4.1. Evaluation
For the following discussion, it should be kept in mind that the lines were calculated in a descriptive way by finding all points, where the tangents to the families of the , , , and curves coincide with the tangents to the contour lines of . 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
curve against the
-value, see the blue line in
Figure 5. It can be seen, that the extremum at the lower
-value of about 1.0 is a maximum, whereas the extremum at the higher
-value of about 1.3 is a minimum. Additionally shown in this figure is the run of the efficiency curves for
and 0.07 (green and orange lines), the first represents
and just touches the “
for
” 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
.
The author of this paper believes that these overlaps of the 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
-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
-value of about 1 and a
ratio of about 1.1. It can be observed that all four lines for
pass (and must pass) through this absolute maximum of
. Even if this point of the absolute maximum of
comes to lie right of and above the set of the
and
curves—and thus is not displayed in the diagram—the four lines for
must still converge towards this single point of the global absolute maximum of
. Following this thought, it is evident that the lines of
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
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
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
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
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
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
stop before they reach the maximum
-value. Yosifov et al. do not mention or explain this behaviour. Those diagrams, where the lines stop far from the maximum
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 · 10
6. 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
-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).