Updated: Aug 28
How so we push a narrowboat?
I spent a long time trying to work out the answer to this question, and it turns out easy to reach a partial answer but very hard to get a full answer. Spoiler alert! The easy bit is dragging the hull through the water, the hard bit is turning the propeller.
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.
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.
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:
That was the easy part. The hard question is if I want to generate a force of, say, 1kN to travel at about 6kts, how much power do I need to put into the propeller?
The difficulty is that a propeller has many parameters which are all interdependent. There are diameter and pitch (the distance the blades will travel through the water if the propeller turns once, like a corkscrew). Then the area of each blade and the number of blades. The speed of rotation and the torque. When the boat is moving, hull drag causes the water entering the propeller to be slower than the actual boat speed, but because the propeller is sucking water in, there is a corresponding increase in drag.
Let's get back to first principles. The whole point of Perseverance is to cruise quietly. Smoothly. Without fuss. Therefore I opted for a 4-bladed propeller as more blades give a smoother ride. I also found that standard propellers have a limited range of area ratios, so that reduces the number of unknowns further.
Knowing the thrust needed to push the hull through the water at different speeds it is possible to use charts to find out the propeller performance. When I say “charts” there are some fiendish ones out there. The common ones look like this:
or like this:
or even this:
But the one I found I could understand was this sort. For a 4-bladed propeller with 70% area ratio:
Also it's very colourful !
From the drag equations, using Hans Otto Kristensen’s formula for wake fraction and Harvald’s formula for drag inflow, it is possible to compute the value of the a parameter shown as the blue numbers. They come out to have values between 1 and 2.5. The thick blue line is the maximum efficiency line for that range of conditions, and from the red solid lines we can find out the optimum pitch:diameter ratio which is about 0.8. We can also read off the advance coefficient, J and torque coefficient, Kq, from the axes. Throw in the odd density of water and we get:
This tells us the fact well known to all boaters, that it takes a lot more power to go a little bit faster.
Electric motors give practically the same maximum torque for any speed (unlike diesel engines where the torque varies significantly with engine speed). The motor I selected for Perseverance gives a maximum of 70Nm torque, and the chart above goes beyond the available torque from the motor. That is, my motor won't drive Perseverance at 7.5kts.
It is possible to compute the maximum speed for the available torque as a function of propeller diameter thus:
So you can have propellers that are too small or too big, given a maximum available torque.
Now, as well as the speed that can be achieved, there is a handling aspect to propellers. If you want to stop quickly, or punch the boat out of a lock, you do just need to move lots of water. Also, I am sure that the wake caused by a large propeller moving more water more slowly will be less turbulent, and hence quieter, than a small propeller moving a little water more quickly. For this reason, I opted for an 18in propeller. The maximum efficiency line from the multi-colour chart above was 0.8, corresponding to a 14in pitch.
Am I Being Daft?
As a sanity check, I asked lots of people for advice, used on-line and downloadable propeller sizing tools, and plotted all the information I could gather on a chart of diameter vs pitch. Fortunately my 18x14 choice was in the middle of the range which provides reassurance.
I could go on and talk about the final choice of supplier and associated stern gear, but “enough!” I hear you cry. OK, I can take a hint.
Just one last thing. The very good chart I used was from an article in the Journal of Marine Science and Engineering, entitled "Surprising Behaviour of the Wageningen B-Screw
Series Polynomials" by Stephan Helma. Who said engineering was dull?