RESULTADO: METODO BIOETICO PARA EL ABORDAJE DE LOS PROBLEMAS AMBIENTALES

We have already seen that, for the solution of many problems in the mechan- ics of fluids, experimental work is required, and that such work is frequently done with models of the prototype. To take a well-known example, the development of an aircraft is based on experiments made with small models of the aircraft held in a wind-tunnel. Not only would tests and subsequent modifications on an actual aircraft prove too costly: there could be consid- erable danger to human life. The tests on the model, however, will have little relevance to the prototype unless they are carried out under conditions in which the flow of air round the model is dynamically similar to the flow round the prototype.

Consider the testing of an aircraft model in a wind-tunnel in order to find the force F on the prototype. Here the force F might represent the overall force on the aircraft, or it might represent either the lift force,

L, or drag force, D, which are components of F. The model and proto-

type are, of course, geometrically similar. Viscous and inertia forces are involved; gravity forces, however, may be disregarded. This is because the fluid concerned – air – is of small density, and the work done by grav- ity on the moving air is negligible. Surface tension does not concern us because there is no interface between a liquid and another fluid, and we shall assume that the aircraft is a low-speed type so that effects of com- pressibility are negligible. Apart from the inevitable pressure forces, then, only viscous and inertia forces are involved, and thus the relevant dimen- sionless parameter appearing as an independent variable is the Reynolds number. The usual processes of dimensional analysis, in fact, yield the

180 Physical similarity and dimensional analysis result F u2_l2 = φ _ul µ  = φ(Re) (5.16)

where and µ respectively represent the density and dynamic viscosity of the fluid used, u represents the relative velocity between the aircraft (prototype or model) and the fluid at some distance from it, and l some characteristic length – the wing span, for instance.

Now eqn 5.16 is true for both model and prototype. For true dynamic similarity between model and prototype, however, the ratio of force mag- nitudes at corresponding points must be the same for both, and so, for this case, the Reynolds number must be the same. Consequently, the function

φ(Re) has the same value for both prototype and model and so F/u2_l2_is

the same in each case. Using suffix m for the model and suffix p for the prototype, we may write

Fp pu2plp2

= Fm mu2mlm2

(5.17) This result is valid only if the test on the model is carried out under such conditions that the Reynolds number is the same as for the prototype. Then

mumlm µm = puplp µp and so um= up _l p lm  _ p m  _µ m µp  (5.18) The velocity umfor which this is true is known as the corresponding velocity.

Only when the model is tested at the corresponding velocity are the flow patterns about the model and prototype exactly similar.

Although the corresponding velocity ummust be determined by equating

the Reynolds number (or other relevant parameter) for model and prototype, there may be other considerations that limit the range of um. When a model

aircraft is tested, the size of the model is naturally less than that of the prototype. In other words, lp/lm> 1.0. If the model is tested in the same fluid

(atmospheric air) as is used for the prototype we have_m= pandµm= µp

Thus, from eqn 5.18, um= up(lp/lm) and since lp/lm> 1, umis larger than up. Even for a prototype intended to fly at only 300 km· h−1, for example,

a model constructed to one-fifth scale would have to be tested with an air speed of 300× 5 = 1500 km · h−1. At velocities as high as this, the effects of the compressibility of air become very important and the pattern of flow round the model will be quite different from that round the prototype, even though the Reynolds number is kept the same. With present-day high-speed aircraft, such difficulties in the testing of models are of course accentuated, and special wind tunnels have been built to address the problem.

One way of obtaining a sufficiently high Reynolds number without using inconveniently high velocities is to test the model in air of higher density. In the example just considered, if the air were compressed to a density five times that of the atmosphere, then um = up(5)(1/5)(1/1) = up, that is,

model and prototype could be tested at the same relative air velocity. (Since the temperature of the compressed air would not depart greatly from that of

The application of dynamic similarity 181

atmospheric air, the dynamic viscosityµ would not be significantly different for model and prototype. The sonic velocity a = (γ p/) = (γ RT) would also be unchanged. Undesirable compressibility effects could thus be avoided.) Such a procedure of course involves much more complicated apparatus than a wind-tunnel using atmospheric air. If such a test facility is not available, complete similarity has to be sacrificed and a compromise solution sought. It is in fact possible in this instance to extrapolate the results of a test at moderate velocity to the higher Reynolds number encountered with the prototype. The basis of the extrapolation is that the relationship between the Reynolds number and the forces involved has been obtained from experiments in which other models have been compared with their prototypes.

For dynamic similarity in cases where the effects of compressibility are important for the prototype (as for a higher-speed aircraft, for example) the Mach numbers also must be identical (Section 5.3.4). For complete similarity, therefore, we require

mumlm µm =

puplp µp

(equality of Reynolds number) and

um am =

up ap

(equality of Mach number) For both conditions to be satisfied simultaneously

_l p lm  _ p m  _µ m µp  =um up = am ap =  Km/m Kp/p 1/2

Unfortunately, the range of values of, µ and K for available fluids is limited, and no worthwhile size ratio(lp/lm) can be achieved. The compressibility

phenomena, however, may readily be made similar by using the same Mach number for both model and prototype. This condition imposes no restriction on the scale of the model, since no characteristic length is involved in the Mach number. If the same fluid at the same temperature and pressure is used for both model and prototype, then Km= Kpandm= p(at correspond-

ing points) and so am= apand um= up. Thus the model has to be tested at

the same speed as the prototype no matter what its size. In a test carried out under these conditions the Reynolds number would not be equal to that for the prototype, and the viscous forces would consequently be out of scale. Fortunately, in the circumstances considered here, the viscous forces may be regarded as having only a secondary effect, and may be allowed to deviate from the values required for complete similarity. It has been found in practice that, for most purposes, eqn 5.17 is an adequate approximation, even though

umis not the corresponding velocity giving equality of Reynolds number.

The important point is this. If forces of only one kind are significant, apart from inertia and pressure forces, then complete dynamic similarity is achieved simply by making the values of the appropriate dimensionless parameter (Reynolds, Froude, Weber or Mach number) the same for model

182 Physical similarity and dimensional analysis

and prototype. Where more than one such parameter is relevant it may still be possible to achieve complete similarity, but it is usually necessary to depart from it. It is essential that these departures be justified. For example, those forces for which dynamic similarity is not achieved must be known to have only a small influence, or an influence that does not change markedly with an alteration in the value of the appropriate dimensionless parameter. If possible, corrections to compensate for these departures from complete similarity should be made. An example of such a procedure will be given in Section 5.8.

One further precaution in connection with the testing of models should not be overlooked. It may happen that forces having a negligible effect on the prototype do materially affect the behaviour of the model. For example, surface tension has a negligible effect on the flow in rivers, but if a model of a river is of small scale the surface tension forces may have a marked effect. In other words, the Weber number may have significance for the model although it may safely be disregarded for the prototype. The result of this kind of departure from complete similarity – an effect negligible for the prototype being significant in the model (or vice versa) – is known as a

scale effect. The roughness of the surface of solid boundaries frequently gives

rise to another scale effect. Even more serious is a discrepancy of Reynolds number whereby laminar flow exists in the model system although the flow in the prototype is turbulent. Scale effects are minimized by using models that do not differ in size from the prototype more than necessary.