*2.4 Scale speed, stability and prop selection
for scale model boats*

*Introduction*

I listen to so many interpretations of the two words “scale model” that I need to say what I mean in order to make a start.

I like to think that my goal is to make a model that looks like some real boat when seen up close and, when it is operated remotely, behaves exactly like the full-size when seen at any distance. I suppose that this definition will find sympathy with many other modellers. To some extent success is dependent on the choice of scale subject, for example a square-rigger as compared with a sleek power-boat but there will be some things that affect all types of boats. In particular scale speed and stability and propeller selection perhaps.

** **

*Scale Speed*

When I am on the coast I see cargo boats a fair way out and I can, by sighting against a fixed mark, get a good idea of how long the boat takes to cover its own length. I can imagine someone saying that a model of this boat should take the same time to cover its length and claim this to be the scale speed. The only problem is that the model will go very, very slowly and not please passers-by or the modeller. So some new idea must be used. For my part I think that one should look at the wake. After all the boat on the move creates a wake that is part of the image of the boat.

I suspect that most modellers never look at the wakes that their boats make. They appear to think that the wake is determined by something akin to divine providence and unalterable. I think that it is possible to get this wake right and the fact that the hull is properly to scale gets us half way there.

However, from the outset, there is a practical problem in that the modeller may be totally unable to observe his prototype under way. So how will he know whether his model behaves like the prototype?

Physics has something for us in connection with scale speed. We can make some progress by looking at the work of William Froude.

There is no doubt that Froude was
the first to show, in about 1880, that it is possible to predict the resistance
to motion of a ship by testing a scale model of the ship in a towing tank. By
“scale model” Froude meant a large piece of wax that had the same geometrical
underwater shape as the ship and floated to the scale depth. The important insight for us is that the
model must be towed at the speed of the full size ship multiplied by the square
root of the scale. So, if the full size made 20 knots, a 1/30^{th}
scale model must be towed at 0.182 times 20 or 3.65 knots. Then Froude saw that
the wave pattern made by the model would be the same as that of the full size
and to the same scale. If this applies to our scale model it is important.

Froude also thought, correctly as it turned out, that this idea of using tests on a model could be used to predict full size performance would not hold for models that were shorter than 6 feet. In fact models shorter than 6 feet do not give reliable results and nowadays we are more likely to use models of 20 feet or longer. Most scale models that amateur builders produce are shorter than 6 feet and most lie in the range of 3 feet to 6 feet with most around 3½ feet. Does this matter? Well the reason for breakdown is that surface tension becomes significant as the model gets smaller and the wave pattern that it creates gets smaller as well. As it stands this is an assertion and most think of the surface tension of water as too small to consider but look at figure 1. This is a jet of water issuing from a tank of water through a hole 12 mm in diameter and the water in the tank is rotating. As a result of the rotation an air-entraining vortex forms in the tank and air passes through the centre of the hole with the water. This causes the jet to spread to produce this hollow shape that is like two wine glasses rim to rim. As it spreads, the area of free surface of the water increases and the surface tension pulls the diverging jet back together and in fact another cycle of expansion and contraction takes place. This shows that the forces produced by surface tension can be significant.

For the model boat builder it happens that the essential character of the wake is correct but no matter how hard you drive a model of a tug of say 50² in length you will not get the foaming bow wave that is characteristic of the full size. The foam will be suppressed, at least in part, by the surface tension.

We make small boats because we have to carry them and the best we can do is to use the idea of the square root of the scale to find a scale speed for our model and then, with the exception of very small boats, if the hull has been made to scale and not distorted in any way there is a very good chance that the wave pattern will be very much the same as that of the full size. We must make the proviso that Froude was dealing with merchant ships and they do not plane so what he said does not carry over to high-speed boats.

There is another important outcome of Froude’s work for displacement boats. He observed that there is a speed at which the resistance to motion becomes excessive and any further attempt to increase speed will just lead to the boat creating a very large bow wave and attempting to “climb” up it. Froude showed that this speed in miles per hour is around 1.5 times the square root of the length in feet. For a model of say 4 feet in length this becomes 3 mph and this is the absolute maximum. The realistic speed for a scale model of a displacement hull of around 4 feet in length is 2 mph or slower.

This means that we have a good idea of the speed range of our model and what its scale speed should be if we know the service speed of the full-size. The figures depend on established data and not on dubious theories.

