Aerodynamics (Lanchester)/Chapter 8

by Frederick William Lanchester

Chapter VIII.

 

The Aerofoil.^[1]

 

§ 172. Introductory.—At some future period it may be found possible to rationalise the treatment of the theory concerned with the form of the aerofoil on a comprehensive basis, so that the sectional form at every point shall be correlated to the pressure reaction and the strength of the cyclic disturbance. At present we are compelled to take our stand on a simplified and somewhat conventional hypothesis.

In the case of the aeroplane, in respect of which a certain amount of experimental data is available, we can at once proceed to apply the fundamental propositions of the preceding chapter, to determine the angle of least resistance, thus:—

Let, as before (§ 163), $x+y$ be the total resistance in the line of flight, where $x$ is the direct resistance (due to skin friction, etc.) and $y$ that due to work expended dynamically.

Then the condition of least resistance is that $x=y.$

Now $x=\xi \ A\ P_{90},$ and $y=\beta \ W=c\ \beta ^{2}\ A\ P_{90}$ (for small values of $\beta$), or $\xi =c\ \beta ^{2}$, that is, $\beta ={\sqrt {\frac {\xi }{c}}}$ where $\beta$ is the angle of inclination in radians; $\xi$ is the coefficient of skin friction (§ 157), and $c$ is the constant according to § 159.

If $\beta ^{\circ }$ be the value of the angle expressed in degrees, the expression becomes $\beta _{c}={\frac {180}{\pi }}\ {\sqrt {\frac {\xi }{c}}}.$

Taking for example the case of a square plane for which the value of $c$ is 2, and taking $\xi =$ .02, we have—

$\beta ^{\circ }={\frac {180}{3.14}}\times {\sqrt {\frac {.02}{2}}}=5.7^{\circ }$ approximately.

A plane of elongate form in pterygoid aspect whose value of $c$ is = 3 would thus have an angle of least resistance of slightly over 4${\tfrac {3}{4}}^{\circ }.$ This is about the minimum value that would in the ordinary way be obtained, assuming that correct values^[2] have been assigned to $c$ and $\xi .$

When we have to deal with an aerofoil of curvilinear section adapted to the form of the lines of flow, we may obtain useful results by adopting the hypothesis of constant sweep (§ 160). According to this hypothesis it is assumed that the support is derived from a layer or stratum of fluid uniformly acted on by the aerofoil, and whose cross-sectional area is constant. This area, for a given plan-form of aerofoil in stated aspect, is equal to the aerofoil area $A$ multiplied by the constant $\kappa$, or, as given in § 160, we have, sweep $=k\ A.$

It will be further assumed that the relation ${\frac {\alpha }{\beta }}$ (§ 161) is constant for any given plan-form and aspect.

§ 173. The Pterygoid Aerofoil. Best Value of $\beta$—.

Let	$\epsilon =$	${\frac {\alpha }{\beta }}$ and, as before,
„	$A=$	aerofoil area,
„	$\kappa A=$	sweep,
„	$\xi =$	coefficient of skin friction.

$C$ is the constant of the normal plane (§ 136).

Now the direct resistance $x=\xi \ A\ C\ \rho \ V^{2},$ and the aerodynamic resistance $y$ is equal to the energy expended aerodynamically per second divided by the velocity, or

$y={\frac {1}{2}}\ \rho \ \kappa \ A\ V\times V^{2}\ (\beta ^{2}\ {\boldsymbol {-}}\ \alpha ^{2})\ {\mbox{÷}}\ V$ $={\frac {1}{2}}\ \rho \ \kappa \ A\ V^{2}\ (\beta ^{2}\ {\boldsymbol {-}}\ \alpha ^{2}).$

But condition of least total energy is that $x=y.$ Let $\beta =\beta _{1}$ and $\alpha =\alpha _{1}$ when $x=y.$

	$\xi \ C={\frac {1}{2}}\ \kappa \ (\beta _{1}^{2}-\alpha _{1}^{2})$	(1)
Now,	${\frac {\alpha }{\beta }}=\epsilon ,$  or  $\alpha _{1}^{2}=\beta _{1}^{2}\ \epsilon ^{2}$
or	$(\beta _{1}^{2}-\alpha _{1}^{2})=\beta _{1}^{2}\ (1-\epsilon ^{2})$
(1) becomes	$\beta _{1}^{2}\ (1-\epsilon ^{2})={\frac {2\ \xi \ C}{\kappa }}$
or	$\beta _{1}={\sqrt {\frac {2\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$	(2)
If $\beta$ is expressed in degrees this becomes—
	$\beta _{1}^{\circ }={\frac {180}{\pi }}{\sqrt {\frac {2\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$	(2a)

Of the quantities involved in this expression, $\xi$ and $C$ are known by experiment; $\kappa$ also may be experimentally determined by the method of superposed planes discussed in §§ 154 and 161; the experimental data are, however, at present wanting. For planes and other forms of aerofoil in pterygoid aspect $\kappa$ is a function of the aspect ratio and is greater when the aspect ratio is greater. The form of this function requires to be experimentally determined and plotted as a curve for certain simple geometrical plan-forms such as the ellipse and the rectangle, the co-ordinates to represent respectively the aspect ratio and the corresponding values of $\kappa .$

The quantity $\epsilon$ is also some function of the aspect ratio, and again we are lacking in experimental information. The values subsequently employed for $\kappa .$ and $\epsilon$ are those which the author is in the habit of using, and which are found to give results reasonably near the truth; they have not, however, been determined or verified by any scientific method, and must at present be regarded as open to suspicion.

