Aerodynamics (Lanchester)/Chapter 5

by Frederick William Lanchester

Chapter V.

 

The Aeroplane, the Normal Plane.

 

§ 128. Introductory.—Any material plane, that is to say any thin rigid plate bounded by parallel plane surfaces, when propelled through the air or held stationary in air in motion, experiences a reaction of greater or less magnitude. Any such "plane " is, from the manner of its employment, termed an aeroplane.

Theoretically an aeroplane is regarded as being material and rigid without possessing thickness. In practice, a certain amount of thickness being necessary, the edges may either be cut square, as in the planes employed by the late Professor Langley, in which case an allowance requires to be made for the edge effect, or, the edges may be carefully bevelled and rounded off, so that the aeroplane becomes an equivalent body of streamline form, in which case it is believed that no allowance is required.

The study of the aeroplane may be said to form the elementary basis of experimental aerodynamics as relating to the problem of flight. Whilst laying due stress on this fact, it may be pointed out that the importance of aeroplane study consists in its educational value and its bearing on certain subsidiary problems, rather than in the direct application of the aeroplane to the main function of flight, i.e., the support of the weight. This statement might appear somewhat unexpected, but it may be explained at the outset that the author does not employ the term aeroplane outside its correct signification, that is to say, to denote other than a true or plane aeroplane; the misuse of the word being avoided by the introduction of the term aerofoil,^[1] to denote a supporting member, or organ of sustentation of undefined form. Thus a plane aerofoil is an aeroplane, or a pterygoid aerojoil is an aerofoil of wing-like form.

There are cogent reasons why the aeroplane should take foremost place in the matter of experimental study. It is recognised as essential to the inductive mode of investigation that, whenever possible, one of the conditions of experiment, and one only, should be changed at a time, and it is primarily on this ground that the aeroplane recommends itself. The aeroplane is possessed of a geometrical definiteness that admits of no ambiguity; a specified contour form, making a definite angle with, and presenting a definite aspect to the line of flight, constitutes (for any given velocity) the whole of the factors by which the conditions of experiment are defined. Beyond this there is (skin-friction apart) a certain obvious relationship between the pressure components about the co-ordinate axes, and the angle of flight, that forms a valuable and instructive link in the interpretation of experimental results.

§ 129. Historical.—Our knowledge of the aeroplane to-day is the result of the work of a number of investigators. The exact date at which the study of the subject was seriously taken in hand is in doubt; a certain amount of experimental work on the resistance of bodies in the air is known to have been done early in the eighteenth century, notably by Sir Isaac Newton (1710), and Dr. Desaguliers (1719), whose observations are, however, believed to have been confined to the motion of spherical bodies. Newton also extended his researches to the theoretical study of bodies of different forms in a hypothetical medium (ref. § 2), and showed that the theoretical and experimental results are not altogether out of harmony in spite of the unreal nature of his hypothesis. Newton further attempted to solve the problem of the normal plane in an incompressible continuous medium (Principia, prop, xxxvii. and cors. 7 and 8, prop, xxxvi). These propositions, resting as they do on the supposition of the congealing of portions of the fluid, are known to be unsound, but the results are not without interest.

The next experimental records chronologically are those of Robins (the inventor of the experimental device known as the whirling table), about the middle of the eighteenth century, and Charles Hutton in and about the years 1787–8; whilst among the most recent may be mentioned the systematic researches of S. P. Langley, 1888–90, and the investigations of W. H. Dines of about the same period. An abridged account of the most important of these investigations, with some criticism of the methods and conclusions, is given in a subsequent chapter^[2] devoted to experimental aerodynamics.

§ 130. The Normal Plane.—Law of Pressure.—The simplest case of the aeroplane is that in which the direction of motion through the air is at right angles to its surfaces.

