Chapter 11    Ship resistance and the concept of model testing

 

Introduction

One might argue that ship resistance is not relevant to the activities of a mechanical engineer. This text is about the construction of an empirical science and the work of Froude (1810 – 1879) is woven into that science because it showed that the testing of models is a way of gathering data about the way that flowing fluids interact with solids to produce forces on them. His work opened the way for wind tunnels and prepared the way for the computer modelling that is now used in tandem with model testing to reduce the amount of physical testing that is needed to achieve some result. It was a seminal piece of work. Froude’s papers have been published and figure 11-1 is the flyleaf of the book. I enjoyed reading them.

 

Froude

Froude’s celebrated work, that was continued very ably indeed by his son, was initiated in 1868 with an application to the Chief Constructor of the Navy for funds to construct a towing tank and employ three people in order to carry out a programme of experiments over a period of two years. Froude was then 58 and an intensely practical engineer with great experience including a very extensive knowledge of ships of all sorts. He was a man of independent means who lived in Torquay in England and gave his time without payment to undertake his programme of testing. Very wisely he stipulated that the programme of testing would be free from any outside direction and be entirely in his control. Equally wisely he also stipulated that his son should work with him on a salary. The Navy gave him £2,000 and got a bargain. The work eventually lasted 14 years when the site on which the facilities had been built was restored to its owner again as farmland. In those years the testing of models had been firmly established as an invaluable aid to ship design and subsequently led seamlessly to the use of wind tunnels for the design of aircraft.

 

It always interests me to try to decide why one man produces a piece of work that changes everything when others who were contemporary did not. Froude graduated in 1828 with a first in mathematics and a third in classics from Oriel College Oxford. His principal interest appears to have been sailing but his real strength was as a very practical engineer. He spent most of his life in and around the River Dart and he must have been well aware of, and interested in, all things to do with ships. His degree shows that he had the intellect to think about mechanical things at a quite different level to that of, say, George Stephenson. In order to place Froude in his context we must have some idea of the state of the knowledge of ship design in the mid 19^th century.

 

I think that it is fair to say that ships are the only form of mechanically-powered transport to have existed for hundreds of years and were designed by various rules of thumb that had grown up over a long time. By the start of the 19^th century science was influencing design of mechanical devices but it was emerging in an era when everyone had grown used to thinking in empirical ways and attempted to deduce everything from observation. Inevitably, during the gradual evolution of science, ships attracted the attention of all sorts of people as being a very interesting subject for the application of the emerging science. After all there was the hull, the engines, the boilers, the propulsion system to be developed and all of them might be, and indeed were, improved by the use of science combined with improved manufacturing methods. All sorts of ideas were advanced to try to improve the design of all the elements of ships and my impression of naval architecture is that many of these ideas still persist long after they should have been discarded. Froude was born into this era with, seemingly, an inbuilt interest in ships.

 

So what was happening in ship design when Froude was a young man? The ss Great Britain was launched in 1843 just 5 years after Froude graduated. It was half as long again as any other ship that had ever been built, it was the first large iron ship and Brunel designed the engines and the newly-invented screw propeller. Brunel knew how to make a very good estimate of the resistance to motion of his ship and how this resistance would vary with speed. His engine worked with a boiler pressure of 5 psi and exhausted to a jet condenser giving a vacuum of 10 psi. Sea water was used in the boilers that were in three sections so that any one section could be blown down to get rid of very saline water that resulted from the evaporation of the water fed to the boiler. The four cylinders were very large because the low-pressure steam in use involved very large volumes and they were of a size (88² diameter by 70² stroke) that came to be used much later as the low-pressure cylinders of compound engines. The “theory” of screw propellers was in existence and Brunel knew about slip and the way the blades should twist. He was able to predict the performance of his ship from engineering principles and he did it very accurately as was proved on the first run of ss Great Britain when it was worked up to its design speed. The ship was launched over 30 years before Froude started his work and many improvements must have taken place in every aspect of ship-building during that time. Froude, who knew Brunel, would have been well aware of the general situation.