*Wave patterns*

I spent a lot of time on wave patterns in the section of this website on the bulbous bow. The important one in this context is figure 2 because it shows that a hull being driven by an airscrew and not a propeller generates a wave pattern from both the bow and the stern. In between the surface is flat. Inevitably for a sailing boat these two waves will be generated but they will be modified by the heel and the leeway. For a powered boat we have the complexity of the propeller that adds to the pattern in figure 2. It is obviously possible for the contribution to the wake from the propeller to be different from a scale version of the full size. If I stop my steam launch in calm water and then suddenly open the throttle it is possible to watch the propeller create its contribution to the wave pattern before the boat gathers any significant way. It is clear that the surface of the water falls ahead of the propeller and rises behind it. The propeller is dragging water down to go through its blades and pushing it out at higher speed. This high-speed flow of water is not sustainable and the water heaps up to flow away in all directions. It becomes a source of another wave pattern to add to the one generated by the hull.

Just look at the picture in figure 3. The propeller is producing all the waves behind the boat and some of those towards the aft of the hull. As this is not a scale model, the propeller is the size that I thought would drive this hull using my engine. There is a complexity here for the scale modeller that is not easy to resolve. Nevertheless it is more interesting that just ignoring the wake as a fact of life.

I look at models of scale boats that are going afloat for the first time and too many are not sufficiently stable. I do not comment but I wonder what the builder intends to do. Some decide to live with it, some place their faith in sheets of lead laid on the bottom of the hull but none of them think that something should have been done during building. I know one modeller who built a yacht from a kit and found that it would not stay upright in anything other than very light winds. When he complained to the manufacturer the reply was “they are all the same”!!! The basic problem is that people do not understand stability. If they look for any useful explanation they find, whether they know it or not, that most received explanations are wrong or half understood by their authors.

Now we run into the particular problem that is thrown up by the stability of the model. The problem is that whilst we can faithfully recreate to scale the hydraulic forces acting on the full size by making the model accurately and then making it float to the scale marks there is nothing in the process of modelling to scale to preserve the position of the centre of gravity. As a result the centre of gravity could be a long way from the best position for the model and, further, that this position will almost certainly not correspond with the best position for the full size. Ask yourself why a cruise liner may have as many as 14 decks all above the main deck yet an oil tanker may be 80% submerged. We need to examine how any boat can be stable.

Let us look at how a model boat behaves. Choose a day without wind and set the boat afloat. Assuming that it does not capsize and sink it will settle down at rest on the water. Examine this state of rest. Is it listing? Is it floating with the bow or stern down relative to the waterline? If it is, move some piece of internal equipment such as a battery to put the matter right. What has changed? Quite clearly the position of the centre of gravity of the whole boat has changed, not by much, but it has moved. The shape of the hole in the water made by the boat has also changed even though the volume is still the same. What has actually happened is that the upthrust exerted on the boat by the water is again vertically in line with the weight of the boat. Everything is in equilibrium and will remain that way unless we choose to upset it.

So the upthrust can move relative to the boat.

Now let us drive the boat. This brings into existence forces and combinations of forces to interfere with this steady balance. The formation of a bow wave may lift the bow. The effect of the propeller may be to lower the stern. The application of the rudder may tend to tip the boat sideways into the turn and the subsequent following of a curved course may tip the hull sideways out of the turn. On another day there may be wind to contend with.

What does the hull do in response? If you forget about friction it tries to move so that the upthrust is still in a vertical line with the weight. In fact the hull is always trying to return to some steady attitude. If we work on a windless day to avoid wind forces we can get rid of the forces due the rudder action and be left with a model moving smoothly through the water in a straight line with no list but with the bow and stern in some different attitude to that when the boat is at rest. The boat is in equilibrium under the gravity and dynamic forces exerted on it. Any boat that is stable when at rest is very likely to be stable when under way.

So how does the upthrust keep moving about?

I have glibly talked of upthrust and it is time to re-examine this idea. The boat floats because the water exerts pressure all over the wetted surface of the hull. The pressure is not the same all over but increases with depth. Pressure is just an idea although it fits in quite nicely with the force exerted on your feet when you wade in deep water in soft waders. This pressure feels just like a force and that is because the pressure is acting over an area. If you chose to think of the wetted area as made up of lots of little areas each with a pressure acting on it you must go on to think of lots of little forces acting on the hull. All these little forces add up to one big one, the upthrust, and this must be vertical. If it were to act in some other direction the boat would move in that direction.

Now we have to think about this upthrust moving relative to the hull when the hull lists. In these days of very high performance computers one might decide to describe the hull to the computer and ask it to find out how the upthrust moves when the hull lists. It is not as easy as it is made to sound. In days gone by people had to find ways of coping and they came up with the concept of a centre of buoyancy. They said that the upthrust acts through the centre of buoyancy, which is exactly the same as the centre of volume of the hole in the water made by the boat, and then set about locating this imaginary point. There was a precedent for this idea. The term “centre of gravity”, that has entered the language as if everyone understands it (which is very doubtful), has its origins in the idea that if you think of a solid as being made up of lots of little bits then you can think of the centre of gravity as being the point where all the little weights of those little bits balance out. Then if you think of a volume as being made up of lots of little volumes the centre of volume is the point where all the little volumes balance out. This turns out to be a powerful idea.