Even under these simple conditions the determination of the pressure-velocity law has not been made without some difficulty, and although the approximate form of the expression, $P$ varies as $V^{2},$ was correctly given by Hutton, Smeaton and others more than a century ago, it is only of recent years that the constant connecting the two sides of the equation has been ascertained with any degree of certainty, and that with a possible error of five per cent, or so. Writing the expression in the form—

$P=k\ V^{2},$

the value of $k$ is variously given by different authorities as from .00166 to .0023 where $P$ is in pounds per square foot, and $V$ is in feet per second.

The experimental basis of the law of the Normal Plane is twofold; tests of wind pressure at known mean velocity, and experiments on the resistance to motion of planes through still air. At first sight there might appear to be no fundamental distinction between these two methods; the difference might be thought to be merely one of relative motion; owing, however, to certain considerations that require to be taken into account, the results obtained by the two methods are strikingly different, and the discrepancy in the value of the constant as given by different writers may be to a certain extent explained.

§ 131. Wind Pressure Determinations.—One of the characteristics of the aerial disturbance which we know as wind is the continual fluctuation both as to direction and velocity; this characteristic is so well known as to have found expression in the vocabulary of every civilised nation—“gust of wind,” “coup de vent,” etc. Wind may be said to consist of a general motion of translation with a superposed motion of turbulence (§ 37), the result being that at no point does the velocity or direction remain constant for any length of time.

One immediate consequence of this variability is that for a wind of known mean velocity $=V,$ the mean value of $V^{2}$ is higher than would be the case if the problem were one of uniform air current having the same mean velocity, and therefore the pressure (which depends upon $V^{2}$) will also be higher. If we neglect the secondary effect due to the components of motion of the air in directions parallel to the pressure plane (§ 146 et seq.), so that the mean pressure on the plane is due only to the normal component of motion of the wind, then it would appear that the pressure will be proportional to the energy per unit volume; for dimensionally:—

	Pressure	$=$ Force $/L^{2}$
	Force	$=$ Energy $/L$
∴	Pressure	$=$ Energy $/L^{3}.$

That is to say, the pressure is proportional to the energy per unit volume.

Now the average energy per unit volume in the wind is the sum of the separate energies of mean velocity and of turbulence (the latter for our present purpose being reckoned only in respect of motion in the direction at right angles to the pressure plane), and in a wind possessing such energy of turbulence, the mean pressure will be greater than would be the case for the simple air current in the proportion that the sum of the energies bears to the energy of mean velocity.

The validity of the above reasoning is unquestionable if we are dealing with a fluid whose properties are those of the Newtonian medium; the energy of turbulence being represented by a variation in the individual velocities of the particles such as will not affect their mean velocity. It is, however, open to question whether it applies rigidly in the case of a fluid possessed of continuity.

Another standpoint from which we may view the present problem is that the turbulence, by effecting a rapid transference of momentum from one part of the fluid to another, acts in effect to augment the apparent viscosity, and in this way adds to the pressure reaction. In any case experiments made on a fixed plane or other body in moving air, cannot be regarded as valid when the conditions are reversed.

Beyond the above there are certain considerations of a practical nature that tend to further invalidate wind pressure measurements as representing aeroplane resistance. It is probable that the maximum pressure on a plane under given wind conditions is not in proportion to the area exposed, and that a small plane is liable to greater extremes, and where a maximum record is made, the absolute area exposed becomes an important factor in determining the pressure per unit.

§ 132. Still Air Determinations.—Under the conditions of experiment in still air, none of the foregoing considerations apply, and it may be safely asserted that the resistance per unit area is approximately proportional to the square of the velocity and is almost independent of the size of the plane. There is some doubt as to the exactitude of the $V^{2}$ law, as in all similar cases of fluid resistance, and it is likely that this doubt will remain until the methods of experiment have undergone refinement; on the one hand, if there is a departure, existing method is too crude to determine its nature, on the other hand it has been shown by Allen (§ 35 et seq.), that where the $V^{2}$ law rigidly applies the resistance is entirely independent of viscosity, a result that would appear to be highly improbable.