It has been assumed in the present section that there is no loss of energy incidental to “handling” the fluid other than that due to skin friction. It is in practice possible that there is some unavoidable loss in eddy making by the aerofoil itself; especially is this probable if the angle $\beta$ is considerable. Apart from the practical considerations introduced by the necessity for thickness in the aerofoil, which probably imposes a minimum limit on its fore and aft dimension, there are reasons (which will be discussed hereafter) for supposing that perfect continuity of motion is not possible under the conditions of finite lateral extent. Whatever defect in the theory may he introduced by considerations of this nature may be legitimately ignored at the present stage.^[3]

§ 174. Grliding Angle.—Let $\gamma$ represent, in circular measure, the gliding angle—that is, the angle of flight path at which the force to overcome the resistance is exactly provided for by the component of gravity in the path of flight. It will be assumed that $\gamma$ comes within the definition of a small angle, i.e., $\gamma =\sin \gamma =\tan \gamma$ with sufficient approximation. Then—

$\gamma ={\frac {\mbox{Resistance}}{\mbox{Weigh}}}.$

Now weight $(W)=\rho \ \kappa \ A\ V^{2}\ (\alpha +\beta )$ and resistance comprises—

(1) Aerodynamic resistance $={\frac {\mbox{Energy per sec.}}{\mbox{Velocity}}}$

$={\frac {\rho \ \kappa \ A\ V^{2}\ (\beta ^{2}-\alpha ^{2})}{2}}$

and

(2) Skin frictional $=\xi \ C\ \rho \ A\ V^{2},$ and

(3) Body resistance $C\ \rho \ \alpha \ V^{2}$ where $\alpha$ is a normal plane area to which the body resistance is equivalent.

Now (3) is a superadded resistance with which for the moment we will not concern ourselves, so that we have—

$\gamma ={\frac {\beta ^{2}-\alpha ^{2}}{2\ (\alpha +\beta )}}+{\frac {\xi \ C}{\kappa \ (\alpha +\beta )}}$ $={\frac {(1-\epsilon )}{2}}\ \beta +{\frac {\xi \ C}{(\epsilon +1)}}.$

But we know that for Least Resistance these terms are equal, consequently under the conditions of Least Gliding Angle ($=\gamma _{1}$) we have $\gamma _{1}=(1-\epsilon )\ \beta _{1},$ that is to say, the least gliding angle will be

$\gamma _{1}=(1-\epsilon )\ {\sqrt {\frac {2\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$

If $\gamma _{1}$ be expressed in degrees this will require to be multiplied by ${\frac {180}{\pi }}.$

§ 175. Taking Account of Body Resistance.—The foregoing investigation has included the temporary assumption that the whole direct resistance is constituted by the skin friction of the aerofoil, as in prop. i. of the preceding chapter. We will now take into account the influence of a resistance independent of the surface of the aerofoil, the body resistance or $x_{1}$ of § 165.
Fig. 111.

We know that such a resistance, which may be-represented by an equivalent normal plane, inevitably results in an increase of the gliding angle; also that this increase will be less the lower the velocity, for, according to the equation of the foregoing section (and Prop. V. of § 166), the gliding angle is constant, neglecting body resistance, so long as the aerofoil is properly designed, and does not depend upon the velocity; we may therefore regard the gliding angle as made up of two parts: the part which is constant in respect of velocity, and the part due to body resistance, which varies as the velocity squared, The resistances, and therefore the gliding angles, may be presented in the form of a diagram (Fig. 111), in which abscissae represent velocity and ordinates the gliding angle; the dotted line represents the constant resistance, and the curve (struck from the dotted line as datum to the equation $\gamma \propto V^{2}$) shows the manner in which the resistance increases with the velocity. Values of $V$ and $\gamma$ have been assigned for a supposititious case.

§ 176. Value of $\beta$ and $\gamma$ for Least Horse-power.—By prop, ii., § 164, we know that the condition for least horse-power is—$y=3x,$ when $y=3x$ let $\beta =\beta _{2}.$

Then, following § 173—

${\frac {1}{2}}\ \kappa \ \beta _{2}^{2}\ (1-\epsilon ^{2})=3\ \xi \ C$

or

$\beta _{2}^{2}={\frac {6\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}$

$\beta _{2}={\sqrt {\frac {6\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}.$

A result that otherwise follows from corollary to prop. iii.—

$\beta _{2}={\sqrt {3\ \beta _{1}}}.$

Let $\gamma _{2}=$ gliding angle for least horse-power. Following § 174 we have—

	$\gamma _{2}={\frac {y+x}{W}}$  where  $y=3\ x$
or,	$\gamma _{2}=1{\frac {1}{3}}\ {\frac {W}{y}}=1{\frac {1}{3}}\ {\frac {\beta _{2}^{2}-\alpha ^{2}}{2\ (\beta _{2}+\alpha )}}$
∴	$\gamma _{2}={\frac {2}{3}}\ \beta _{2}\ (1-\epsilon )$
or in terms of $\beta _{1}$—
	$\gamma _{2}={\frac {2}{\sqrt {3}}}\ \beta _{1}\ (1-\epsilon )$
or,	$\gamma _{2}={\frac {2}{\sqrt {3}}}\ \gamma _{1}=1.155\ \gamma _{1}.$ (approx.).

In Fig. 112 the $x$ and $y$ resistances are shown as curves separately and superposed. In the lower portion of the figure

Fig. 112.

abscissae represent velocity and ordinates $\gamma$ values. It is evident that so long as we are confined to the small angle hypothesis the resistances may be thus represented and the sum of the separate $\gamma$ angles will give the resultant angle.

In the upper part of Fig. 112 the $\gamma$ angle is represented graphically, it being supposed that the aerodrome is launched from the point $a$.