 

I think that Froude tells us much about the state of ship design in the second half of the 19^th century in his application for funds to the Chief Constructor of the Navy. This is an extract :-

 

Moreover while it was found in my experiments that on comparing the performances of a 3 foot and 6 foot similar model, the former exhibited a slight excess of proportionate resistance possibly attributable to the above cause; between the 6 foot and 12 foot models no such difference was observable. Lastly this is worth notice. Calculations based on a large number of indicator diagrams taken from the largest well-proportioned vessels of the wave-line type, at such speeds as not to create notable waves, have satisfied me that we may regard from 23 lb. to 24 lb. per square foot of mid-ship section as the resistance of such vessels at 10 knots, this figure, reduced by the rule of the squares of the velocities, which my experiments shew to be fairly correct below such speeds as create notable waves, would give 0.74 lb. to 0.78 lb. per foot as the resistance due to 1.8 knots. Now, at that speed, the 6 foot wave-line model makes, according to my experiments, a resistance of just 0.75 lb. per square foot.  Whereas, if a 6 foot model were considerably affected by the action referred to, its resistance should be markedly greater.

 

It appears then that so far from the foregoing objections having any real foundation, we are able, in fact, to frame from the results of small scale experiments by a precise and simple method an accurate prediction of the performance of full size ships.

(ii) The existing state of knowledge as to the merits of particular forms in ships is very unsatisfactory.

 

(a)  It is generally held that Vessels of what is called the "wave-line" type offer the least resistance, and there is nothing either in the reasonings or the statements of its advocates to limit its superiority to the condition of comparatively moderate velocity.

 

Now, if it is allowed as I have contended that the performance of such models as were used in my experiments is precisely similar to that of similar ships, it is clear from those experiments, that at high speeds the wave-line form is markedly inferior to a form totally dissimilar to it in appearance and utterly opposed to it in principle

 

(b)   An opinion is beginning to gain ground and, with good reason, that surface friction forms a large share of the resistance of a ship, yet concerning the nature of this force much misconception plainly exists; for by the best authorities who have treated of the subject it has been held as the basis of calculation that a long plane surface of say one foot in width moving endways through the water encounters the same resistance of friction for each square foot throughout its length. Now it is quite clear that the first square foot in its length in experiencing frictional resistance from the particles passing it, and thus impressing force on them in return must communicate to them some velocity in the direction of its motion. The particles in contact with the second square foot having thus already received forward velocity must exert less frictional resistance upon it, unless the degree of that frictional resistance be independent of the degree of velocity with which the particles traverse the surface; a supposition manifestly erroneous, and quite as subversive of received opinions as the alternative conclusions, I advance that surface friction is not uniform for every part of the surface?

 (c) In the existing state of knowledge there is no guide for determining the influence upon a ship's resistance of the ratio of her length to her breadth and to her depth; a question of the most vital importance in the construction of ships of war, where by increasing the length though we no doubt increase the carrying power, we increase also in a greater degree the weightiest and costliest parts of the structure, adding at the same time to the difficulty of handling the ship; whereas by increasing the breadth we can add almost as largely as we please to the carrying power with an increase only to the less weighty and less costly parts of the structure, leaving the handiness unimpaired.

 

(d) Though there are current very precise rules by which it is said we can determine the limit of speed beyond which a vessel of given length cannot economically be driven, these rules have been fixed by an (as it appears to me) arbitrary and not strictly relevant application of the theory of wave-motion. Moreover, the rule implies that the limit will be at the same speed for all ships of the same length, but this according to my experiments is obviously not the case, for instance in the experiments on the two 6 foot models* the same in length, breadth and displacement, one has the limit at a speed fully 8 per cent, higher than the other, making at that speed 20 per cent the less resistance of the two though at the lower speeds it had made the greater resistance.

 

I have here enumerated certain deficiencies and inaccuracies in the received views on the subject. These alone are manifestly of very high importance, moreover we may reasonably expect that an extension of the inquiry will discover many other deficiencies and errors equally serious.

 

We see quite clearly that the science of ship design was attracting rival theories just as yacht design does now. Some of these theories were downright destructive, (see the activity of Lardner on page 11 of “The Iron Ship” by Ewan Corlett.) and others were insidious in leading to the design of ships that were much less safe than they would have been had not the spurious “theories” been accepted. Froude was a far-sighted man.