So, if we can find the centre of volume of the water displaced by the hull, we can locate the upthrust.

The idea underlying our understanding of ship stability is that when a hull heels the shape of the displaced volume changes and the centre of buoyancy moves relative to the hull. This means that the upthrust moves and the stability of the hull depends on where the centre of gravity of the boat is and whether the upthrust moves in such a way that the heeled boat tries to return to an even keel. If it does it is stable.

It might have been very convenient if full-sized boats and models of them needed to be designed in the same way but modellers want their models to be stable and have no problem with the strength of their hulls whereas the stability of boats and ships must be such that the stabilising forces give an easy ride and do not strain the structure of the ship. So I will consider the stability of models

In figures 5 I have drawn a section of a model boat having a beam of 10², a rectangular section, and floating at a depth of 4². These dimensions are towards the high end for model boats. In figure 6 I have drawn the same diagram for a model of semi-circular cross section. I have in mind that most displacement hulls lie somewhere between these two shapes so that these are representative of all the others. Let me look at the diagrams in turn.

For the rectangular hull the centre of buoyancy B will be half way between the bottom and the waterline. Now suppose that the hull can be made to heel thorough 10°. The effect is to submerge one triangular volume of the hull and to raise an equal volume above the water line. This changes the shape of the hole made in the water by the hull and the centre of buoyancy moves to B¢. I did not guess the position of B¢. It can be calculated and it is as accurately placed as I can draw.

Now, in both positions, the upthrust, equal to the weight of the boat, acts vertically upwards through the centre of buoyancy and the weight acts vertically downwards through the centre of gravity, G, which I have placed in an entirely arbitrary position. When the hull is floating on an even keel the upthrust and the weight are in line but, when it is heeled, the upthrust and the weight are not in line and the two forces act to push the hull back to its original position. They are said to form a couple in this case a restoring couple. The hull is stable.

The centre of gravity could be anywhere on the centre line but it is now quite obvious that as the centre of gravity goes higher the restoring couple gets smaller and if it goes lower the restoring couple gets larger. It cannot go above point P where the upthrust cuts the centre line without the hull becoming unstable.

Now for the semicircular section we can do the same thing but this time the shape of the hole in the water does not change. The upthrust acts through the centre of the arc whatever the heel may be. So P is at the centre of the arc. The centre of gravity, again shown in an arbitrary position, has swung to the left and again we have a restoring couple. Again the hull will be stable provided the G does not go above P. B is located accurately on the diagram.

In both cases it is apparent that the hull will increase in stability as the distance between P and G increases. But it is also apparent that, for these quite heavy hulls, the point P is not very high up when we would like it to be several inches higher. We could increase the restoring couple by increasing the weight of the boat, because the upthrust and the weight are larger for essentially the same distance between them. This necessarily means that the hull will sink deeper into the water and well below its proper marks before any useful change occurs but as it also means that the hull has greater inertia and consequently responds more slowly and looks to be very sluggish. It is worth noting that if the hull of rectangular section were to be say 50² long it would weigh about 60 pounds. This is not the way to go. I think that the best we can do is to aim to load to the same waterline as the prototype and, at the same time, attempt to increase PG.

This forces us to think whether we can move P upwards and/or G downwards.

It turns out that P can only be raised for a given hull by increasing the beam and that would take us away from the scale fidelity. So we cannot move P and must concentrate on G.

In my view the time to think about lowering G is at the start of building. Absolutely everything above the waterline must be as light as possible consistent with it looking like the full size. This calls for an attitude of mind that includes weight in the decision on how to manufacture the various parts. Keep in mind that the bits that you think are too small to weigh anything add up to a significant weight in the end.

But what happens if, despite all the weight saving aloft, the boat is still not stable enough when it is loaded down to its marks with ballast in the bottom of the hull?

There is a sort of article of faith that sheet lead can be placed in the bottom of the hull until the model becomes stable. In fact, as sheet lead is added, there is a sort of race between the stability becoming adequate and the model sinking. The mechanics is against it. The increase in stability for a given amount of added ballast is very small.

Others, who have been caught on this hook before, have chosen, at the building stage, to distort the hull downwards by the addition of an inch or two below the waterline. This makes the hull heavy, spoils the wave pattern and the whole model now fails to give the correct impression of the original boat to anyone who is not familiar with the prototype. On top of that the boat is sluggish.