Thus, if $a\ b$ represent the gliding angle of least resistance (shown for example as 10°) then $a\ c$ will represent the gliding angle for least power, the angle being 11°.55. If we suppose that two aerodromes are launched simultaneously from the point $a$ (of equal weight and “sail” area and plan-form), the one being designed for least resistance and the other for least power, their respective trajectories will be the two straight lines $a\ b$ and $a\ c,$ and their positions after the lapse of a certain definite time will be given by the points $b$ and $c$ where $a\ b$ is to $a\ c$ in the relation ${\sqrt[{4}]{8}}:1$ (prop, iii., § 164). We may draw a curve $e\ c\ b\ d$ through the points $c$ and $b$, which will represent the position occupied by aerodromes simultaneously launched from the point $a$, for other values of $\beta .$

Now since the angle of least resistance is the minimum gliding angle, the line $a\ b$ will be a tangent to the curve $e\ b\ d$ at the point $b,$ and since the least power expenditure corresponds to the slowest rate of fall, the tangent to the curve at the point $c$ will be horizontal; we have thus defined the character of the curve in question, which represents the simultaneous loci for similar aerodromes of different $\beta$ values.

The existence or otherwise of body resistance does not affect the problem as here presented; it is included in the plotting as one of the resistances that vary as $V^{2}.$

§ 177. The Values of the Constants.—The paucity of reliable data has already been made the subject of comment, and the values of many of the constants here given can only be regarded as rough approximations. To prevent misapprehension on this point, the tabulated figures,, where not considered reliable, have been entitled “plausible values,” and accompanied by a sign of interrogation (?).

It has been demonstrated in § 161 that, on the hypothesis of constant sweep, the constants $C\ c$ and $\kappa$ are related to one another according to the equation—

${\frac {\alpha }{\beta }}={\frac {c\ C}{\kappa }}-1,$

or, employing as before, the symbol $\epsilon$ to denote $\alpha /\beta ,$ we have—

$\epsilon ={\frac {c\ C}{\kappa }}-1,$

that is to say, theory supplies us with a link connecting the whole of the constants involved in the equations of best value of $\beta$ and least value of $\gamma .$

Of the above constants the value of $C$ is known for planes of different aspect ratio from the experiments of Dines, the results being given in the form of a curve in Fig. 89 (Chap. V.). These values tabulated are as follows:—

Table I.

Aspect Ratio "$n.$"	Constant "$C.$"
3 4 5 6 7 8 — 10 — 12	.685 .70 .71 .72 .725 .73 — .74 — .75

"$C.$" is defined by equation—
$P_{90}=C\ \rho \ V^{2}.$

The values of $c$ are less authentically known; they must be regarded at present merely as plausible values^[4]; they have been given plotted for different values of aspect ratio in Figs. 105 and 106, and, tabulated, are as follows:—

Table II.

Aspect Ratio "$n.$"	Constant "$C.$"
3 4 5 6 7 8 — 10 — 12	2.16 (?) 2.27 (?) 2.38 (?) 2.48 (?) 2.55 (?) 2.62 (?) — 2.73 (?) — 2.80 (?)

§ 178. On the Constants $\kappa$ and $\epsilon$.—Of the constant $\kappa$ we know but little with certainty. Langley's experiments with two pairs of planes four inches by fifteen inches superposed (Fig. 102) suggest that for planes whose aspect ratio is about 4 or 5 $\kappa$ has a value of somewhat less than 1. For reasons given in § 161 the actual value is probably somewhat greater than that ascertained experimentally for pairs of superposed planes.

On the value of $\epsilon$ we are entirely without information so far as direct experiment is concerned. If the value of $\kappa$ were known for an aerofoil of given aspect ratio, the value of $\epsilon$ can be obtained from the equation given in the preceding article, i.e., $\epsilon ={\frac {c\ C}{\kappa }}-1.$

We may provisionally assume that $\kappa$ is a function of $n$ and constant in respect of other variables. It is true that we have taken no account of the influence of plan-form, but we may legalise our position in this respect by specifying some standard form such as a rectangle, and leave the onus of drawing up tables of equivalent proportions in any other form to future experimenters.

At present the quantitative data are in so unsatisfactory a state that it is almost unnecessary to specify the precise form to which $n$ values are supposed to relate; we may take it that we are dealing with a rectangular plan-form, and that n denotes the lateral breadth in terms of the fore and aft dimension; thus, for planes in pterygoid aspect $n$ has a value greater than unity, and for planes in apteroid aspect, less.

Now if $\kappa$ is a function of $n$ alone, $\epsilon$ is also a function of $n$ alone, and if we can by experiment or theory establish a form of expression in the one case, the other follows from the equation.

It is evident that the circumstances determining $\epsilon$ are foreign to our present hypothesis, and we shall require to temporarily take our stand outside this hypothesis in order to investigate the question.

Fig. 113.§ 179. An Auxiliary Hypothesis.—Let us suppose an aerofoil represented in plan in Fig. 113 supported in a continuous medium; then if $M_{a}$ be the upward momentum communicated to the air passing between planes represented by the lines $b\ b$ and $b_{1}\ b_{1}$ at the time when its upward momentum is a maximum—that is, when it comes within the direct influence of the aerofoil (the descending field of Chap. IV.); then, assuming a field of force symmetrical about the plane $z\ z,M_{\alpha }$ will also represent the upward momentum communicated to the air in partially arresting its downward motion during recess.

