No dynamical equation depending on the quantities mentioned earlier can exist which is not included in (3). For the purposes of comparison of resistances it has been found convenient by the
aerodynamical laboratories to tabulate the value of F for various
bodies and to use the symbol k D for it. Equation (3) may then be written alternatively as
and in this form several points of importance are evident. To make the case specific, consider the resistance of a sphere in air as obtained from a wind-tunnel measurement. If the dimension / be identified
with the diameter d of the sphere it will be noted that -=-% is an
experimentally determinate quantity and from it values of ko are determined. An examination of the dimensions of k will indicate that they are zero ; the coefficient is therefore a pure number and so of international validity. Another method of statement would be to say that the numerical value of k is independent of the system of units used so long as the system is self-consistent. Meas- urements of force may be made in dynes, mass in grammes, lengths in centimetres and time in sees, to meet the standards of the physicist. Alternatively the engineer may use the force unit of Ib. weight, the slug as a unit of mass, the foot for length and the sec. for time, or, if he prefers it, the force unit of poundal, the mass unit of pound, and the foot and second. In all cases the tabulated values of k D would be identical. There are further advantages of the system; k D is independent of the air density and for most aeronautical purposes almost independent of size and speed, so that comparison between model and full scale is readily made by comparison of the corre- sponding values of k D . The extent to which the two agree is a meas- ure of the utility of experiments on models.
Equation (4) also shows that k D depends on a single variable
not separately on v, I or v. On theoretical grounds alone therefore we may say for our special assumptions that k D will not change if the velocity of the same body be doubled in a fluid having twice the viscosity. The kinematic viscosity of air is 12 or 13 times that of water at ordinary temperatures and hence the resistance coefficient will be the same if the velocity of air be 12 or 13 times that of water. Stanton has shown that this is true for smooth and rough pipes by testing with the two fluids in the same apparatus. 1 The law was used in the calibration of the pitot-static pressure tube.
may be kept constant in many other ways; if air be the fluid
used in two experiments, then v and / may vary so long as the product is constant. A model aeroplane to one-tenth scale would give a resistance coefficient on test equal to that on the aeroplane at one- tenth the speed. Since the speeds of flight reach 200 ft. per sec. this law is inapplicable to the complete aeroplane, for compressibility of the air would be very important in the model test at 2,000 ft. per second. In testing streamline struts or wires, it is easily possible to make models larger than the reality and so to extend the equivalent speed from that of the wind tunnel to that of flight.
It should be noted, however, that failure to satisfy the law of corresponding speeds, i.e. t)/ = constant, does not necessarily imply failure to obtain similarity of flow between model and full scale. In most of the experiments known to us, resistance varies very closely as the square of the speed and the hypothesis that an exact law existed is worth examination.
Since R varies as v* it follows from (4) that k D is independent of
v and further that fa> must then be a constant for all values of
In such a case the law of corresponding speeds is of no importance, for kn can be deduced from a test at any speed on any size of body. It needs little effort to see that if R varies a little from proportion- ality to f 2 the motions in model and full scale will be nearly similar
and that the function is relatively unimportant. It is on this
variation from strict theory that aeronautics depends in many applications of model results. Since there is no absolute theoretical sanction except in the case of corresponding speeds, the identity of the values of k D on the model and full scale must be tried out in a sufficiently large number of typical cases if reliability is to be estab- lished. This has in effect been done for aeroplane wings.
It is exceedingly difficult to determine from flight experiments the resistance of the wings of an aeroplane, for the flying apparatus must be complete with body, undercarriage, airscrew and engine, all of which materially affect the resistance of an aeroplane. The com- parison of the pressures at chosen points on an aeroplane wing in flight and on a model of it in a wind tunnel is far less difficult and has been made. 2 The theory which led to equation (4) leads also to the conclusion that the pressure divided by air density and square of
1 Advisory Committee for Aeronautics, 1911-2, p. 41.
2 Report, Scale Effect Sub-Committee, A.C.A., 1917-8, R and M, No. 374.
speed is a function only of -- Special photographic anemometers 3
were made by the Royal Aircraft Establishment for use in flight and the pressures over a section of the upper and lower wings of a biplane were measured.