 

Froude clearly recognised that if it were to be possible to test scale models of the hulls of ships in some sort of towing facility and to predict, from the outcome of these tests, the behaviour of the full-sized ships many of these ill-founded ideas would be proved to be wrong. I think that we may take it for granted that the design and construction of the towing facility would not be seen by Froude, with his undoubted skill as an engineer, as anything more than an interesting challenge and he was very well suited to such activity. However the scientific basis for the testing was not readily obvious and had been the subject of much debate over the years. As I read his papers I sense that he seems to have been acutely aware of the acrimonious nature of the scientific debate that was going on and knew that the standard of his work would have to be beyond criticism. He succeeded.

 

A towing tank had first been used in France 100 years earlier. Unhappily the idea seems to have been that there was some underlying relationship between speed, resistance and size that would be more or less the same for all ships. This sounds a bit naïve but ships at the time were all the same shape.[1] The idea was to take measurements of resistance of arbitrarily shaped models when towed and then to think of a way to use these measurements to predict the resistance of any other full-size ship. It could not work even though several celebrated men tried to make it work. A result of this activity and of other attempts at towing was a split in naval thinking between those who favoured testing full-sized ships as a means of improving ship design and those who favoured model testing. The testing of full-sized ships would have been very expensive, exceedingly difficult, as Froude discovered, and very slow to yield results. The testing of models would be, by comparison, cheap and speedy and not subject to the vagaries of weather, wind and tide. Froude found a way to use model testing and, probably more importantly, a way to prove the method cheaply, so that even the most sceptical could not really say that it was not worth the very tiny investment involved. Froude appears to have been either very shrewd or knowledgeable about the ways of men. I like the sound of him.

 

All this tells us why Froude needed to be able to follow his own agenda of testing and why he gave his time to the project as distinct from seeking to be paid. Had the Admiralty funded the programme in its entirety it is almost inevitable that a committee would have been formed to oversee it and that members of that committee would have wanted Froude to pursue their ideas and not his own. As it was, £2,000 was a very modest sum to set against the immense rewards that followed from the investment.

 

The outcome was that Froude built a very large towing tank of 278 feet long 36 feet wide and 10 feet deep and showed that it is possible to predict the behaviour of a full-sized ship from towing tests on a scale model of the hull and then went on to pursue a programme of investigation to create a more coherent body of empirical knowledge about ship design. He was the instigator of a programme of investigation that was pursued by his son R. E. Froude for more than three decades and made them architects of this branch of our empirical science. It was a great achievement.

 

So what was his idea? It is contained somewhere in his published papers but it is difficult to winkle out and I suspect that every reader of those papers will come to a slightly different view of the work. I shall give mine.

 

The background to model testing

There can be no doubt that Froude’s first task had to be to show that tank testing of models worked. There was only one way and that was to test a model of an existing ship, predict its resistance to motion, and then go out and measure the resistance of the full-sized ship.

 

It all seems very simple but one must consider the situation in the mid 19^th century. We are accustomed to seeing photographs of vessels of all sorts under way. The photographs are taken from the air and, even if they are not taken with the study of the waves and the wake in mind, the surface effects are there for anyone to study. Froude would have had to look at bow waves and wakes as they were being made and never from a useful observation point. Nevertheless it seems that Froude over the years had recognised that the resistance to motion of a ship was caused partly by friction between the hull and the water that created a wake in which energy was stored and then lost in random eddies and partly by making waves that very obviously carried energy away from the ship.[2] [3]

 

Figure 11-2 is a photograph taken from an advertisement and shows a racing sculling boat on the move. The use of oars means that, with the exception of the circular waves on the right, the effects that we see on the surface are entirely due to the action of the hull on the water and not confused by the presence of a propeller. We can see the wave pattern coming from the bow and another wave being generated at the stern and we can see the confused wake caused by friction between the hull and the water. In the extract above Froude has commented on all these things. I think that one can have some sympathy with the people who were trying to think about the problem of ship resistance in the 19^th century.

