Chapter 12   Dimensional analysis, rational expressions and model testing

 

Introduction

It is evident that the treatment of channels and Froude’s work on ships were conducted in the same way as one might explore some new territory by the obvious method of going to see what is there. At the very least, in science, some means of exploration that was a little less dependent on trial was needed. Froude had shown that for the troublesome problem of ship resistance where there is viscous friction and wave-making operating together there is an underlying order so there was every reason to expect to find order throughout the science of fluid flow. To some extent there was already some order in what has become known as the physical properties of fluids, for example, density, viscosity and surface tension, and these had been expressed in terms of more fundamental units of mass, length and time. It was the relationship between these physical properties and the spatial properties of the system like length and diameter and speed that was not clear.

 

It must have been apparent that a whole science was growing up that was centred on the flow of water and air round and through all sorts of solid bodies, for example ships and channels. That science would depend on the intelligent use of Newton’s laws, mathematics and experiment and Froude had shown that all three used together were very powerful indeed. One might add that attempts to use experiment alone have not produced a coherent body of knowledge[1] and nor have attempts to use mathematics alone.[2] 

 

I have described Rayleigh’s contribution to this process in Chapter 7 pages 8 to 10. Now I want to explain dimensional analysis as a fall-out from Rayleigh.

 

Dimensional analysis

The first thing to accept is that, in order to use dimensional analysis you must have a good idea how the system that you want to investigate actually works. The method is reactive and not proactive. It is best explained by example.

 

The resistance to motion of a ship is the most recent example of fluid flow in this text. Let me use that as my example.  The first thing to do is to write down all the physical quantities that are relevant to ship resistance. In order to do so we have to be clear how the resistance is actually created.

 

The resistance is due in part to wave-making and in part to friction between the hull and the water. The resistance caused by wave-making is dependent on the additional pressure created over the hull by the raising of waves above the normal level of the free surface. This additional pressure depends on the size and speed of the ship, on the density of water and on the acceleration due to gravity and it acts on some unspecified area. The skin drag will depend on the speed and the wetted area and on the density and the coefficient of viscosity of the water.

 

We can now easily list some of these quantities. They are speed , acceleration due to gravity , density  and viscosity . This leaves a set of areas of the hull and the length of the hull and it is not obvious how these might be included. For the purposes of dimensional analysis it is supposed that the shape of the hull can be determined from one dimension just as archaeologists can draw a medieval ship from one dimension of the rudder. The dimension that is normally chosen is a typical length like the overall length .

 

Then we could say that the resistance to motion  depends in some way on . You could write that  where  stands for “some function” that will have to be determined by experiment. The terms in the bracket are not multiplied together; they are just a list. They look to be very daunting because we now have to think of a way to set up the experiments. Experiments often throw up lots of experimental data and the experiment is wasted if the data is not stored in some way that makes it available for use in the future. Dimensional analysis can help us to group these variables in recognised ways and perhaps to go on to devise experiments that will lead to a satisfactory method of data storage.

 

I have pointed out that the function of this group of variables must be found by experiment and it is normal to record the function in a graph or family of graphs. Then a graph like, say, Froude’s graph for the frictional resistance of a board when towed edgewise is such a graph. There the graph can be represented by a polynomial expression to considerable advantage. But the graph of the resistance of the whole ship against speed cannot be represented by a single polynomial. However it is possible to represent it by lots of polynomials added together to give the correct shape. This fact permits us to make progress with our dimensional analysis by writing:-

                            In this series   is just a term that is typical of the many terms in this series.

 

Now this equation is part of a description of a physical relationship between well-known quantities and it must be in consistent units so that, whatever the combination of the variables may be, they reduce in the end to those of force. Furthermore every physical quantity can be expressed in some consistent system of units but, more usefully to us, it can also be expressed in fundamental units of mass, length and time. I will give some examples :-

Length

Acceleration

Time

Force

Mass

Density

Velocity

Viscosity

 

Now the equation for the typical term can be expressed in these fundamental units. I have shown it in the first line of the table below. As the equation has every physical quantity given to some power, equations can be drawn up for the sums of these indices. We can have up to three equations, one each for mass, length and time but one might add other dimensions such as temperature for, say, heat transfer. I have put the three equations into a table because this is just a manipulation exercise that can go wrong very

 

 

=

  1

=

  

 

 

 

 

  1

=

 

  -2

=

 

 

 -

easily. At least in this tabular form it is easy to check the equations in their columns before you proceed. I use the same layout for longhand.

 

 

 

Now we can simplify.