It has been proved from the behaviour of projectiles in flight that the $V^{2}$ law breaks down when the velocity of sound is approached, and without doubt this applies also to an aeroplane when a similar velocity is reached. The defect that manifests itself at these high velocities is that the pressure becomes considerably greater than the law would indicate, or as it may be expressed, the value of the index increases, the expression being written: $P=k\ V^{n}.$ On the other hand, at very low velocities at which the influence of viscosity makes itself felt the law becomes modified in the opposite direction, the value of the index diminishes.

It is probable that in actuality these two influences correct one another over a fairly wide range, so that the $V^{2}$ law may become a far closer approximation than would otherwise be the case.

§ 133. Quantitative Data of the Normal Plane.—The following are the generally accepted data of the Normal Plane, the authority being stated where known:—

Wind Pressure.—$P=k\ V^{2}$ where the constant $k=.0023,P$ is in pounds per square foot, and $V$ feet per second.

The value of $k$ given is that usually accepted, and will be found in the majority of text-books, also in the “Encyclopaedia Britannica” under the article on “Wind.” Molesworth, in his “Pocket Book,” gives the figure .002288, but his authority is not disclosed, neither are particulars given of the method by which accuracy has been obtained to so many places of decimals.

Still Air Data.—Form of expression and units as before. Hutton is quoted as giving $k=.0017.$ This result at the time of his experiments (1787–8) must be considered quite remarkable, in view of the fact that one of the most recent determinations, the corrected mean of a great number of experiments made by Professor Langley, exactly a century later, gave an almost identical result.

Langley, in presenting his final result, $k=.00166$ as the corrected mean of his experimental records, states that the possible errors of experiment are such as to leave a probable uncertainty of about 10 per cent. The temperature and pressure corresponding to the above value are given as 10 degrees $C.$ and 736 m.m. mercury; if we reduce to sea level we obtain Hutton's result, $k=.0017,$ almost exactly.

Dines^[3] has shown that the pressure depends not only upon the velocity but also upon the shape or “contour form” of the plane, and that the pressure is least for planes of compact outline, such as a square or circular disc. In his experiments he obtained values for a rectangle 16 inches $\times$ 1 inch greater in the proportion of 8 to 7 than for a square of equal area. The value of $k$ given by Dines for planes of compact form is about 6 per cent, below that of Langley; the latter value is approximately equal to Dines' result for a rectangle of 4 : 1 ratio. This 6 per cent, difference is an actual disagreement. The planes employed by Langley for his determination were of square form.

§ 134. Resistance a Function of Density.—Employment of Absolute and Other Units.—In order that the expression $P=k\ V^{2}$ should be dimensional the constant $k$ must include a quantity of the dimensions ${\frac {m}{l^{3}}}.$ This can be eliminated by introducing the density of the fluid into the expression.

Employing British Absolute Units, let:—

$P=$ pressure in poundals per square foot.

$V=$ velocity feet per second.

$\rho \;=$ density of fluid, lbs. (mass) per cubic foot.

$C=$ constant.

The expression then becomes:—

$P=C\ \rho \ V^{2},$

in which $C=32.2\ k/\rho ,$ and for air^[4] at 10 degrees C. and 760 mm. pressure we have $\rho =.078,$ whence,—

$C={\frac {32.2\times .0017}{.078}}=.7.$

The equation thus becomes:—

$P=.7\ \rho \ V^{2}$

The equation is identically the same in C.G.S. absolute units, and the constant is of the same value; that is to say, $P=$ dynes per square c.m., $V=$ c.m. per second and $\rho$ grammes per cubic c.m.

If we express $P$ in grammes per square c.m., and $V$ in metres per second, and substitute for $\rho$ for air at 10 degrees C., we obtain the equation in the form:—

$P=.009\ V^{2}.$

If the velocity is given in English miles per hour it is sometimes convenient to have the expression in the form:—

$P$ (pounds) $={\frac {V^{2}}{275}}.$

§ 135. Fluids other than Air.— If the whole physical properties of a fluid were represented by the symbols in the equation, or if, the equation being as it is, the fluid were incompressible and of zero viscosity, the constant $C$ would be the same for different fluids.

