# Human Powered Boat: How do I calculate the speed curve with a linear increase of thrust, weight & length? (e.g 8 vs 4 vs 2 person rowing shell)

In competitive rowing, the more people in a boat, the faster it goes, but what is the ratio of additional human to speed, and what is the theoretical limit?

If I could fabricate a boat where each additional person added 100kilos, 100watts of power, and 1 meter of length to the wetted surface, how could I calculated when the curve starts to flatten, and how it is affected by changes to those variables?

Baluncore
Welcome to PF.
The boat you describe is a displacement hull, so wave generation is very important.
The wetted surface drag is proportional to the length and the square of the speed.
The speed of the displacement hull is proportional to the square root of the length.
https://en.wikipedia.org/wiki/Hull_speed

256bits
Gold Member
In competitive rowing, the more people in a boat, the faster it goes, but what is the ratio of additional human to speed, and what is the theoretical limit?

If I could fabricate a boat where each additional person added 100kilos, 100watts of power, and 1 meter of length to the wetted surface, how could I calculated when the curve starts to flatten, and how it is affected by changes to those variables?
Curve flattens when the power needed to stroke the nth extra 1m through the water matches the power produced by the nth additional rower. Each previous rower can power his 1m, and have some left over to power the bow and the stern at the extra speed. Excuse the liberal usage of the term power, but I think you can see what I mean.

anorlunda
Staff Emeritus
That's a very complicated question. The answers provided by @Baluncore and @256bits are valid, but there are still more factors to consider.

If you could hold the weight of the boat plus crew constant while doubling the waterline length, it should go faster. So where's the limit? In competitive situations, the racing association typically has mandatory design constraints. Most famous was the old fashioned 12 meter rule used for America's Cup boats for many years.

If there was no rule to constrain me, I would investigate foils that lift the hull out of the water. Could a human powered rowboat do that? I don't know, but it would not violate the laws of physics.

jrmichler
Mentor
The resistance vs speed of a displacement hull, as stated above, is complex. Skene's Elements of Yacht Design, 8th Edition, has the following graph for hull resistance vs speed. It's for sailboats, and will probably change for the long, slim hull of a scull.

Human powered hydrofoils have been built. This link shows one in action: http://www.ihpva.org/water.htm. This link has a theoretical analysis of human powered hydrofoils: http://www.ihpva.org/hparchive/pdf/19-v6n1-1987.pdf. And there is a Wikipedia entry: https://en.wikipedia.org/wiki/Human-powered_hydrofoil.

anorlunda
berkeman
Mentor
If I could fabricate a boat where each additional person added 100kilos, 100watts of power, and 1 meter of length to the wetted surface, how could I calculated when the curve starts to flatten, and how it is affected by changes to those variables?
It seems like you are leaving out something fundamental in going from a 4-seat to an 8-seat skull, no?

pbuk
Gold Member
The calculations for a boat with the fine entry and long, narrow hull of a racing shell are very different from a normal displacement vessel: at racing speeds of 5-6m/s they are already above 'hull speed'. Also, the skin friction of an extra metre of hull is not really significant: the main penalty to adding an two extra rowers is moving 100 200kg more water out of the way every 3.5s or so. (Edited because the main penalty to adding one extra rower is going in circles).

A good general introduction to rowing physics is here, with more detail on drag vs speed at http://eodg.atm.ox.ac.uk/user/dudhia/rowing/physics/rowing.pdf.

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The calculations for a boat with the fine entry and long, narrow hull of a racing shell are very different from a normal displacement vessel: at racing speeds of 5-6m/s they are already above 'hull speed'.
This certainly is not obvious. Do you have a source for this assertion?

pbuk
Gold Member
This certainly is not obvious. Do you have a source for this assertion?
Which one?
The calculations for a boat with the fine entry and long, narrow hull of a racing shell are very different from a normal displacement vessel:
This is covered in section III of the paper in the second link in my post.
at racing speeds of 5-6m/s they are already above 'hull speed'
I calculate hull speed in knots as ## 1.34 \sqrt{l_{wl}} ## with ## l_{wl} ## in feet. Lets take the waterline length of an eight as 60 foot so this gives hull speed as 10.4kts or 5.35ms-1. At 6 minutes for 2k (world record is under 5:20) average speed is 5.58ms-1.

If that's a bit close, look at a coxless four at 44' long: hull speed is 8.89kts or 4.57ms-1, 6:20 for 2k (world record under 5:40) average speed 5.26ms-1.

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OK, thanks for the additional info.