The fundamental principles of model testing of ships

It seems to me as I read Froude’s papers that he was very conscious of criticism, either real or imaginary, of his work and that this colours the way that everything is presented. In this respect he had much in common with Darwin and was probably only at peace when he was working at his self-appointed task.

 

I will not give Froude’s arguments for his method of testing because it is not in the style that we now use. The steps in the argument are these.

 

The resistance to motion of a ship is caused by two effects, wave-making and viscous resistance. Whilst these are generated in different ways they coexist and must to some degree interact. Froude, being an engineer, made the eminently sensible decision to treat them as separate effects that could be measured and scaled up independently. Then  :-

                                                       

where  = total resistance of ship or model,  is the component of resistance due to wave-making and  is the component due to skin friction.

 

Froude saw that he would have to test his model at a speed that gave the same size bow wave in proportion to the length of the model as the full-sized bow wave bore to the length of the ship. Then the two systems would look the same and behave in the same way. He deduced that the speeds of the model and full-size would be in proportion to the square roots of their lengths and he called these corresponding speeds. He then said that if, in some way, he could determine the resistance due to wave-making for the model he could scale it up by multiplying by the cube of the scale ratio, that is, the cube of the ratio of the lengths. 

 

We need some satisfactory explanation for this. Figure 11-3 shows a model boat under way that was 4 feet long with a swim head that is obviously a temporary attachment. The water ahead of the bow is clearly lifted and it has the profile shown in figure 11-4.

The obvious effect of the motion of this swim head bow is to push water forwards and upwards and it is evident from the bubbles in the picture that the wave in front of the bow is close to rolling just as it might in an open channel. It also follows that water must flow under the boat and that somewhere there is a horizontal division between water that will go upwards and forwards and water that will go backwards under the hull. The momentum in the vertical direction of the two flows must be equal and opposite because there can be only one value for the net force between them. The result is that the water with higher speed flowing under the boat will experience a drop in pressure as evidenced by the trough on the side of the boat.[1] (Had this been a channel we would have talked of constant specific energy.) The rise in the water level ahead of the bow will increase the static pressure all over the head by an amount equal to  and this pressure acting on the area of the front of the bow will produce an additional force on the boat that has a component that resists the motion. The rise in level  where  is the speed of the boat. The water must also split in the horizontal direction to produce the vee shaped bow waves and it seems to me that too much time is spent on these waves instead of looking at the process of generating the wave pattern as a whole. Then bulbous bows make sense.

 

One way or another this will happen for bows of any shape whether they are sharp like a warship, blunt like a tanker or fitted with a bulbous bow. A rise in level will occur and it will alter the pressure acting on some difficult-to-define area around the bow to produce the resisting force. The central argument of model testing is that, if the value of  is proportional to the length, the pressures will be in proportion to the square of the speed and the areas that are affected will be the same shape and proportional to the square of the length. Then if two hulls of the same shape but of different sizes running with the same trim at the corresponding speeds the resistances will be proportional to the cubes of their length.

 

Froude argued that if the skin friction could be calculated for the model it could be subtracted from the total resistance of the model to give the wave-making resistance. This wave-making resistance could then be scaled up to give the full size value and then a calculated value of the full size skin friction could be added to give the total resistance. That is reasonable but now we must find a way to determine the skin friction.

 

Having given priority to wave-making we cannot now measure skin friction using the model. We need to be able to calculate the skin friction and we know that the magnitude of the skin friction is, in part at least, dependent on the nature of the surface on which the water acts. Previously the solution was to describe the surfaces for example, brick work in good condition for a channel or new drawn tubing for a pipe. Froude introduced the use of paraffin wax as a very tractable material for making models. It is still used. So for the model we have only one surface to deal with but for the ship there may be some considerable variation.[2]

 

In the extract from Froude’s paper above paragraph (b) sets out the problem as he saw it and nothing has changed except that we now talk about boundary layers as a normal feature of flows round solid bodies. The friction force at any point on the wetted area expressed as a force per unit area is not the same all over the wetted surface of a ship or its model.

 

Froude investigated this by towing wooden boards of about 3/16² thick (4.75 mm) and 19² wide on edge and measuring the drag at various speeds. It sounds straightforward but if the dynamometer used to measure the drag is to measure only the drag the boards have to be buoyant and naturally float on edge with almost neutral buoyancy. Furthermore the boards were of lengths of 1 foot, 3, 8 15, 25 and 50 feet[3]. They were coated to give what would have been well known surfaces such as tallow, calico, Hay’s composition and Peacock’s composition or varnish (Presumably Hay’s composition and Peacock’s composition were anti-fouling paints.) This is the same strategy as was used for quantifying friction of channels and subsequently of pipes.

 

Froude must have made hundreds of test runs with these boards but when he presented a paper on his work he gave us an overview. Graph 11-2 is from Froude’s paper but has been reworked to make the legend readable on a screen. The graph is really a derivation from the original data that was for boards of the specified length but 19² wide and, of course, double-sided. The boards were coated with tallow etc., and it seems that these graphs are for some mean value of the resistance. The resistances have been quoted as if for a single-sided board having a width of one foot.

 

Inspection shows that that, at a given speed, the resistances are not proportional to length just as Froude predicted. What is not evident is that this family of graphs can be represented very accurately by expressions of the form  and that  is about 1.85[4].

 

Froude also presented a plot of the resistance against length for various speeds and this is graph 11-3. Every graph in this family becomes straight at about 30 feet at the low speed and at about 50 feet for the high speeds. This means that the resistance for longer boards could be found by extrapolation.

 

Froude had established the characteristics of the drag on the surface of boards when towed on edge at various speeds. It was, and is, of a very convenient form if it turned out to be directly transferable to the surface of a ship.

 

Froude had come far enough for the admiralty to support a programme of measurement of the resistance to motion of a full-sized ship. The ship used was HMS Greyhound. It was 180 feet long, screw driven with a round bilge hull. Testing went on for six weeks and involved towing Greyhound  with another ship and making measurements of the drag. This proved to be difficult even though the weather was good for most of the time and Froude made other measurements that were of interest. His account is worth reading and should have put an end to the ideas of using full-sized ships in preference to model testing. Ultimately Froude could compare his prediction for Greyhound with direct measurement. I have copied his graph and enlarged the legend to a readable size in graph 11-4a and graph 11-4 legend.

 

The curves are strikingly similar and the displacement between them shows us how important it was to be to be able to assess the state of the ship’s bottom. It seems that the bottom was covered with copper but Froude did not know its condition. He opines that it might have been corroded but it seems to be optimistic to think that it was much the same as varnish. What seems like a small change to treating the bottom as part varnished and part calico gives total agreement between model and full size.