The experimental determination in the case of sea-water has been made by Captain Beaufoy, and independently by R. E. Froude, the results being in close agreement. In absolute units we have:—

$P=.55\ \rho \ V^{2},$

that is to say, the value of the constant is ${\frac {55}{70}}$ or approximately four-fifths of that in the case of air.

This difference is undoubtedly due to the lower kinematic viscosity of water, which is less than air in the ratio of 1 : 14. The nature of the relationship connecting the function kinematic viscosity and the changes in the value of the constant, is not very clear; the existence of 3uch changes shows the form of the expression to be inexact, for, according to Allen (§§ 35 and 42), under these circumstances the $V^{2}$ law cannot strictly apply. It might, without departing from the form of the expression, be possible to establish an empirical relationship, and it is in any ease of interest to endeavour to ascertain the probable magnitude of the constant for the particular case when viscosity becomes vanishingly small.

There is no fluid known of which it can be said that viscosity is a negligible quantity; neither is it possible to deduce from the data of known fluids what the behaviour of such a fluid would be. We have consequently to fall back on pure theory.

§ 136. Normal Plane Theory Summarised.—Several methods of computing the pressure on a normal plane have been proposed; up to the present none of these can be considered entirely satisfactory.

1. The Method of the Neutonian Medium.—The theory of the Newtonian medium has been already discussed (§ 4); it has been shown that on this hypothesis we have two possible results: (a) if the particles are elastic, $P=2\ \rho \ V^{2};$ (b) if the particles are inelastic, $P=\rho \ V^{2}.$

Both these results are higher than that given by experiment for a viscous fluid, a defect that is due to the faulty hypothesis, the Newtonian medium possessing no continuity. Newton was fully conscious of this fact.

2. The Neutonian Method (Book II., Section VII. prop. xxxvii.).—In this proposition,^[5] Newton arrives at a result for a fluid possessing continuity the equivalent of which is:—

$P=.25\ \rho \ V^{2}$

3. The Torricellian Method is here so named merely as a matter of convenience as being based on the Torricellian principle, and not as due to Torricelli himself.

In a continuous fluid, the theorem of Torricelli, which is founded on the Principle of Work, shows that a given pressure is capable of generating a certain definite velocity in the fluid; thus, representing the pressure of the fluid by its hydrostatic head, the latter gives the height through which a body would require to fall in order to acquire the corresponding velocity. If we arrogate that the converse is true, i.e., that a given velocity is capable of generating the corresponding pressure, and that the conditions present in the case of the Normal Plane are such that this corresponding pressure will be generated, then we obtain the result: $P=.5\ \rho \ V^{2},$ for, if $s=$ “head,” we have: $s={\frac {V^{2}}{2\ g}}$ and mass whose weight constitutes pressure $=p\ s,$ or pressure $=g\ \rho \ s=g\ \rho \ {\frac {V^{2}}{2\ g}}=.5\ \rho \ V^{2}.$

4. The Helmholtz Kirchhof Method.—This method is based on the theory of Discontinuous Motion (Chap. III., § 97); the solution is only known in the case of a lamina bounded by parallel lines of infinite length; in this case the expression is: $P=44\ \rho \ V^{2}.$

The Helmholtz theory of discontinuous motion is in all probability the correct theory of the fluid whose viscosity is vanishingly small; and the above result may therefore be taken as rigidly accurate. It is unfortunate that the mathematical difficulties of this method have only been overcome in a few isolated cases.

To summarise, we have:—

(1)	Newtonian medium (a) elastic particles	$C=2$	All shapes.
	Newtonian medium (b) inelastic particles	$C=1$	’’      ’’
(2)	Newtonian method, prop, xxxvii.	$C=.25$	’’      ’’
(3)	Torricellian method	$C=.5$	’’      ’’
(4)	Helmholtz-Kirchhoff	$C=.440$	infinite lamina.