And the air outside the surfaces $b\ b$ and $b_{1}\ b_{1}$ will be receiving upward momentum the whole time it is in the field. Let the sum of this momentum be denoted by $M_{1}.$

Now, since the total residuary momentum must be zero (§§ 5 and 117) the downward momentum remaining in the air between the surfaces $b\ b$ and $b\ b$ is also equal to $M_{1}$, and if $M_{\beta }$ be the downward momentum in the air when it quits the descending field we shall have: $M_{\beta }=M_{\alpha }+M_{1}.$

But according to the main hypothesis we may represent ${\frac {M_{\alpha }}{M_{\beta }}}$ by ${\frac {\alpha }{\beta }},$ that is

$\epsilon ={\frac {M_{\alpha }}{M_{\beta }}}={\frac {M_{\alpha }}{M_{\alpha }+M_{1}}}.$

It remains for us to assess the value of $M_{1}$ in terms of $M^{\beta }.$

Let us suppose that, in a manner analogous to the limitation of the sweep, the air external to the surfaces $b\ b\ b_{1}\ b_{1}$ be represented by the limited region cut off by two further surfaces $c\ c$ and $c_{1}\ c_{1};$ then it is evident that the distance separating these surfaces will be greater the greater the lateral extension of the aerofoil.

Calling the fore and aft dimension of the aerofoil unity, so that its lateral dimension will $=n,$ let us assume that the distance between $b\ b$ and $c\ c$ is proportional to $\kappa ,$ and let it be denoted by $a\ \kappa .$ We have no direct means of testing the accuracy of this assumption; we can only say that it is a reasonable assumption, since the conditions that influence the depth of the layer of air acted upon obviously affect the extent of the disturbance of the fluid in other directions.

Then the upward momentum received by the air at the time of its crossing the plane $z\ z$ is, from considerations of field symmetry, just half the total eventually imparted, that is $={\frac {M_{1}}{2}},$ but we are supposing that the ah- that reaches the plane $z\ z$ between $b\ b$ and $c\ c$ has received momentum pro rata with that within the region bounded by $b\ b\ b_{1}\ b_{1},$ that is to say:—

	$M_{\alpha }\left/{\frac {M_{1}}{2}}=n\right/2\ \alpha \ \kappa ,$
or	$M_{1}=M^{\alpha }\ {\frac {4\ \alpha \ \kappa }{n}}$
that is, by (1)	$\epsilon ={\frac {M_{\alpha }}{M_{\alpha }\ \left(1+{\frac {4\ \alpha \ \kappa }{n}}\right)}}$
or	$\epsilon ={\frac {n}{n+4\ \alpha \ \kappa }}.$
We may take a constant $\epsilon$ to represent $4\alpha ,$ and our expression is—
	$\epsilon ={\frac {n}{n+e\ \kappa }}.$

§ 180. $\kappa$ and $\epsilon$, Plausible Values.—We are now able to find an expression for $\kappa$ in terms of $c,\ C,$ and $n$, for we already have the equation—

	$\epsilon ={\frac {c\ C}{\kappa }}-1,\quad$ ∴ $\quad {\frac {n}{n+e\ \kappa }}={\frac {c\ C}{\kappa }}-1.$
or	$c\ C={\frac {2\ \kappa \ n+e\ \kappa ^{2}}{n+e\ \kappa }}$
∴	$c\ C\ n+c\ C\ e\ \kappa =2\ \kappa \ n+e\ \kappa ^{2}$
∴	$e\ \kappa ^{2}+2\ \kappa \ n-c\ C\ e\ \kappa =c\ C\ n$
or,	$\kappa ^{2}+\left({\frac {2\ n}{e}}-c\ C\right)\ \kappa ={\frac {c\ C\ n}{e}},$
whence—
	$\kappa =$ $\pm \ {\sqrt {\left\{{\frac {c\ C\ n}{e}}+\left({\frac {n}{e}}-{\frac {c\ C}{2}}\right)^{2}\right\}}}$ ${\boldsymbol {-}}\ {\frac {n}{e}}+{\frac {c\ C}{2}},$

the rest is a matter of choosing such a value of $e$ as fits in best with experience. The author has taken $e=3.3,$ and this is the basis on which the following Table of plausible values of $\kappa$ and $\epsilon$ is founded.

§ 181. Best Value of $\beta .$ Least Value of $\gamma .$—Assuming the values given in the Tables (I., II. and III.), we are now in a position to obtain numerical values for the best values of the angle $\beta$ for aerofoils of different aspect value. Table IV. illustrates the process of calculation in the case of the pterygoid aerofoil $\xi$ being taken as = .03.

Table III.

Plausible Values, $\kappa$ and $\epsilon .$

$n.$	$\kappa .$	$\epsilon .$
3 4 5 6 7 8 — 10 — 12	1.00 (?) 1.03 (?) 1.064 (?) 1.10 (?) 1.12 (?) 1.14 (?) — 1.175 (?) — 1.195 (?)	.48 (?) .54 (?) .59 (?) .62 (?) .65 (?) .68 (?) — .72 (?) — .75 (?)

Table IV.

$\beta$ values for $\gamma =$ minimum.

