Chapter 5       The resistance to the motion of a surface vessel

Introduction

Most of us have opportunities to look at the motion of boats of all sorts on the surface of water. We become accustomed to the general features of the wave motion and wakes yet never find a way of sorting out how those features come about. Our special interest is the motion of the hulls of model yachts but, as a group, these are just a special case of the whole family of hulls. If we look at full sized hulls first the surface features can be interpreted and the reader may find it easier to learn. Then we can look at our model hulls.

A look at the water after a rowing skiff has passed by tells us that we have two major effects to think about. The water surface will show a pattern of surface waves which spread out and a wake of eddying water that does not spread very much. These two effects appear to behave independently but, at the time that they are being generated by the hull, they must, to some extent, interact.

A look from the rowing skiff at the water near to the hull but some way aft will reveal that the boat is dragging water along in a fairly thin layer (an inch or two). This water mixes with water surrounding it to produce the eddying wake. Energy has been given to the water to give it this swirling motion and the oarsman provides the energy. From his point of view there is a drag on the boat. This is called skin drag. The waves formed in the water also carry energy away (as can be seen when the waves from a ship roll up a sloping beach) and this must also require an effort from the oarsman to sustain. So we talk about a wave making drag. Those who thought about this originally did not make this distinction and made little progress in coming to understand ship resistance. Then William Froude separated these two effects and set about dealing with them independently and laid the ground rules for estimating the resistance to motion of ships.

It would make sense for us to follow Froude and think about these two effects separately using information available from the study of full sized boats. We can usefully start by looking at the several modes of motion of surface vessels.

Forms of motion of surface vessels.

It seems that boats of all sorts can have up to three forms of motion each distinguishable by its wave pattern. They have been called drifting, sailing, and planing. Drifting is a proper form of boat movement even though it is very slow. It produces almost no wave motion at all and the resistance to motion in drifting is almost wholly due to water friction. At one time large freight boats drifted on continental rivers being controlled by poling and, on the large rivers, the boats may acquire an extra speed of up to 2 knots over the speed of the current of 4 knots as a result of the slope of the surface. The fact that these large vessels can make 2 knots on the slope of the surface of the rivers tells us immediately that the frictional resistance is very low, certainly of the order of 1/10,000 of the weight of the vessel.

When sailing (not necessarily under wind power) the boat moves with enough speed for wave making to become important. Then the proportion of the resistance to motion of the boat due to wave making increases with increasing speed. At low sailing speeds a bow wave and a stern wave form and several waves in between. All these waves spread out sideways in a chevron pattern. As the speed increases the number of intermediate waves becomes fewer until there is one wave at the bow and one at the stern and a hollow in between. A further increase in speed causes the bow to lift and the stern to fall as the boat leaves the stern wave behind and moves forward on the back of the bow wave. This is the start of planing.

High powered cruising boats plane. They produce a very large bow wave, a hole in the water at the transom and a large detached following wave. It is exciting but not very elegant and pretty inefficient.

If the boat is suitably shaped and light enough it can, with sufficient power, lift up and over the bow wave and glide across the surface with very little disturbance. This may be regarded as a fourth motion but does not interest sailors of model yachts.

Most model yachts either drift or sail with an occasional lift on to the plane.

Skin Drag

We need to have a clear idea of what we are actually concerned with. Picture 5-1 was published in New Scientist and it is a photograph taken using an electron microscope of the skin of a shark. As sharks mostly swim deeply submerged the resistance to their motion is wholly friction drag. No doubt there is an evolutionary advantage to the shark to reduce this drag to a minimum. This photograph shows the remarkably complex surface that has evolved to do that. It is clear that the shark skin is not just a super smooth surface; those flukes do something to the layer of water immediately adjacent to the skin and, whatever it is, this shape is needed to do it. It is hard to imagine any process by which such a surface could be manufactured. It is clearly quite fragile and any man-made version of it would not have the self-repairing properties of nature. By comparison any man-made surface is exceedingly clumsy. At best a man-made surface may be highly polished but this is still very rough by molecular standards. This means that we are interested in the internal friction of water surrounding a moving surface especially when that surface is smooth to the touch.

We have already noted that a rowing boat drags water along as it moves. It would seem to be obvious that science should by now have an understanding of the mechanism that produces this effect. The truth is that we have no fundamental explanation. All we can do is say that when a surface moves through a liquid (or a gas) a layer of the liquid appears to “stick” to the surface and move with it and that a force like a friction force acts on the surface. Generally, in science, this is called a viscous force but it is good enough for us to call it a friction force.

A racing eight is very long and slender and it produces almost no wave pattern. Its resistance is almost wholly due to skin friction. It has a surface that is very like that of a model yacht. Regardless of the fact that the surface is smooth and covered with a water- repelling wax the water in contact with it still appears to become attached to the surface and to move with it. The layer of water affected by the boat, that is the layer in which the water is actually moving, starts off being very thin at the bow (See Figure 5-2) and gets thicker and thicker as we look towards the stern. In the region of the stern the flow breaks down into all sorts of swirls and eddies and the eight leaves an eddying wake. A significant weight of water has been given kinetic energy by the efforts of the oarsmen and this will now be lost in friction in the water in the wake.

People define a velocity gradient as the change in velocity divided by the distance over which it changes regardless of whether the boundary layer is orderly or eddying. In fluid flow a large velocity gradient is associated with high skin drag. In the boundary layer the velocity gradient decreases as the boundary layer thickens and therefore the skin drag is greatest at the bows where the boundary layer is thin and least at the stern. This means that, as the hull gets longer the average skin drag becomes lower. It is hardly surprising therefore that yachts designed for sustained high speed have to be very large with long slender hulls.