And experimental determinations in ordinary viscous fluids as follows:—

Air	(Hutton)	$C=.7$	Shape not stated.
’’	(Langley)	$C=.7$	Square planes.
’’	(Dines)	$C=.66$	Circular and other “compact” forms.
’’	(Dines)	$C=.76$	Rectangular lamina 16:1 ratio.
Water	(Beaufoy, Froude)	$C=.55$	Shape not stated.

§ 137. Deductions from Comparison of Theory and Experiment.— The method of the Newton medium may he dismissed on the grounds of faulty hypothesis; the prop, xxxvii. method may be discarded as being certainly unsound; the Torricellian method is based on a tacit assumption that the fluid in proximity to the front face of the plane is destitute of velocity, which we know is not true, except at one point or on one line. The Helmholtz method alone stands on a scientific basis, and at present this gives a result in but one special case.

We are in want of data; let us assume data and develop the method. The results can be corrected for more reliable figures when such have been ascertained.

Data assumed:—

Helmholtz' result for infinite lamina	$C=.44.$
Dines' determination for plane 16 inches $\times$ 1 inch assumed as for infinite lamina	$C=.76$	(Air).
Beaufoy's result augmented 10 per cent. for infinite lamina. (Water)	$C=.60.$

These values are plotted in Fig. 87, in which abscissae represent viscosity (kinematic), and ordinates values of $C$; that is, where $V$ is constant, ordinates are proportional to kinematic pressure.

Drawing tangents to this curve at $a$ and $b$, we can deduce the values of the indices $q$ and $r$ in the general equation of §§ 35 and 42,—

$R=c\ \nu ^{q}\ l^{r}\ V^{r}.$

The result given by the curve as drawn is as follows:

Water: $R=c\ \nu ^{.07}\ l^{1.93}\ V^{1.93}.$

Air: $R=c\ \nu ^{.1}\ l^{1.9}\ V^{1.9}.$

These figures must be regarded as a mere illustration. Not only are the data unreliable but there is considerable doubt attaching to the accuracy of a curve drawn through three points only. Observations are wanted made with planes of one standard size and shape, and at a standard velocity in fluids of different $\nu$ value, in order that this indirect method of estimating the index values should be really effective.
Fig. 87.

§ 138. The Nature of the Pressure Reaction.—The resistance experienced by an aeroplane in motion is due to the difference of pressure between its anterior and posterior faces, and it is the integration of this difference into the area that has hitherto been referred to as the pressure on the plane and denoted by the symbol $P.$

By the theory of Helmholtz, the pressure difference in a fluid of zero viscosity is entirely due to the excess of pressure on the front face, the “dead-water” being supposed to carry the ordinary hydrostatic pressure of the fluid.

In real fluids there is a viscous drag at the surface of discontinuity, or stratum of turbulence, across which a continuous communication of momentum takes place. This constitutes a force acting rearward on the dead-water as a whole, the reaction of which appears as a region of decreased pressure (a partial vacuum) on the rear face of the plane.

Dines^[6] has investigated this point experimentally, and has found that for a one foot square plane in air at 60 m/h., the deficit of pressure on the rear face is approximately one-half the excess pressure on the front of the plane, the measurement being made in the centre of the plane. A similar proportion was found to obtain when the pressure on the mouth of a tube was measured pointed towards and away from the relative wind direction.

Lord Kelvin has pointed out (Nature, p. 597, 1894) that the pressure recorded by Dines in the above experiment, i.e., 1.82 inches of water, corresponds exactly to that given by the Torricellian method, that is to say, that the excess pressure that occurs at the centre of a normal plane for any given velocity is that of the corresponding hydrostatic head. This fact is fully consistent with hydrodynamic theory. If the stream lines could be plotted it is evident that at the point on the face of the plane where the stream divides, the velocity of the fluid will be nil, therefore by § 82 the pressure at this point will be in excess of that at a distance away by an amount corresponding to the head due to the relative velocity of the fluid.

We have here a definite proof that the Torricellian method is inapplicable in the determination of the constant $C,$ for at every other point on the face of the plane than that at which the stream divides the fluid is possessed of velocity, and consequently its pressure is less than that given by the calculation on the basis in question.

