In this post, I postulated that skin drag amounts to about 80% of drag and simply divided the skin drag by 0.8 to obtain the total drag.

Bill Atkinson pointed me to this work by Tuck and Lazauskas. First of all, form drag is estimated there by an equation from a paper by Scragg:

with

where and are the half angles (in degree) of the bow and stern, respectively. Tuck and Lazauskas do a complete modeling of wave drag as well, and come up with something that we could almost call a phase diagram for rowing shells. Looking at the figure 2b here, the wave drag is around 6% of total drag for shells optimized for racing speed, a Froude number around 2 in the figure. Of course, wave drag varies quite strongly with the shell’s velocity. However, the shell velocity varies during the stroke, and it seems a good approximation to use an average value for wave drag which is only a minor portion of the total drag. Doing the integrals to calculate the instantaneous value would slow down the simulation software too much, for a relatively small improvement in accuracy.

As far as the bow and stern angles are concerned, I collected data from boat manufacturers sites about the boat length, beam and draft as a function of the displacement.The results are depicted in figure 1. The solid line is a linear fit, showing that all boats are basically of (more or less) the same shape. Assuming a parabolic boat shape, the bow half angle would be 8 degrees, with an elliptic shape around 6.3 degrees. This would give a value of between 0.12 and 0.15. So, the total drag (wave drag, skin drag and form drag) would be about 1.2 to 1.22 times the skin drag. Or, in other words, skin drag accounts for about 82% of the total drag.

Figure 1 Shell length vs beam (both in m) for Vespoli and Empacher shells

Regarding the scaling equation used for different boat types, it is clear that if all boats are basically scaled versions of the same shape, then beam, draft and length should be proportional to the 1/3 power of displacement. This is clearly shown in figures 2 and 3.

Figure 2: Beam as a function of displacement for different makes and types of shell.

- Figure 3: Shell length vs displacement for different makes and types of rowing shells

Finally, figure 4 shows the wetted area as quoted on various manufacturer’s websites as a function of the displacement. The solid line shows the scaling that I use in my model.

Figure 4: Wetted area as a function of displacement for rowing shells of various manufacturers

So, in conclusion, the mystery isn’t a mystery any more. The lower value quoted by Slasias and Tullis is just the skin drag portion calculated from a 5m² wetted area. In reality, the wetted area of a four is slightly higher, and one has to add 20% for wave drag and form drag. Looking at the scatter of the data in the various graphs, I would also say that the accuracy of my model is good within about 10%, not better. This is not a disaster, as the relative results predicted by the model are still valid. Finally, boat design is found to be extremely important, as a 5% reduction of skin drag can means a mean velocity increase of 2.6%. This amounts to more than 2 seconds per 500m for an eight.

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