Calculated from Equation, $\beta _{1}={\sqrt {\frac {2\ \xi \ C}{\kappa \ (1\,{\boldsymbol {-}}\,\epsilon ^{2})}}}$ for $\xi =.03.$

$n.$	$\kappa \,(?).$	$\epsilon \,(?).$	$\epsilon ^{2}.$	$1-\epsilon ^{2}.$	$\kappa (1{\boldsymbol {-}}\epsilon ^{2}).$	$C.$	$\xi .$	$2\,\xi \,C.$	$\beta _{1}^{2}.$	$\beta _{1}.$	$\beta _{1}^{\ \circ }.$
3 4 5 6 7 8 — 10 — 12	1.00 1.03 1.064 1.10 1.12 1.14 — 1.175 — 1.195	.48 .54 .59 .62 .65 .68 — .72 — .75	.23 .291 .348 .384 .422 .462 — .518 — .562	.770 .709 .652 .616 .578 .538 — .482 — .438	.770 .730 .695 .678 .648 .614 — .567 — .523	.685 .700 .71 .72 .725 .73 — .74 — .75	.03 .03 .03 .03 .03 .03 — .03 — .03	.0411 .0420 .0426 .0432 .0435 .0438 — .0444 — .0450	.0534 .0575 .0612 .0638 .0671 .0715 — .0783 — .0861	.231 .240 .247 .252 .259 .268 — .280 — .293	13.2° 13.75° 14.14° 14.4° 14.8° 15.0° — 16.0° — 16.8°

Table V. gives the results for values of $\xi$ equal .025, .020, .015 and .010 in the respective columns; both Tables IV. and V. being values of $\beta$ appropriate to least gliding angle, that is, for $\gamma =$ minimum.

Table V.

Values of $\beta _{1}$ (continued).

$n.$	$\xi =$ .025	$\xi =$ .020	$\xi =$ .015	$\xi =$ .010
3 4 5 6 7 8 — 10 — 12	.210 = 12.0° .219 = 12.5° .226 = 12.9° .230 = 13.2° .236 = 13.5° .244 = 14:0° — .256 = 14.7° — .268 = 15.3°	.189 = 10.8° .196 = 11.2° .202 = 11.6° .206 = 11.8° .212 = 12:15° .218 = 12:5° — .228 = 13:0° — .239 = 13.7°	.163 = 9.3° .169 = 9.7° .174 = 10.0° .178 = 10.2° .183 = 10:5° .189 = 10.8° — .198 = 11:3° — .207 = 11.8°	.133 = 7.6° .138 = 7.9° .142 = 8.1° .145 = 8:3° .149 = 8:5° .154 = 8.8° — .161 = 9:2 — .169 = 9.7°

Table VI.

Least Gliding Angle ($=\gamma _{1}$) (Theoretical).

Minimum value of $\gamma$ or $\gamma _{1}=(1-\epsilon )\ {\sqrt {\frac {2\ \xi \ C}{\kappa \ (1-\epsilon ^{2})}}}$

$n.$	$\xi =$ .030		$\xi =$ .025		$\xi =$ .020		$\xi =$ .015		$\xi =$ .010
3 4 5 6 7 8 — 10 — 12	6.85° 6.32° 5.8° 5.5° 5.2° 4.8° — 4.5° — 4.2°	1:8.3 1:9 1:10 1:10.4 1:11 1:12 — 1:12.8 — 1:13.7	6.25° 5.75° 5.3° 5.0° 4.7° 4.5° — 4.1° — 3.8°	1:9.2 1:10 1:10.8 1:11.5 1:12.2 1:12.8 — 1:14 — 1:15	5.6° 5.15° 4.75° 4.5° 4.25° 4.0° — 3.65° — 3.42°	1:10.2 1:11.1 1:12 1:12.8 1:13.5 1:14.4 1:15.8 — 1:16.8	4.8° 4.4° 4.1° 3.9° 3.6° 3.4° — 3.2° — 3.0°	1:12 1:13 1:14 1:14.7 1:15.9 1:16-8 — 1:17.9 — 1:19	3.95° 3.65° 3.40° 3.20° 3.00° 2.80° — 2.60° — 2.40°	1:14.5 1:15.7 1:16.8 1:17.9 1:19.1 1:20.5 — 1:22 — 1:23.9

Table VI. gives the theoretical values of $\gamma$ corresponding to the ascertained values of $\beta$, that is to say, this Table represents the theoretical minimum gliding angle for the values of $\xi$ employed.

In Tables IV. and V. the angle is given both in circular measure and in degrees; in the case of the gliding angle Table VI. the equivalent is given in the inverse form, i.e., as a gradient. In all cases the assumption is that of the small angle as already stated.

In actual aerodrome models, owing to the necessity for organs of equilibrium the resistance is greater than that due to the considerations taken into account in the foregoing Table; there is additional resistance due to the added surface, or body resistance.

Table VII. (Aeroplane).

Least Resistance. Values of $\beta _{1}^{\ \circ }.$

From Equation—$\beta _{1}^{\ \circ }={\frac {180}{\pi }}\ {\sqrt {\frac {\xi }{c}}}.$

Values of $c$ assumed from Table II. (Plausible Values).

$n.$	$\beta$ for values of $\xi$ as follows:—
	.030	.025	.020	.015	.0125	.010
3 4 5 6 7 8 — 10 — 12	6.76° 6.60° 6.43° 6.30° 6.22° 6.13° — 6.00° — 5.93°	6.16° 6.02° 5.87° 5.76° 5.67° 5.59° — 5.48° — 5.41°	5.51° 5.38° 5.25° 5.15° 5.07° 5.00° — 4.90° — 4.34°	4.77° 4.66° 4.55° 4.45° 4.39° 4.33° — 4.24° — 4.19°	4.35° 4.25° 4.15° 4.07° 4.00° 3.95° — 3.87° — 3.83°	3.90° 3.80° 3.71° 3.64° 3.58° 3.54° — 3.46° — 3.42°

Owing to this and other causes which will be explained later, the gliding angle is never found to be as low as the theoretical value, and in the most carefully made model is usually at least 50% greater than theory would indicate.