There may be a small departure from the maximum pressure law due to viscosity, but there is every reason to suppose that in fluids of moderate viscosity such error may be ignored; the motion in advance of the plane may be looked upon as irrotational.

The Helmholtz-Kirchhoff result shows that the distribution of pressure over the front of the plane is fairly uniform over the central part, falling off rapidly near the edges; this is evident from the fact that a maximum of .5 is associated with a mean = .440 in the case of the infinite lamina. In the case of a plane of compact outline it is probable that the Helmholtz hypothesis would give a considerably lower figure, about .40 or somewhat less; the maximum, however, will be the same as for the infinite lamina, so that it may be anticipated in this case the pressure will fall off more rapidly towards the periphery.

§ 139. Theoretical Considerations relating to the Shape of the Plane.—The influence of the shape of the plane is most conveniently studied in the two extreme cases to which we have already directed attention, i.e., the compact form (a square or circular disc) and the parallel strip or infinite lamina. The former is a symmetrical case of three-dimensional motion; in the latter the motion takes place in two dimensions only.

It has been sometimes suggested that since the pressure increases with the relative periphery, the pressure is greatest in the peripheral regions; we have already seen that such is not the case. The true reason is to be found in a complication of causes.

(1) The congestion of fluid that gives rise to the pressure region is less when the fluid can escape laterally in two dimensions than when its “spread” is confined to one dimension.

(2) The “spurting” of the lines of flow past the edges of the plane will be greater when the access of the fluid to the “hinter-land” is the more complete. Thus in an infinite strip of width $=b$ the layers of fluid adjacent to the face of the plane are fed by a much greater stream area than in the case, say, of a circular disc of which $b$ is the diameter, and the spurting past the edge of the plane will be correspondingly the more vigorous; this is represented diagrammatically in Fig. 88. It is evident that the plane that causes the greater displacement of the lines of flow will experience the greater pressure.
Fig. 88.

(3) The viscous drag on the dead water will be greater when the periphery is greater. Thus, the pressure on the rear face of the plane will be less, that is, the vacuum will be greater for planes of elongate or erratic form.

§ 140. Comparison with Efflux Phenomena.—An analogous case illustrating the foregoing principles is to be found in the efflux of fluids under pressure (§§ 95–96). In the case of a jet issuing from a simple circular orifice we have a case of three-dimensional motion, and as the flow takes place inwardly the layers of fluid in the vicinity of the orifice will be fed by a greater stream area than would be the case if the orifice had the form of a slit and the motion in two dimensions; the “spurting” at the edge will therefore be more vigorous and the contraction of the jet will be greater. This is found experimentally to be the case, the coefficient of contraction being usually taken, for the circular aperture, as from .615 to .620, whereas for a slit aperture it is found to be about .635, On the principles discussed in § 95, the greater the jet contraction the less the pressure is relieved on the wall of the vessel in the vicinity of the orifice.

We may follow the comparison further. In the case of the Borda nozzle the access of the fluid to the jet is improved by the arrangement of an inwardly projecting “lip” so that the pressure on the wall of the vessel undergoes next to no reduction, and the coefficient of contraction becomes (theoretically) = .5 or by experiment .515.

By similarly fitting a lip or a projection to the edge of the normal plane, opposed to the relative direction of the wind, its pressure constant can be considerably increased. If the lip be of sufficient height to render motions of the fluid adjacent to the plane itself very small, so that the square of such velocity as it may possess may be everywhere negligible, then the pressure on the face of the plane will, on the hydrodynamic principle already cited (§ 138), be everywhere that due to the Torricellian head, and the pressure constant will be .5; it would appear to be impossible for it to rise above this value.

In viscous fluids there would be doubtless some departure from strict theory, owing to the fact that the fluid in advance of the plane has rotation impressed upon it by viscous stress, and the hydrodynamic principle assumes irrotation; in ordinary fluids the error due to this cause should not be great. Beyond this there is the separate phenomenon of the suction on the back of the plane, which may be regarded as supplying an added constant, the sum of this and the pressure constant making the $C$ of the equation.

