Preamble
This blog used to be part of "Pick a Propeller" but it became apparent that readers interesed in hull drag were not finding it hidden there. Hence this is now in a separate post.
Dragging the Hull through the Water
If we’re not going to exceed the canal speed limit, we are not going to be planing or using hydrofoils. Narrowboats are displacement boats for which two forms of drag dominate; friction drag and wave making drag.
Viscous Drag
Let’s consider friction drag first. Imagine you are on a boat moving very slowly down the canal. You are looking down at the side of the boat to where the smooth hull slides gracefully through the gleaming blue water.
OK, a bit too poetic but you’re looking at the right place. The water touching the boat is moving with the boat but the water a little way away is not moving. Keep looking down and imagine a series of layers of water between the boat and the canal, each being sheered between its neighbours. This sheering motion leads to viscous friction which is trying to bring the boat to a halt. The viscous friction depends upon (1) the viscosity of the liquid you’re moving through, (2) the wetted surface area of the boat and (3) the speed. Fortunately water is quite runny stuff – imagine if we had to cruise through treacle!
This reminds me of the “world’s longest running lab experiment”, the pitch drop experiment where the mindbogglingly viscous pitch is “flowing” through a funnel. The experiment started in 1927 and so far, nine drops have fallen. That makes pitch 230 billion times more viscous than water, so we would die waiting for the lock to empty. But I digress.
For a narrowboat, you can work out the surface area from the length and weight of the boat, because the width is fixed and the more it weighs the deeper it sits in the water. Curiously the affect of the shape of the hull makes practically no difference to the friction drag. The graph below is computed for Perseverance at an estimated 14 tonnes and 62ft.
Super-nerds will ask if the coefficient of viscosity is a constant. Well, it reduces slightly as the speed increases, but the overall drag is based on the coefficient times the square of the speed and it is the speed which is dominant. The groovy thing about drag being a function of the square of the speed is that it’s the same going backwards as forwards – that is, if the boat is moving backwards, we get the same drag because (-5) x (-5) = 25 = 5 x 5. Makes sense, otherwise we would all go around in reverse and have perpetual motion.
A huge assumption in this calculation is that the boat is travelling in deep water well away from the banks. As soon as the canal becomes shallow, or you travel close to other surfaces (tunnel walls, narrow bridge holes or locks) the drag increases markedly because the velocity gradient is higher so the sheer forces and hence drag are much higher. Still, this gives us a “best case” scenario.
Wave Drag
I said the other main element of the drag was the wave making drag. Going back to our imaginary boat trip, if we let the speed increase you will see ripples coming from the bow and stern. As the speed increases the ripples grow into waves and at some point they start breaking on the bank and we quickly throttle back and look around to see if we’ve been spotted.
The effort used to make these waves is affected by the shape of the boat under the water, and this is one of the places where the skill of a good boatbuilder comes into play. As well as the shape of the boat there is a dependency on the length of the boat compared with the inertia of the water, as originally identified by William Froude. In the 1860s, he worked out how to relate model ship tests to full scale ship performance, by towing model ships in long tanks to measure their drag. In fact, some water from his original towing tank is kept by current ship designers at Gosport and there is a tradition that whenever a new towing tank is commissioned, anywhere in the world, a little Froud water is added to the tank.
Now, the wave making drag coefficient increases with speed and is then, like the viscous drag coefficient, multiplied by the square of the boat speed. At low speeds the wavemaking drag is negligible, but increases rapidly with speed so that by about 7.5 kts it is as large as the viscous drag. (7.5 kts is the highest speed covered by the performance charts I used).
There are some other components to drag and if we add them all together we get this plot:
Here I have included a small term for the actual friction at the surface itself, a factor for being of relatively broad beam for the draft, and an estimate of the drag of the bow thruster tube. These are all added together to give a total drag.
Studious readers of these blogs will remember that power is force times speed, so we just multiply up the drag by the velocity to get the power required to propel the hull through the water thus:
I like pictures, but I know some people like to see the numbers, so here are the values in tabular form:
Summary
The key thing to remember is that hull drag increases as the cube of the speed. To go twice as fast takes eight times more power, or, more helpfully, to go 25% faster requires twice the power.
You are making good progress, and keeping to the 5kt limit allowed by the CRT signboards. You come up behind a slowcoach making only 4kts. With this speed difference it would take a minute and a half to overtake, or you could slow down and half your power requirements.
See Also
Halving the power required to drive the hull does not halve the fuel consumption - see Why Waste Money to see the effect of reduced power on diesel efficiency.