Reverting to the simple case of the plane aerofoil, or aeroplane (§§ 162 and 172), we have seen that the value of $\beta$ for least resistance, that is to say, least gliding angle, is given by the expression $\beta _{1}={\sqrt {\frac {\xi }{c}}},$ and since the aerodynamic and direct resistances are equal we have least value of $\gamma =2\ {\sqrt {\frac {\xi }{c}}}.$ In Tables VII. and VIII. the calculated angles are given for values of $\xi$ ranging from .01 to .03.

Table VIII. (Aeroplane).

Theoretical Least Gliding Angles. ($\gamma =$ min.).

$n.$	$\xi =$ .030	.025	.020	.015	.0125	.010
3 4 5 6 7 8 — 10 — 12	13.52° 13.20° 12.86° 12.60° 12.44° 12.26° — 12.00° — 11.86°	12.32° 12.04° 11.74° 11.52° 11.34° 11.18° — 10.96° — 10.82°	11.02° 10.76° 10.50° 10.30° 10.14° 10.00° — 9.8° — 9.68°	9.54° 9.32° 9.10° 8.90° 8.78° 8.66° — 8.48° — 8.38°	8.70° 8.50° 8.30° 8.14° 8.00° 7.90° — 7.74° — 7.66°	7.80° 7.60° 7.42° 7.28° 7.16° 7.08° — 6.92° — 6.84°

§ 182. The Aeroplane. Anomalous Value of $\xi .$—The actual behaviour of an aeroplane presents an anomaly with regard to the value of $\xi .$ It has been remarked in the previous section that in the case of the pterygoid aerofoil the theoretical results can never in practice be fully realised, owing partly to the necessity of added surface and partly to other causes. In the case of the aeroplane, in spite of the fact that the same values of $\xi$ are employed in the calculation the reverse is the case, and investigation shows that in effect the value of $\xi$ is considerably less in the case of an aeroplane than its ordinary value, and may amount to no more than half the coefficient as ordinarily found applicable.^[5] The result is that the aeroplane shows results far better than theory would indicate unless a diminished value of $\xi$ be employed in the equation.

The probable explanation of this anomaly is to be found in the supposition that for the angles investigated the flow is of the Rayleigh-Kirchhoff type, as illustrated diagrammatically in Fig. 98 ($a$), the result being that the effect of skin friction is only felt on the one face of the plane instead of on both faces, as would be the case if the flow were conformable, and consequently the apparent value of $\xi$ is only about half its real value.

There is a serious but not insuperable difficulty attached to the foregoing explanation. It would appear that since the “dead water” is itself subject to a tangential drag at its free surfaces, and since, as a whole, it has no influence to keep it in position other than the reaction of the aeroplane itself, this frictional drag must be transmitted to the aeroplane, and so in some way take the place of the missing skin friction.

On examining the matter in greater detail, it is evident that the form of the dead water region is determined primarily by the dynamics of the live stream, and if the fluid be supposed frictionless the dead water will extend indefinitely rearward, and its pressure will throughout be uniform. If now we take into account the effects of viscosity there will be a frictional or viscous drag acting tangentially at the surface of discontinuity between the dead water and the live stream, and referring to Fig. 114, it is evident that the cumulative effect of this drag will be to create a pressure gradient, the pressure at $A$ being less than that at $B$, and that at $B$ less than that at $C$, and so on. In consequence of this pressure difference the dead water will become the seat of a lively circulation as indicated by the arrows, the motion of the fluid in the vicinity of the plane being in the direction of flight, and that in the vicinity of the free surface being in the opposite direction. Now the result of this will be to produce a tangential drag in a forward direction; in fact, any skin friction experienced on the upper face or "back" of the plane will be of negative sign; we are thus unable to attribute the “retention” of the dead water to the direct influence of the plane.

On following the matter further it is evident that it is the partial vacuum in and about the region a a a that supplies the necessary
Fig. 114. reaction to prevent the dead water from being washed away, the lines of flow being at this point in close proximity to one another, as indicated in the figure. We thus find that the back of the plane is not only apparently, but is really, relieved of the frictional drag, which is actually borne in some way dynamically by the fluid itself.

§ 183. Aeroplane Skin Friction. Further Investigation.—The present stage of our explanation cannot be regarded as entirely satisfactory. It would appear to be essential, if we suppose the aeroplane to be maintained in steady motion by an applied force, that all reactions experienced by the fluid must be eventually traceable to the applied force. In the case under consideration we have traced the action and reaction merely from one part of the live stream to another part. It remains to be shown in what manner the motion of the fluid in the region $a\ a\ a$ (Fig. 114) is counterpart to some component resistance in the line of flight not otherwise essential.

Let us suppose a limited stratum of fluid to be dealt with, firstly, by a system of superposed aeroplanes; and secondly, by a kind of honeycomb of curved tubes whose leading orifices point in the direction of flight, and whose trailing or discharge orifices make the same angle with the line of flight as the angle $\beta$ of the aeroplane system. Then, if $W$ be the total weight supported in either case, the resistance in the case of the aeroplanes will be (for small values of $\beta$) $=W\ \beta ,$ but in the case of the curved tubes it will only amount to half this quantity, or $=W\ {\frac {\beta }{2}},$ the operating surfaces in either case being supposed frictionless.

It is therefore evident that the aeroplane involves twice as great a resistance to traction as that aerodynamically necessary,^[6]and from what we know of the Kirchhofl' form of flow we can see that this added traction is employed in generating and maintaining the spurting forward of the fluid round the leading edge, indicated by the lines $a\ a\ a\ a$ in the figure. When the work expended in traction is entirely devoted to diverting the stream, as in the theoretical case of the curved pipe system, then there is no spurting forward of the fluid, and no discontinuity in the system of flow; and on the other hand, the operating surfaces are fully exposed to frictional resistance. "When the stream is brusquely diverted by an aeroplane there is an aerodynamic resistance involved in excess of that necessary to divert the flow, and this, by giving rise to a form of flow of the discontinuous type, diminishes the frictional resistance.