 

I have wondered why Froude was so confident about his plans. He was not a young man so he had had time to gather a great deal of information and to experiment to clear the way for his major experimental programme. One criticism of the testing of models was the effect of surface tension that was obviously a factor for a small model but equally obviously not a factor for a ship. Froude had cleared this out of the way by experimenting to find the minimum length of a model for the effect of surface tension to be small enough to be ignored. He decided that this length is about 6 feet (2 metres) and all his models were longer than this and in some cases much longer at 18 feet. These days the length might be 20+ feet but towing tanks are somewhat larger. But there is another factor that I think that Froude would have known about and it would have given him some confidence. Steam engines in those days were slow speed machines that were made to quite high standards[4] and the behaviour of steam in the cylinders was very predictable. This meant that the power given by the steam to the pistons could be calculated quite accurately as the indicated horsepower. The mechanical efficiency of these engines was known within ordinary engineering limits so the power going to the propeller could be calculated and finally the propeller efficiency was virtually fixed if the propeller was made to reasonable standards. The net effect is that the power required to drive a given ship when it was under way could be calculated with some confidence. A graph could be drawn of the power needed to drive the ship versus speed.

When the ss Great Britain was recovered from the Falkland Islands a model of its hull was tested to assess its design in relation to modern designs. Graph 11-1 gives that graph redrawn from Corlett’s book in order to restore the origin and the proper proportions to the x axis. The red lines show the power required to drive the ship at the service speed of 12 knots. The power required would have been 900 horsepower. At 810 horse power the speed would have been 11.6 knots and at 990 horse power 12.3 knots. So a 10% difference in the power only changed the speed by 3+%.

 

In engineering terms these are acceptable figures. So Froude did not need to make predictions that were highly accurate in order to determine the power needed to drive a ship. Nevertheless he went on to make his measurements to the best of his considerable ability and the outcome was fundamental to the subsequent evolution of the science of fluid flow.

 

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.[5] (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.[6]

 

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[7]. 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[8].

 

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.

 

 

 

Implications of Froude’s work

It seems to me that Froude showed us something fundamental about the flow of fluids that had not been demonstrated before. In this chapter I wrote this (and I reproduce it in red because it is easier than scrolling back to find it):-

 

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.[9] (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.

 

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.

 

The curves in graph 11-4 are so totally similar that there can be no doubt that the pressures do rise over the hulls in the same way and that the pressures do act on the various areas of the hull in the same way to produce forces that can be related by simple scientific methods. The fluid flow over hulls is repeatable and by extension will be repeatable for aeroplanes, rivers, water turbines and every other similar application.

 

A direct consequence of this is that if you choose a category of ship, for example an oil tanker, the goals of designers are always the same and the resulting designs will tend to a best shape that will be substantially the same, if not identical, for all tankers regardless of size. The service speed and the power requirements will change with size in a predictable way. We have potential for a systematic storage of design data. The same thing will be true for other devices that are made in a range of sizes.

 

This is important and it came about through the skill and persistence of one man. It is appropriate that his name is immortalised in the non-dimensional parameter that bears his name.

 

Critical speed and the Froude number

The Froude number is just a device that we choose to use when it suits our purpose. It has no counterpart in reality like a depth. It is dimensionless because it is the ratio of two velocities one of which, the numerator, is normally an actual velocity. The denominator is usually derived from a length and is a notional velocity that is used as a reference value and possibly called a critical value because it might be the line of demarcation between two regimes of flow.

 

For displacement hulls a critical speed is worked out for the speed of a wave that has a wavelength equal to twice the length of the waterline when stationary. The idea behind this is shown in figure 10-14 where the model yacht is on a wave with crests at bow and stern. There is an expression for the speed of this wave and it is :-

                                   speed of the wave =  in some consistent units. It is often called the critical speed but it can be exceeded by a small amount without the need to plane but progress at speeds in excess of this critical speed involve a pronounced bow up attitude and very high resistance. The critical speed concept is a good guide to the maximum practical speed that can be attained by a displacement hull.[10] Obviously a dimensionless group can be constructed from the ratio of actual speed in knots over hull speed in knots but ordinary ships do not operate at speeds anywhere near to their critical speeds (Or any other consistent units.) and, at the critical speed, the number will be 1. This group has been called the Froude number in recognition of Froude’s contribution to ship science.