                                              

         

We can now go back and re-write the typical term :-

         

This says that the resistance is equal to  times a function of Reynolds number and the Froude number. The group  is, in fact the product of a pressure and an area, the group  being the pressure that is more recognisable as the stagnation pressure , and  is the area. It looks quite exciting but if you ask whether this has improved our position relative to our knowledge of ship resistance it is hard to see what has changed.

 

Dimensional analysis has sorted out the variables but we are no nearer to designing an experiment to determine the function . However if you follow Froude’s argument we can see that we might write :-

          and then  and  and, if you take the line that some espouse, that this separation is not justified, this change does not makes it any more legitimate. Froude chose an experimental plan that was correct in the event whatever objections there may be.

 

So we have a way to sort out relevant physical quantities but it is not a sure-fire method and must be used with care. Try doing a dimensional analysis for a sharp-edged orifice in the base of a cylindrical tank to see what I mean. This dimensional analysis is just another tool in the locker of those constructing an experimental science. It has its greatest value for pipes, as we have seen, for objects over which air flows steadily like aerofoils and objects that move underwater like torpedoes and hydrofoils.

 

Now there is a need for some examples to practice on.

 

Tutorial examples on dimensional analysis

The following examples have the main objective of giving practice in the manipulative phase of dimensional analysis.  None of the examples involves a decision on the relevant physical quantities in any of the given physical system and the combinations chosen may not be complete. The style of the questions shows to some extent the way in which dimensional analysis is applied.

1.  This question is about the orifice tank that is described in the laboratory sheets at the end of this whole text. It is evident that the principal physical quantities involved are, the orifice diameter, , the head above the orifice , the density of the fluid , its viscosity , and the acceleration due to gravity g.  If we take account of the diameter  of the tank we can say that the flow from the tank

                                                    

Show that :-                    .

Compare this result with the rational expression for the steady flow through an orifice.

2.  When a ship moves in water there is a resistance to motion.  This resistance may be attributed to two main effects, the viscous force between the underwater part of the ship and the water, and the disturbance of the free surface that causes waves which carry energy away from the vicinity of the ship.  We consider this system by the method of dimensions by supposing that the resistance to motion  depends on :-

(i)     the shape and size of the ship which we represent by one dimension, its length ,

(ii)      the speed of the ship ;

(iii)     the viscosity of the water :

(iv)   the density :

(v)    the acceleration due to gravity , because the energy of the wave is

stored in the gravitational field.

Show that :-                              .

3.  Suppose that a sphere of radius  was allowed to fall through a fluid at rest which has density  and viscosity .  There would be a phase during which the sphere would be accelerating to its terminal velocity. We could say that, at some instant during this phase, when the velocity is  and the acceleration is , the drag is given by :-

                                            

Text Box: 4   An aeroplane wing has a special shape to its cross-section called an aerofoil section. When this shape is made to move through the air at a steady speed , a force is produced on the wing. This force is clearly dependent on the size of the wing, that is, on the length of the aerofoil  and on the span of the wing . It also depends on the angle  that the aerofoil makes with the direction of motion. Then the force might be considered to be a function of  where  are the density and viscosity of air.

Show that:-                        

 

Text Box: 5  A dashpot is a device that is used to provide a resistance to motion. It is often used as a damper, that is a device to damp out unwanted oscillations as in a vehicle suspension system. The essential features of a dashpot are, a cylinder that is filled with oil and a loose-fitting piston that moves coaxially with the cylinder in the oil. When an axial force is applied to the piston, oil is made to flow through the annular gap between the piston and the cylinder and a pressure difference develops between the two sides of the piston and a net force on the piston.

 

It may be argued that the force  on the piston is a function of the diameter  of the piston, the length  of the piston, the annular gap , the speed of the piston  and, of course, the viscosity  and the density  of the oil. Show that :-

                                       .

 

Text Box: 6   Figure 6 represents a journal bearing. Such bearings are normally used in pairs to support a loaded shaft that is required to rotate. The shaft is plain, that is, it is cylindrical and it runs in two bearing brasses that together form a cylindrical sleeve round the shaft. The brasses are supported and retained by the bearing block. These bearings are fed continuously with oil that fills the space between the shaft and the brasses. This space is nominally specified as a radial clearance  but in fact the shaft does not run coaxially with the bearing. It runs with some eccentricity  that leads to a wedge action in the oil thay separates the shaft and the brasses to produce only viscous friction and no solid friction.

 

It is clear that there must be some relationship between the load  that can be supported by the bearing and the shaft diameter , the length of the bearing , the viscosity , the eccentricity , the speed of rotation  and the radial clearance .