§ 141. The auantitative Effect of a Projecting Lip.—For planes of compact outhne. Dines obtained the following results:—

Plane 1 foot diameter, circular.

Projection of lip or rim.			Percentage increase.
$\scriptstyle {\frac {1}{8}}$	inch		6	per cent.
$\scriptstyle {\frac {3}{8}}$	”		10	”
$\scriptstyle {\frac {5}{8}}$	”		14	”

We have stated that the probable value of the pressure constant on the Helmholtz basis for a plane of compact outline is about .40 or somewhat less; this would give a possible augmentation of 25 per cent, or somewhat more (the limit being .5 according to the preceding article); but Dines' result is the percentage on the whole constant and requires to be multiplied by 66/40, so that his figure for a 5/8 inch rim becomes 23 per cent. This result is in harmony with the theory, but would seem to point to the probability of a lower value than .40 for the Helmholtz constant, in view of the probability of higher resistances being experienced with greater depths of rim. It is worthy of remark that Dines obtained, for a hemispherical cup, pressures about 16 per cent, greater than for a plane circular disc.

In general, if we neglect the influence of rotational motion within the stream, let, as before, $C$ be the experimentally ascertained constant for any plane, and $c$, the pressure constant on the Helmholtz hypothesis, and let $C+n\ C$ he the total augmented pressure for the same plane fitted with a deep rim, we shall have the relation:—

$c_{1}=.50\ {\boldsymbol {-}}\ n\ C\quad$ or, $\quad n\ C=.50\ {\boldsymbol {-}}\ c_{1}.$

If we apply this to the two-dimensional case of the infinite strip, we have Kirchhoff's determination of the Helmholtz constant $c_{1}=.440,$ and Dines' experimental result $C=.76,$ so that—

$.76\ n=.50\ {\boldsymbol {-}}\ .44\quad$ or, $\quad n=.08.$

That is to say, the maximum possible addition to the pressure is, in this case, about 8 per cent.

Dealing with this problem in an analytical investigation, Love^[7] has shown that if $\epsilon$ be the ratio of height of lips to breadth of plane, the pressure will be increased approximately by an amount $={\frac {6}{5}}{\sqrt {\epsilon }}$ of that for the same plane without the lips. It is evident that this expression only holds good for small or moderate values of $\epsilon$, for the limiting value would otherwise be exceeded. This would occur when ${\sqrt {\epsilon }}={\frac {5}{6}}\times .08=.0666$  or, $\ \epsilon =.258,$ so that Love's approximate equation will be perceptibly in error for some considerably smaller value.

§ 142. Planes of Intermediate Proportion.—We have so far dealt with the two extreme cases of contour form typical of two- dimensional and three-dimensional motion, the infinite strip, and the plane of compact outline.

Our knowledge of other forms is at present somewhat limited. It may be fairly assumed that, just as the value of the constant is, within the limits of observation, the same for the circular as for the square form, so for an ellipse or other tolerably regular elongate form it will be the same as for a rectangle of like proportions. We will therefore confine our attention in the present section to rectangular planes of different proportions.
Fig. 89.

We have to rely chiefly on the observations of Dines for data. Fig. 89 gives the value of $C$ plotted as a function of the length of the plane in terms of its breadth, the form of the plane being represented graphically by the shaded area. The small circles denote the observation data on which the curve is based. The curve is not carried beyond the ordinate proper to the square plane, as it obviously repeats itself, the corresponding abscissae being in arithmetical and harmonical progression respectively.

For planes of highly irregular form no definite rules can be laid down. An assumption that such planes are built up of simpler components will sometimes enable the value of $C$ to be assessed; but as the whole range of $C$ values lies almost within the admittedly possible allowance for experimental error, our want of knowledge on this point is not so serious as might otherwise appear.