William Froude experimented to find an empirical expression for skin drag, R, for ships and models of ships made from wax. The expression he produced is still used. It is :-

R = f A.v  where A is the wetted area, f is a coefficient which, as we might expect varies with length and the smoothness of the surface, v is the velocity and n is about 1.82. As this is an empirical expression f depends on the units employed for A and v. If we work in square feet and feet per second and want R in pounds, f varies between 0.005 for short vessels (6 feet or less) and 0.0035 for very long vessels (this reflects the reduction in average skin drag with length).

This expression is valid for model yachts and in Graph 5-3 the resistance per square foot of wetted surface for typical yacht speeds is given.

Resistance to motion resulting from wave making.

When a ship moves across the surface of water the disturbance of the surface is very easily seen. Everyone must have seen the bow waves produced by boats of all sizes and by swimming birds. There is just one flow pattern. The water in front of the boat or bird is lifted and, in the absence of constraints spreads out sideways. One might expect a simple continuous V shaped wave to form but in fact the wave is split into short waves which look like overlapping tiles on the end of a roof. A ship generates just the same wave pattern and when the ship moves down a river the wave pattern will spread and eventually reach the foreshore. There the waves will roll and break and the energy content is obvious. This energy is transmitted to the water by a force pushing the hull which, in its turn, pushes the water out of the way forming the waves. We have to understand this wave pattern.

We can start by observing that we do not see a groove in the water to mark the passage of the ship. The water somehow moves in every direction to make a space for the ship to move into and then moves back again to fill the space behind. Ultimately the surface of the water shows no evidence at all for the passage of the ship. The continuous movement of water from in front of the ship to behind is not a simple process and its effects spread over very large areas of the water surface. We must take the explanation in steps.

So what happens? Probably the best picture to look at is Picture 5-6. This is the wake produced by a stick poking through the free surface of a river. As the water approaches the stick, the surface rises as the water immediately upstream of the stick slows down giving up kinetic energy. Once past the stick the water is then like the pendulum before it is released, it has potential energy greater than normal, and like the pendulum its subsequent motion can only disappear as a result of friction. Water cannot be heaped up like this without it flowing away in all directions in a wave pattern to exchange the potential energy that it has relative to the surface for kinetic energy. But, in this case, it flows away in a stream of water and so we see the combination of two flows. The water flows round the stick and then starts to drop gaining kinetic energy over and above that of the undisturbed stream as it does so until it reaches the original level when it has excess kinetic energy. Then it goes on down until the kinetic energy is stored again as a reduction in pressure but now below the original surface level and then it comes back up again gaining kinetic energy to repeat the cycle and form the wave. Meanwhile the water is flowing sideways and the two flows combine to form the arrow-like waves. The first wave after passing the stick is different to the others. The surface comes up as one might expect to form a wave but the sideways flow is about to split the wave in two. This it does before the next crest is reached and, as there is nothing to stop these waves moving sideways, they split leaving a confused wake in the middle that just gets wider. The surface goes on oscillating up and down in the middle but the energy it has is spread wider and wider so that its effect disappears. The other waves continue to spread and are well defined.[1]

The wave pattern in the middle takes place under the hull of a boat and it is mainly the train of oblique waves off from the bow that is obvious. Sometimes the whole pattern can be seen as in Picture 5-7 where the transverse waves between the separate oblique waves are evident. The boat is in fact turning to the left on a calm surface and this is preventing the wave pattern from changing too quickly for us to see its character.

So how precisely does this wave pattern cause drag? We can look at the two pictures of a small yacht under way at a slow speed in Picture 5-8a and moving quickly in Picture 5-8b. In 5-8a the water is hardly disturbed by the movement of the yacht and is effectively floating as if it were at rest. We can see the position of the normal water line. We also know from Chapter  3 that there is then no net force on the hull from the pressure acting on it. When we look at 5-8b the water has risen up above the water line at the bow as far back as about a third of the length. In the middle the level has dropped below the original waterline and not much has changed right at the transom. Now we have to decide what change has taken place in the pressure forces on the hull. In Chapter 4 we recognised the problem and said that we could take the pressures to be proportional to depth as for water at rest. We do know that the pressure on the surface of the water regardless of its shape is atmospheric. It follows that where the surface has come up the pressures on the hull and therefore the forces will have been increased and vice versa. Between 5-8a and 5-8b a significant force has been created on the bow and no comparable force on the stern. The hull experiences a resistance to its motion.

No doubt the alteration of the flow pattern will alter the frictional resistance but it is not hard to accept that the total resistance to motion is that due to the formation of the waves plus that due to friction. The variation of this total resistance with speed for this yacht is given in Chapter 7 as a graph of thrust versus speed.

# The idea of a critical speed

Picture 5-8b shows the yacht to be moving at a speed which gives a crest at both bow and stern. The speed of the boat is the same as the speed of a wave with a wavelength equal to the length of the boat. There is an expression for this speed and, if we call it the critical speed , it can be found from  where  is the length of the hull. We shall find that this speed is useful when considering graphs such that of resistance against speed but it must not be assumed that this is the maximum speed that the yacht can reach. It can go faster but the resistance to motion grows very rapidly above the critical speed.

[1] Very large ships produce large waves that may travel many miles and cause consternation when they break on a bathing beach.