\mathit{T}}_{\mathit{n}}\right|}_{\min }$	${\Delta }{\mathit{T}}_{\mathit{n}}$	${\displaystyle \widehat{\raisebox{1ex}{$\mathit{P}$}\!\left/ \!\raisebox{-1ex}{$\mathit{D}$}\right.}}$	${\left.{\mathit{P}}_{\mathit{D}}\right|}_{\min }$	$\mathsf{\Delta }{\mathit{P}}_{\mathit{D}}$	${\displaystyle \widehat{\raisebox{1ex}{$\mathit{P}$}\!\left/ \!\raisebox{-1ex}{$\mathit{D}$}\right.}}$	${\left.{\mathit{P}}_{\mathit{n}}\right|}_{\min }$	${\Delta }{\mathit{P}}_{\mathit{n}}$
B2-30	––––––			––––––			––––––			––––––
B2-38	––––––			––––––			––––––			––––––
B3-35	0.94	0.155	0.182	1.14	0.102	0.043	0.93	0.033	0.049	1.14	0.021	0.010
B3-50	0.89	0.167	0.225	1.13	0.114	0.042	0.88	0.037	0.065	1.12	0.025	0.010
B3-65	0.86	0.228	0.385	1.12	0.162	0.056	0.86	0.054	0.127	1.11	0.037	0.014
B3-80	0.86	0.384	0.932	1.11	0.266	0.094	0.85	0.101	0.389	1.11	0.063	0.025
B4-40	1.02	0.272	0.134	1.18	0.177	0.042	1.01	0.063	0.039	1.17	0.039	0.010
B4-55	1.04	0.229	0.100	1.23	0.144	0.018	1.04	0.052	0.028	1.22	0.031	0.004
B4-70	1.10	0.229	0.064	1.31	0.138	0.004	1.10	0.053	0.018	1.31	0.030	0.001
B4-85	1.21	0.254	0.027	––––––			1.21	0.060	0.008	––––––
B4-100	1.38	0.286	0.000	––––––			1.38	0.070	0.000	––––––
B5-45	1.21	0.301	0.028	1.30	0.191	0.008	1.21	0.071	0.008	1.30	0.043	0.002
B5-60	1.25	0.231	0.015	1.35	0.141	0.001	1.25	0.052	0.004	1.35	0.031	0.000
B5-75	1.32	0.195	0.003	––––––			1.32	0.043	0.001	––––––
B5-90	––––––			––––––			––––––			––––––
B5-105	––––––			––––––			––––––			––––––
B6-50	1.37	0.246	0.001	1.40	0.165	0.000	1.37	0.057	0.000	1.40	0.037	0.000
B6-65	1.37	0.210	0.001	––––––			1.37	0.047	0.000	––––––
B6-80	1.40	0.197	0.000	––––––			1.40	0.044	0.000	––––––
B6-95	––––––			––––––			––––––			––––––
B7-55	1.39	0.204	0.000	––––––			1.39	0.047	0.000	––––––
B7-70	––––––			––––––			––––––			––––––
B7-85	––––––			––––––			––––––			––––––

shows an overview of all these values for all propellers in the Wageningen B-screw Series. The table clearly shows that this overlap does not just sporadically occur, but that it is a widespread phenomena.

An overlap can only be observed for the “${\eta }_{o,max}$ for $T_D$, $T_n$, $P_D$, and $P_n=\mathrm{const}$” lines, but never for “${\eta }_{o,max}$ for $J=\mathrm{const}$” or “${\eta }_{o,max}$ for $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.=\mathrm{const}$”.

4. Discussion

4.1. Evaluation

For the following discussion, it should be kept in mind that the ${\eta }_{o,max}$ lines were calculated in a descriptive way by finding all points, where the tangents to the families of the $T_D$, $T_n$, $P_D$, and $P_n=\mathrm{const}$ curves coincide with the tangents to the contour lines of ${\eta }_{o,max}=\mathrm{const}$. Mathematically speaking, this is equivalent to solving an extrema problem.

To investigate the solutions, we take the propeller used in the example in Section 3.2 and plot the efficiencies along the $P_D=0.15$ curve against the $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-value, see the blue line in Figure 5. It can be seen, that the extremum at the lower $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-value of about 1.0 is a maximum, whereas the extremum at the higher $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-value of about 1.3 is a minimum. Additionally shown in this figure is the run of the efficiency curves for $P_D=0.1$ and 0.07 (green and orange lines), the first represents ${\left.P_D\right|}_{\min }$ and just touches the “${\eta }_{o,max}$ for $P_D=\mathrm{const}$” curve in the apex; the second comes to lay right of the apex. For these two cases, it is apparent that the propeller with the highest possible efficiency is situated beyond the boundary of $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.=1.4$.

The author of this paper believes that these overlaps of the ${\eta }_{o,max}$ lines causing the ambiguities described are physically not explainable.

There are the obvious reasons: Firstly, the open-water efficiency drops and starts to climb again in the region of the overlap. Secondly, there is no optimum except at the boundary of the available data, right of the overlap.