Show that :-                  

 

 

 

 

 

 

Solutions to examples on dimensional analysis

 

 

The relationship between dimensional analysis and rational expressions

If you have worked through the examples above or even if you just read the questions you must have realised that dimensional analysis did not change the world and make everything in fluid flow simple. Look at question 1 on the orifice tank. The outcome of the dimensional analysis is scarcely recognisable when compared with the rational expression that we normally use. In truth if the rational expression had not existed we would have had to create it from the result of the dimensional analysis in order to make progress. I do not think that the result for ship resistance has much to commend it over Froude’s method. In fact when you look at the uses that have been found for the Froude number they look as though they were “forced” uses and did not come naturally from some analysis or experimental plan.

 

The fact is that the important fields of fluid flow like pipes, channels, sewerage, ships, lubrication and hydraulic machines had all been sorted out quite satisfactorily before dimensional analysis came along. All this experience was not going to be replaced by new methods based on non-dimensional groups unless there was some very significant advantage. The great fields of activity that had yet to develop when dimensional analysis appeared was aerodynamics and model testing and, as luck would have it, they are the ones that are most suited to the use of non-dimensional numbers.

 

We now have non-dimensional groups in profusion all over science and it is hard to escape the thought that some have been devised to ensure that someone’s name will survive into the future. In my view one should always start to think about something that is new to you by deciding how it works. Dimensional analysis will not tell you nor will mathematics. When I start, I follow a golden rule: “look for the forces”. It always works where momentum and the like just hides the real modus operandi.

 

However dimensional analysis should not be ignored. It is always there and might fit in with what you are doing. But do not fall for the idea that only mathematics is acceptable and that somehow data that has been gathered by experience and experiment and then systematised for storage and retrieval is second best waiting only for something better to be devised. It is almost certainly correct and have stood the test of time and is available in engineering handbooks now and not some time in the future.

 

The juxtaposition of dimensional analysis and rational expressions leads to examination questions that may or may not have some value to the practising engineer. I give some here with their solutions.

 

Tutorial examples on dimensional analysis and rational expressions

1 (a) A sphere of diameter  and surface roughness  moves at a uniform velocity  in a fluid of density  and viscosity . If the resistance to its motion is taken to be some function of  show by the method of dimensions that :-

                                        

 (b)(i) A smooth sphere of material of density  falls at its terminal velocity  through a liquid of density  and of viscosity . If the drag on the sphere is given by the rational expression :-

                                               where  is the projected area of the sphere,  is its velocity,  is the density of the fluid and  is the coefficient of drag show that  :-

                                              

    (ii) A smooth sphere of 30 mm diameter and made of steel falls through water at 10°C and reaches its terminal velocity. Given that, in the range of Reynolds number, defined as , from 1,000 to 100,000,  = 0.5 show that the terminal velocity will be about 2.3 m/s.

 

For steel take  = 7,800 kg/m3 and for the properties of water use the data in the steam tables. 

 

2(a)(i) A sphere of radius  moves in a straight line through a fluid of density  and viscosity . If the drag  exerted on the sphere by the fluid is taken to depend on the velocity  and the acceleration  of the sphere as well as on  show that :-

                                       

       (ii) Measurements are made of the velocity of small steel balls falling vertically in oil and these show that,  at the terminal velocity, the drag is proportional to the product of  and . Show from this observation and the outcome of dimensional analysis above that the drag is proportional to .

 

 (b) A steel ball has a diameter of 1.59 mm (1/16²) and, in oil having density 850 kg/m3 and viscosity 0.188 kg/ms, has a terminal velocity of 51 mm/s. If the drag on the ball is given by the rational expression :-

                                                     Drag =  find the value of .

For steel take  = 7,800 kg/m3.

 

3(i)  When a fluid flows at a steady rate through a long horizontal pipe the force exerted on the water to sustain the flow  must be resisted by a tangential force acting on the inner surface of the pipe. Some would argue that there is a shearing stress  exerted on the wall of the pipe. Show that :-           where  is the pressure drop over a length  of pipe of diameter .

  (ii)  The rational expression due to Darcy for friction loss in pipe is :-

Use this and the expression for  to show that :-      

  (iii) Using dimensional analysis show that for the pipe in (i) :-

           and that  where  are the density and viscosity of the fluid,  is the mean velocity and  is the surface roughness.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solutions to examples on dimensional analysis and rational expressions

 

 

 


[1] Just look at sailing to see what I mean.

[2] Mathematics as it is practiced bears comparison with socialism. In both cases, if it does not work, the practitioners attribute the failure to not applying it rigorously enough and redouble their efforts. The basic premises are the last things to be reconsidered.