 

But Froude, as we have seen, did not use this idea. He needed only to claim that if two hulls of exactly the same shape but of different sizes move at speeds that are proportional to the square roots of their lengths their bow waves would be exactly proportional to their lengths. There is the common element here that speed and length are related by a square root relationship. This is hardly surprising when one considers Bernoulli with his interchangeability of kinetic energy and potential energy. We have a ratio

         wave height on the model/wave height on the prototype = Ölength model/length prototype.

 

Froude used speeds up to 12 knots but now we routinely think of speeds of twice this for tankers and higher for passenger ships. These higher speeds are the result of using much larger ships and driving them with high-powered, efficient engines running on cheap oil.[11] Typically a tanker will be up to 350 metres long and it will run at 25 knots.

 

Now we need some idea of the variation of critical speed with length. It is shown in graph 11-5. Immediately it is evident that the tanker is operating at about ½ the critical speed as indeed was ss Great Britain.

 

As I showed in graph 11-1 the power to propel ss Great Britain

is increasing very rapidly at speed of 12 knots and there is quite clearly a speed that gives the best return on the money that has been invested in the operation of the ship. Presumably this applies to all ships and the most important item in that financial compromise is probably the cost of fuel and, if it is not now, it soon will be.

 

It seems that comparison of ship performance is made easier if one thinks of the operating speed as a Froude number rather than a real speed because it tells us where the ship is operating on the graphs of resistance and power versus speed. The range of Froude number within a group of ships designed for the same purpose is really quite small. Oil tankers run at about 0.5 and cruise liners at about 0.7. Size-for-size the power installed in a tanker is about 70 % of that installed in a cruise liner and this reflects the different economics and the difference in the way in which they are used.

 

Geometrical and dynamic similarity

This text has thrown up a need to describe the conditions that must be satisfied in order to compare the behaviour of two solid bodies interacting with a flowing fluid. First there is the question of shape. The two bodies must be the same shape[12] exactly except perhaps for the surface texture and they must also be set in the same attitude relative to the undisturbed fluid that flows over them. They are then said to be geometrically similar as a shorthand description just as we use one-dimensional flow as shorthand in pipe and channel flow. There is still the question of speeds and when the two operate at speeds that give the same flow patterns and, in turn, forces that can be related by simple mathematics the two are said to be dynamically similar. The two speeds are said to be corresponding speeds.

 

We can carry these descriptions on into aerodynamics.

 

Endnote

I included this chapter on ships primarily because of the importance of Froude’s work in finding a way to refute the many spurious ideas that were prevalent in his day and in doing so show us that we could go forwards knowing that fluid flow is repeatable in the best scientific meaning of the word.

 

It is also interesting to me to read current authors still bemoaning the fact that Froude chose the treat the effects of skin friction and wave-making as independent. The success of his method seems to be an affront in some way. I think we are back with “slaves do arithmetic”. This is great pity really because Froude’s method can very often get you past seemingly insuperable obstacles.

 

I have said nothing about ship science as it is practiced because it is too arcane for me to understand it.

Examples on model testing

1)  A ship has a length of 200 m and is designed to operate at 12.5 m/s. A model is made to a length of 7 m and is to be tested in a towing tank. When towed at the corresponding speed for wave-making the total resistance to motion is 82.5 N.

 

The skin drag of the model is given by  and for the ship by  where the resistances are in pound force and  is in square feet and  is in knots.

 

Calculate the corresponding speed and the resistance to motion of the ship if it has a wetted area of 5,000 m².

 

2) A ship has a length of 100 metres and a wetted surface area of 2,600 square metres. It is to be driven at 0.55 of its critical speed. A model is to be tested and the chosen scale ratio is 25. The total resistance to motion when tested under dynamically similar conditions is  28.5 N

 

Calculate the speed of the model, the resistance to motion of the ship at its design speed and the power needed to drive it..

 

For the model use skin drag  N ;  for the ship skin drag  N where  is in square metres and  in metres/second.

 

 

Solutions

1) 1 knot = 0.515 m/s. Therefore speed of ship  knot.

 square feet.  lbf  kN.

 and then

We need . .          knot

Therefore  lbf  N

 

And  N

 and from this  kN.

Total resistance = 980 + 663 =1,643 kN.  Power needed = 20.5 MW

 

2) Critical speed . So speed of ship:-

                      knots.

 Then

 and  N

 N   from which  N.

Resistance to motion of the ship is  N.

Power to drive ship