Owing in part to the return current in the dead water region, and in part to the forward motion of the fluid on the front face of the plane in the vicinity of its leading edge, and again in part to the slowing of the flow over the remainder of the face (owing to its being a pressure region), it would appear that the net skin friction might even be less than that computed on the basis of a single surface.

§ 184. Some Consequences of the foregoing Aeroplane Theory.—The consequences of the peculiar behaviour of the aeroplane in respect of skin friction are of considerable moment.

The aeroplane, thanks to its power of evading a considerable portion of the resistance due to skin friction, is capable of being utilised for the support of the load without any very great loss of efficiency. Considered thus, and compared to an aerofoil of pterygoid form, it is found to give results that are really remark- able. Experimenting on a small scale, it is difficult to construct a model with a pterygoid aerofoil that, so far as gliding angle is concerned, will perform better than a ballasted aeroplane of the most crude description. An analogous example is found in the case of the screw propeller. Most of the theory relating to the aeroplane, wing form, and peripteral motion, finds its analogue in the theory of the screw propeller (Chap. IX.), and it is well known to designers of the latter that, so long as the pitch is rightly chosen in view of the torque and thrust, and provided that the angle, area, and proportions, of the 'blades are suitable, there is but a moderate gain in efficiency to be obtained by departure from the simple helical form of blade.

It is probable that the relative advantage of the pterygoid form becomes greater when the size of the aerodrome is increased, owing to the relations of weight and area discussed in § 196, and the relatively less importance of skin friction. If this should prove to be the case the present theory would account for the remarkable difference between the flight and wing form of birds and insects, showing in detail that which was anticipated in Chap. II. (Compare § 196.) In general the wings of flies, dragon flies, moths, etc., are approximately flat—they are in fact aeroplanes; and further, when flexed by the pressure to which they are subjected in flight it is probable that they actually present a convex surface to the “wind.” On the contrary, the wings of birds are always concave on the under side and convex above; they are in fact true pterygoid forms. This is not only the case when the wing is quiescent but is visibly the case when the bird is in flight. It is of particular interest that some of the larger butterflies and moths—for example, many of the ornithoptera—show clearly a rudimentary development of the dipping front edge, proving that this feature is not merely an incident of a different method of construction.

§ 185. The Weight per Unit Area as related to the Best Value of $\beta .$—We may now resume the main subject from the point to which it was carried in § 181, and we can show that the value of $\beta$ corresponding to a minimum gliding angle denotes a definite relationship between the area $A$, the velocity $V,$ and the load carried $W.$

According to the hypothesis of constant sweep, we know that the mass dealt with per second is given by the expression $\rho \ \kappa \ A\ V,$^[7] and the velocity of the up-current is a $V,$ and that of downward discharge $=\beta \ V,$ on the assumption that we are dealing with small angles.

Consequently the weight supported ($W$) which is equal to the momentum communicated per second, will be $\rho \ \kappa \ A\ V^{2}\ (\alpha +\beta );$ but we have $\alpha =\epsilon \ \beta ,$ so that our expression becomes—

$W=\rho \ \kappa \ A\ V^{2}\ \beta \ (\epsilon +1),$

or—

${\frac {W}{A\ V^{2}}}=\rho \ \kappa \ \beta \ (\epsilon +1),$ which is constant.

Now ${\frac {W}{A\ V^{2}}}$ may be written ${\frac {P_{2}}{V^{2}}}$ where $P_{2}$ denotes pressure, i.e., weight per unit area sustained by the aerofoil. In Table IX. are given values of ${\frac {P_{2}}{V^{2}}}$ for aerofoil of pterygoid form and of different aspect ratio, calculated from values of $\beta$ given in Tables IV. and V., for $\xi$ taken as .03, .025, .02, .015, and .010 in the respective columns.

Table IX.

Pterygoid Aerofoil.

Values of $P_{2}/V^{2}$ for Least Resistance.

$n.$	$\kappa \,\rho$ $(\epsilon {\mbox{+}}1).$	$P_{2}/V^{2}$ for values of $\xi$ as follows:—
		$\xi =$ .03	$\xi =$ .025	$\xi =$ .02	$\xi =$ .015	$\xi =$ .010
3 4 5 6 7 8 — 10 — 12	.1154 .1236 .1319 .1390 .1440 .1493 — .1574 — .1630	.0266 .0296 .0326 .0350 .0372 .0400 — .0440 — .0477	.0242 .0270 .0298 .0320 .0340 .0364 — .0403 — .0437	.0218 .0242 .0266 .0286 .0305 .0326 — .0359 — .0390	.0188 .0209 .0230 .0247 .0263 .0283 — .0311 — .0337	.0153 .0170 .0188 .0204 .0215 .0231 — .0254 — .0275

In the employment of this Table it must be remembered that we have to deal with British absolute units, so that $P_{2}$ will be poundals per square foot; thus, supposing- it were desired to design an aerofoil of aspect ratio $n=$ 10 to travel at a velocity of 40 feet per second, then the value of $P_{2}$ for least resistance will be = 1,600 $\times$ .0440 (taking $\xi$ = .03), or 70.4 poundals or 2.2 lbs. per square foot (approximately).

In Table X. the appropriate load per square foot is given for velocities from 10 to 80 feet per second for various values of “$n,$” the value of $\xi$ has been taken as .03, .02, and .01.