Generally, we can argue that we can extend the propeller series to even higher $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values. Eventually we will arrive at a pitch setting, where the open-water efficiency will become zero, since such a propeller would have blades perpendicular to the section inflow and hence would not be able to accelerate water in the axial direction. Thus, it is not unreasonable to assume that a pitch to diameter ratio must exist, where the open-water efficiency is globally at its highest. Indeed, this can be seen in both Danckwardt diagrams for the Wageningen B2-30 and B2-38 propellers (see Appendix B): a peak of the open-water efficiency can be noticed at a $J$-value of about 1 and a $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$ ratio of about 1.1. It can be observed that all four lines for ${\eta }_{o,max}$ pass (and must pass) through this absolute maximum of ${\eta }_o$. Even if this point of the absolute maximum of ${\eta }_o$ comes to lie right of and above the set of the $K_T$ and $K_Q$ curves—and thus is not displayed in the diagram—the four lines for ${\eta }_{o,max}$ must still converge towards this single point of the global absolute maximum of ${\eta }_o$. Following this thought, it is evident that the lines of ${\eta }_{o,max}$ can not bend back, as can be seen with certain propellers, and this behaviour is deemed as physically inexplicable.

4.2. Implications

Admittedly, paper charts are seldom used nowadays in propeller design work, but depending on the computer algorithm used for automatically searching the propeller with the highest achievable efficiency, the following problems can be encountered: Firstly, in the region of the overlap, the computer program could pick the solution on the upper branch of the ${\eta }_{o,max}$ curve, if no appropriate checks are implemented. It could also jump between the two extrema. Equipped with the knowledge of the described behaviour, the algorithm can be tweaked to find the correct solution. Secondly, in the region right of the overlap, where there exists no optimum, the algorithm could calculate the optimum propeller to have a pitch ratio of 1.4, which is right on the boundary of the available data. As can be clearly seen in Figure 5, the line for $P_D=0.07$ still climbs in the vicinity of the boundary, indicating that there exist propellers with even higher efficiencies beyond the boundary of the tested propellers. It has to be emphasised that, based on the polynomials, there simply does not exist an optimum propeller in this region, where such an optimum propeller must exist. Thirdly, it can be argued that there exists an optimum propeller up to the point where the ${\eta }_{o,max}$ curve doubles back. Nevertheless, special care must be taken in the region of the apex, because the line of optimum efficiency already starts to swerve away.

4.3. Accuracy of the Polynomials

As the issue of the doubling back of the ${\eta }_{o,max}$ lines cast some doubts on the accuracy of the underlying polynomials, the question of the overall accuracy of the polynomials also arises. Whereas Figure 6a,b show a good agreement of the ${\eta }_{o,max}$ lines between the original and the recreated diagrams for the whole range but the area of doubling back, Figure 6c,d show a big discrepancy between the polynomials published in 1969 and 1975. These deviations can be of varying significance, depending on the subsequently employed and more detailed optimisation procedures. When the Wageningen B-screw Series is used to find the optimum propeller and the thus obtained dimensions will be kept fixed and only small corrections are applied to them—as is common practice at the smaller end of the market—it is of paramount importance that this optimisation routine based on the polynomials gives consistent and accurate results.

For large propellers, the outcome of the optimisation based on the Wageningen B-screw Series polynomials is used as the starting point for further and more detailed optimisation. An accurate starting point would speed up the subsequent full optimisation processes, but should not change the outcome. In reality, the optimised variable D or n found in the first step is very often kept fixed and will not be optimised further in the final optimisation. In this case, it is again of highest importance to get accurate results from the polynomials.

4.4. Provenance and Causes of Overlaps

It must be emphasised, that the overlaps observed are not a feature of the presentation in the form of efficiency maps, but of the underlying data, i.e., the polynomials (5) and (6) published by Oosterveld and van Oossanen in 1975 [2]. It should also be noted that at the time when Danckwardt published his diagrams—which show no overlaps at all, see Figure A1a,b—the and polynomials were not known yet. The design charts published by Yosifov et al. are already based on the polynomials [9]. On all their efficiency maps, the lines for ${\eta }_{o,max}$ stop before they reach the maximum $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-value. Yosifov et al. do not mention or explain this behaviour. Those diagrams, where the lines stop far from the maximum $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$ ratio, are for the same propellers, where we have identified an overlap.

Nonetheless, it is not clear where these ambiguities were introduced during the process of manufacturing, measuring, fairing, scaling to uniform Reynolds number, and calculating the regression polynomials. Without further investigation into all of these steps, the source of this behaviour is not known, but some possibilities spring to mind: Between the testing of the first and the last propeller, a time span of more than 30 years passed. During this time span, it can be assumed that the manufacturing of the model propellers and the testing technology improved. The propellers were tested at different basins and also at different Reynolds numbers, and were only later corrected to a uniform Reynolds number of 2 · 10⁶. Even the numerical regression used to calculate the polynomials could have introduced this behaviour. Helma shows in [3], for selected propellers, that the lines of maximum efficiency in the recreated Danckwardt diagrams follow the published lines by Danckwardt at low $\raisebox{1ex}{$P$}\!\left/ \!\raisebox{-1ex}{$D$}\right.$-values (see Figure 6a,b). However, the regression curves given by van Lammeren et al. for propellers with four blades [5] already exhibit a troublesome behaviour at higher values of the pitch to diameter ratio (see Figure 6c,d).

5. Conclusions

With the help of the alternative presentation of open-water characteristics as efficiency maps, it was shown that the current set of polynomials for the Wageningen B-screw Series, as published by Oosterveld and van Oossanen in 1975 [2], shows some troublesome behaviours for higher pitch to diameter ratios for many propellers of the series.

Considering the widespread use of these polynomials, it is suggested to revisit the originally tested data and check all steps involved in the processing of the data sets for the deduction of the polynomials.

The propeller designers would be very well advised to take caution when designing propellers whenever any line of ${\eta }_{o,max}$ doubles back.