Drag revisited

A discussion on the Concept 2 UK Forum reminded me of the unsolved mystery of the amount of drag experienced by a rowing shell. So I went back to my old textbook on Fluid Mechanics (Robert A. Granger, “Fluid Mechanics”, Dover 1995). Assuming fully turbulent boundary layer flow, the drag coefficient for skin drag is a function of the shell’s velocity or Reynold’s number. Granger recommends the following equation:

C_{D_f} = \frac{0.455}{(\log R_l)^{2.58}}

for Reynold’s numbers R_l larger than 10^7. The  International Towing Tank Conference (ITTC) recommends a slightly different equation:

C_{D_f} = \frac{0.075}{(\log R_l - 2)^2}.

For a single with a length of 8.2m and a velocity of 8.42 m/s (1:43 per 500m), the Reynold’s number amounts to 3.3\times 10^7 for water of 10 degrees C. This gives C_{D_f} = 2.459 \times 10^{-3} with the ITTC equation. Assuming the wetted area for a single with a rower weighing 80kg to be 2.25 m² as claimed by Cambridge Racing Shells to be the value for a Fluidesign single, I obtain \alpha_{\rm skin} = 2.77 N s²/m². Using Granger’s equation, I get 2.81 N s²/m². For a four with a wetted surface of 6 m², I get 6.7 N s²/m² using the ITTC equation.

Skin drag is not the only form of drag. There is also form drag and wave drag. It is said that skin drag accounts for about 80% of the drag (Ana Dudhia). So, dividing my number by 0.8 I get a value of \alpha amounting to 3.51 for the single scull at race speed. Pretty close to the value I have used so far. The difference is a slight dependence on the shell’s velocity for the value of \alpha. In figure 1, the drag force using the ITTC equation is compared with the simplified method of taking constant drag coefficient, for a single scull with a rower weighting 80kg (wetted surface of 2.25 m²). The differences are small but at maximum velocity they can be of the order of 10%.

Figure 1: Drag Force vs velocity for a single scull with a rower of 80kg. Green line: Constant drag coefficient approximation. Blue line: ITTC 1957 drag coefficient

The division by 0.8 to account for other forms of drag seems a bit arbitrary. Interestingly, when I try to validate the model on the Lucerne 2010 (no wind) boat speed and ergometer scores of Ondřej Synek, the 2010 world champion in the single, I get a consistent story. Synek’s Lucerne achievement was a power of 586W on the 2k (calculated using the ITTC equation value for drag). Assuming 25W is consumed in moving a 98kg heavy athlete up and down the Concept 2 slide, the remaining ergometer power of 561W corresponds to a 2000m time of 5:42 on the ergometer. Synek is known to have rowed 5:41.8 (Czech National record).

As Bill Atkinson rightly pointed out, the scaling equation I used to calculate for different boat types, depends heavily on the wetted shape. I took the \propto m^{2/3} intuitively without giving it much thought, except that Anu Dudhia uses it here. The numbers given for wetted area on the CRS site seem to confirm this. Also, my validation with the 2010 Lucerne results gives confidence, but, admittedly, I have not given it enough thought from the basic physics side. To be continued …


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