So what can be done? Physics tells us to fit a weighted fin because a relatively small weight carried under the boat can produce a greater improvement in stability than a large mass of lead carried internally. This opens the way for a stable model floating at its correct marks.

Figure 7 shows the weighted fin for my paddle steamer. It contains 2 pounds of lead at the bottom. I had a choice, I could have a top deck and a weighted fin or no top deck and no fin. As I wanted that top deck and you cannot just increase the stability of a paddle steamer by adding lead in the hull because the paddle wheels will then be too deep, there was no choice.

Once the choice was made it became possible to design the fin to assist the rudder in directional control in the absence of a propeller wash. In fact it was designed to give a turning radius of 15 feet at normal speed and that is what it does.

I am all too well aware that most scale modellers would just sniff and turn away but I am an engineer and I use engineering solutions to my problems and I know that the laws of physics permit no exceptions. Fins can be removed for display purposes but I think that they should be on display to show how the model is actually sailed.

Warships are usually built for speed and are normally fine boats. Models are often tender. I have never built a scale warship so I do not know the wrinkles used there for building lightly but I do know that I would make provision to fit a weighted fin to the hull so that I could correct things without the model going too deep if it proved to be too tender.

In summary there can be no doubt that a weight carried under the hull is the best way to improve stability. Weight at a practical depth can be 5 or 6 times as effective as the same weight in the bottom of the hull. As a bonus the fin can be of an accurate aerofoil shape and improve the steering.

In my opinion a significant proportion of model boats would perform much better if they were fitted with a weighted fin especially models of the early paddle steamers. However there is an irrational built-in resistance to this idea so they are not fitted.

I have yet to talk about sail-driven models. The very fact of using sails adds another complexity that will not be resolved by the act of building a model to scale. The model must sail in winds of the same strength as the prototype albeit on much smaller expanses of water on which waves do not form quite so readily. I have dealt with the methods used to cope with this extensively in sections 2 and 4 of this website.

*Screw propulsion.*

My experience involves displacement hulls and I am not willing to speculate about prop riding or other fast boats systems. It seems to me that modellers do not look at what is going on at the sterns of their power-boats. It is pertinent to try to decide how a propeller actually works and I must refer you to my section on propellers in this site that was written to go with this section.

I think that the clear message from that section is that on full-sized boats matching the propeller to the hull and the engine is not at all easy. Although steam power is possible it is likely that our model boat will be propelled either by an IC engine or by an electric motor. Both turn at high speed compared when to the speed of rotation of the propeller of the full-sized boat. The easy thing to do is to use a direct drive, a propeller of small diameter and of small pitch. Some modellers may use a speed reduction system of some sort, for example, gears or belt drives and then the propeller size will increase and the pitch be greater. In practice for lots of models only dedicated modellers will place much emphasis on having a propeller of scale size. However there are some models like tugs and steam launches that must have appropriate propellers and then matching the propeller to the engine becomes unavoidable.

For such boats the diameter of the propeller will be settled from that of the prototype and the scale ratio. We are left with the shaft speed and the pitch as variables. For steam it is unlikely that a step down in speed from engine speed will be needed but, for tugs, a step down is inevitable. Then it comes down to deciding whether to settle on a speed reduction system and then go and buy a range of propellers or to buy one propeller and have some way of altering the step down ratio. Either way we need to have a fairly accurate knowledge of the shaft speed to narrow down the pitch of the propeller. Tachometers are about in most model clubs so measuring the shaft speed is not impossible.

Then it all comes down to deciding on a pitch. For that we need to have an idea of the forward speed of the boat. Most modellers are old men and they walk at about 2 knots. This is a typical value for a displacement hull when going at its normal top speed. Two knots equates to about 1,200 inches per minute. Then, if the shaft speed in revs/minute is known the advance per revolution of the prop is given by 1200/N where N is the shaft speed. So for a shaft speed of 1,000 rpm the advance is 1.2².

This has now to be related to pitch. I think that it would be reasonable to expect a model propeller to have a slip of perhaps 25% then the required pitch would be 1.6² or 40mm. This would give a starting point in the process of selection.

The pitch of a propeller is not always known but it can be found quite easily from its tip angle and its diameter. In figure 9 the red propeller has been clamped to a piece of sheet aluminium using its hub as a reference surface. A piece of card was be cut by trial with scissors until its slant surface lined up with the tip of the blade when fitted as shown. The angle can then be measured using a protractor.

The graph in figure 10 comes directly from the geometry of the helix. The lines are for different pitches. To find the pitch look up the line for the diameter until you reach the tip angle and read the pitch off from the coloured lines.

This graph reduces the guesswork when an existing propeller is not correctly matched to a hull and engine and a new one has to be purchased.