The types of variation of pressure on the full scale are faithfully reproduced by the model and in three of the four comparisons the actual numerical agreement is complete within the accuracy of measurement. The difference on the fourth comparison has not been explained and some doubt exists as to its reality. Repetition of the experiments has not yet been made. Generally, however, it is clear that in heavier-than-air craft the use of models is amply justified. For airships the lack of full-scale experiment precludes any statement of value.
In the course of the investigations of the variations of &D with speed and size it was found that changes of appreciable magnitude occurred at the lower speeds of wind tunnels but that the values tended to a limit. It is the value of ko at the limit of capacity of wind tunnels which is taken in default of correcting factors determined from a comparison between full-scale and model experiments. On the score of cost it is not practicable to increase the size of wind channel or the speed of the wind indefinitely and the highest value of v I appears to be obtained most economically by large size rather than high speed. There are some other advantages of size; the com- pleteness of detail possible increases with the size of model and one of the claims in favour of the large 7 ft. x 14 ft. channel at the Nation- al Physical Laboratory is that the model will be so large that an air- screw can be fitted to it and the combination of airscrew and aero- plane tested under conditions very closely resembling those in flight.
The Effect of Compressibility on the Motion of Air. The law of
corresponding speeds expressed by the relation = constant is
peculiar to the assumptions made in obtaining (4) as to the experi- mental factors which have appreciable effects on resistance. There is an indefinitely large number of laws of corresponding speeds, each law being applicable under limited conditions. The method of find- ing the appropriate law is clear; the process begins with a statement of the physical quantities and measurements involved and concludes when an equation of the correct dimensions has been found. The conditions may be so complex that the answer, when obtained, is of little value; in general the theory of dynamical similarity is useful only when the number of important variables is less than five.
The difficulty here indicated can be seen, if, instead of limiting the problem to a fluid characterized by density and viscosity only, an extra property defining its compressibility is included. There are various ways of expressing compressibility and the most obvious would be through an elasticity modulus. Density is included already in the properties considered, and the velocity of sound in a fluid is determined by the ratio of the modulus of elasticity to the density. It has then come to be usual to assume that the velocity of sound a is a convenient variable when investigating the effects of compres- sibility of a fluid on the resistance to the motion of a body through it.
The equivalent to equation (2) for the extended problem is
R=/, (p, I, v, v, a) - (5)
and restricting the form of / to that which satisfies the theory of dimensions R
To satisfy the theoretical conditions which guarantee the con- stancy of k a it is necessary to satisfy simultaneously the equations
v I 11
= constant, =constant
for such variations of size, speed and fluid as are at disposal. Once the fluid is specified, v and a are given and no law of corresponding speeds exists. Various proposals have been made to use a gas such as carbonic acid in one experiment and air in another, but little use appears to have been made of (6) in the form given.
The formula for the pressure due to a pitot tube anemometer (i) is a particular case of (6). That the form of (i) agrees with (6) can be seen by an expansion of the functional operator of the latter
in powers of , using Maclaurin's theorem. Such an expansion will be useful so long as the effect of compressibility is small and the argument small. There is a further simplification in the case of the pitot tube since the resistance does not depend measurably on . From experiments on the issue of steam from the nozzles of
turbines and the measurements of pressure on a shell in flight it appears that in many cases
_5_ , F 2 ( ^\ - (7)
p/V \a J
is a type of formula applicable to the maximum possible pressure on a moving body for speeds ranging from a few in. per sec. to 2,000 ft. per sec. and upwards.
3 Report, A.C.A., R and M, No. 287, p. 504, 1916-7.