§ 143. Perforated Plates.—Dines has investigated the effect of perforations as affecting the resistance of the normal plane. In one case a plane one foot square was taken and eight circular holes, each one square inch area, were punched, as illustrated in Fig. 90; no difference of pressure could be detected whether any or all of the holes were covered or open. Mr. Dines remarks:
Fig. 90. “The eight holes together take away more than 5 per cent, of the plate, yet a difference of 1 per cent, in the pressure, had it existed, would certainly have been apparent.” Further experiments were made with two kinds of perforated zinc, the one sample, holes .08 inch diameter 77 per square inch, having only 61 per cent, of the total area, was found to give 91 per cent. of the total pressure; another sample with perforations .22 inch diameter, 11 or 12 per square inch, possessing only about 66 per cent, of the total area, gave 80 per cent, of the pressure on a solid plate.

The curve given in Fig. 91 is deduced principally from Dines' experiments. Abscissae give percentage area removed, and ordinates show the corresponding pressure as a percentage of that on the same area intact. It is supposed that when the percentage of area remaining becomes small, the perforations are of square form as indicated.

The anomalous behaviour of the perforated plate is perfectly explicable on theoretical grounds.

The region in the rear of the plane is occupied by the turbulent “dead-water” at a pressure below that of the undisturbed fluid. When a hole is made in the plane, the air flows through from the front to the rear under the influence of the difference of pressure between its two faces. The stream of air finding its way through the perforation carries with it an amount of momentum per second, equal to the force of which the plane is relieved. If there were a conduit to carry this efflux air away without interfering with the dead-water,
Fig. 91. then the plane would show itself relieved of pressure to the extent of the area taken away together with an addendum due to the fall of pressure adjacent to the perforation, in accordance with the well-known principles of efflux theory. But there is no conduit to carry away the efflux air, which consequently passes into and becomes mixed up with the turbulent dead-water; and the efflux air carries with it its momentum, which is communicated to the dead-water, and momentum communicated to the dead-water appears as negative pressure on the rear face of the plane, since it is the plane itself that prevents the dead-water from being washed bodily away. Consequently the vacuum on the rear of the plane is increased to just the same extent as the pressure on the front is diminished, both quantities being measured by their integration over the respective faces of the plane; that is to say, the existence of a perforation has no influence on the total reaction on the plane.

When perforations are made of great size in proportion to the dimensions of the plane, we can conceive of the efflux stream passing en masse through the dead-water without parting with the whole of its momentum, so that in such a case the plane will be relieved of a portion of its resistance. The same may be supposed to happen if the perforations become sufficiently numerous.

Note.—In the present chapter the discussion is based principally on the result of Mr. Dines' investigations, the value of $C$ for the plane of compact form being taken at .66. The author has been influenced in this partly by the fact that in all probability Dines' results are nearer the truth than those of Professor Langley, but more particularly by the consideration that when instituting a comparison it is safer to confine one's attention to the work of a single investigator, and Langley's experiments with the normal plane were not carried far enough to give the information required.

For general employment in the subsequent volume (“Aerodromics”) the value of $C$ is taken as .7, which is the result given by Langley for a plane of square form and corresponds with the result given by Dines for a plane 4 × 1.^[8]

In adopting this value it has been borne in mind that it is desirable to have a general average figure that can be used with safety without specifying the exact form of the plane, and, taking $C$ as .7, it will not matter seriously whether Langley's or Dines' result should ultimately prove to be the nearer to the truth.

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1. From Gr. ἀέρος and φυλλον (lit. an air-leaf).
2. Chap. X.
3. Quarterly Journal, Royal Met. Soc., Vol. XV., No. 72, October, 1889.
4. From the determination of Regnault.
5. See § 129.
6. “On the Variations of Pressure caused by the Wind blowing across the Mouth of a Tube,” Quarterly Journal, Royal Met. Soc. XVI., No. 76, October, 1890.
7. “Theory of Discontinuous Fluid Motion,” Proc. Camb. Phil. Soc, VII., 1891.
8. An erratum published in Volume 2 has been applied: "P. 199, line 7 from foot, for '4 × I' read '4 × 1.'" (Wikisource contributor note)