§ 186. Aeroplane Loads for Least Resistance.—The pressure, or load, per unit area that an aeroplane will economically sustain is considerably less than that tabulated in the preceding section for aerofoils of pterygoid form.

Table X.

Pterygoid Aeeofoil.

Load (pounds) per Square Foot for Least Resistance.

	Ft. per Sec.	Values of “$n$”:—
		3.	4.	5.	6.	7.	8.	10.	12.
$\xi =.030.$	5 10 15 20 25 30 35 30 50 60 70 80	.020 .082 .186 .331 .516 .743 1.01 1.32 2.06 2.97 4.05 .5.29	.023 .092 .207 .367 .574 .827 1.12 1.47 2.30 3.31 4.50 .5.89	.025 .101 .228 .405 .633 .911 1.24 1.62 2.53 3.65 4.96 .6.48	.027 .108 .245 .435 .680 .978 1.33 1.74 2.71 3.91 5.32 .6.95	.029 .115 .260 .462 .721 1.04 1.41 1.84 2.88 4.15 5.64 .7.39	.031 .124 .280 .497 .777 1.12 1.52 1.99 3.11 4.47 6.09 .7.95	.034 .136 .307 .546 .852 1.23 1.67 2.18 3.41 4.91 6.70 .8.73	.037 .148 .333 .593 .926 1.33 1.82 2.37 3.70 5.33 7.25 .9.48
$\xi =.020.$	5 10 15 20 25 30 35 30 50 60 70 80	.017 .068 .152 .270 .390 .610 .830 1.08 1.69 2.44 3.32 4.33	.018 .075 .169 .300 .433 .676 .920 1.20 1.88 2.70 3.68 4.81	.020 .082 .186 .330 .475 .743 1.01 1.32 2.06 2.97 4.05 5.30	.022 .089 .200 .355 .511 .800 1.08 1.42 2.22 3.20 4.35 5.70	.023 .094 .213 .379 .545 .852 1.16 1.51 2.37 3.40 4.64 6.07	.025 .101 .228 .405 .582 .911 1.24 1.62 2.53 3.64 4.96 6.47	.028 .111 .250 .445 .641 1.00 1.36 1.78 2.78 4.01 5.46 7.13	.030 .121 .272 .484 .697 1.09 1.48 1.94 3.03 4.36 5.93 7.75
$\xi =.010.$	5 10 15 20 25 30 35 30 50 60 70 80	.012 .047 .107 .190 .298 .427 .582 .760 1.18 1.71 2.32 3.04	.013 .053 .119 .211 .331 .475 .647 .845 1.32 1.90 2.58 3.38	.014 .058 .131 .234 .366 .526 .717 .935 1.46 2.10 2.86 3.73	.015 .063 .142 .258 .396 .570 .777 1.01 1.58 2.28 3.10 4.05	.016 .066 .150 .267 .418 .601 .820 1.07 1.67 2.40 3.27 4.27	.018 .071 .161 .287 .450 .645 .880 1.15 1.79 2.58 3.51 4.59	.019 .079 .177 .315 .495 .710 .967 1.26 1.97 2.84 3.86 5.05	.021 .085 .192 .341 .535 .769 1.04 1.36 2.13 3.07 4.18 5.45

For the aeroplane we know that the weight supported per unit area is (for small angles) given by the expression—$c\ \beta \times C\ \rho \ V^{2},$ which is $=P_{3}.$

We therefore have ${\frac {P_{3}}{V^{2}}}=\rho \ c\ C\ \beta _{1}.$

For values of $\xi$ respectively .02, .015, .0125, and .01, and taking $\rho$ as before = .078, the values of ${\frac {P_{3}}{V^{2}}}$ for least resistance are given in Table XI.

Table XI.

Aeroplane.

Values of $P_{3}/V^{2}$ for Least Resistance.

$n.$	$\xi =$ .020	$\xi =$ .015	$\xi =$ .0125	$\xi =$ .010
3 4 5 6 7 8 — 10 — 12	.0111 .0116 .0121 .0124 .0127 .0130 — .0135 — .0138	.0096 .0100 .0104 .0108 .0110 .0112 — .0116 — .0119	.0087 .0091 .0095 .0098 .0100 .0102 — .0106 — .0109	.0078 .0082 .0085 .0088 .0090 .0092 — .0095 — .0098

In Table XII. the foregoing results have been interpreted as pounds per square foot for different values of $V$ ranging from 5 to 80 feet per second, and for values of $\xi =$ .02, .015, and .01.

It is scarcely necessary to remark that the values given in the preceding Tables are not based on sufficiently reliable data to justify their being carried to so many places of decimals; the figures as tabulated have a probable error of 10 per cent, or even 20 per cent, one way or the other, and the employment of the third significant figure is only justified as a means of showing the relation of any one value to those adjacent to it in the Table.

It has not been thought necessary to re-tabulate in metric

Table XII.—Aeroplane Data.

Pounds per square foot for different values of $V$ for Least Resistance.

	Feet per Second.	Values of “$n$”:—
		3.	4.	5.	6.	7.	8.	10.	12.
$\xi =.020.$	5 10 15 20 25 30 35 30 50 60 70 80	.0086 .034 .077 .138 .215 .310 .421 .550 .861 1.24 1.68 2.20	.0090 .036 .081 .144 .225 .322 .440 .576 .900 1.29 1.76 2.30	.0094 .037 .084 .150 .234 .337 .460 .601 .940 1.35 1.83 2.40	.0096 .038 .086 .154 .240 .346 .471 .616 .963 1.38 1.88 2.46	.0098 .039 .088 .158 .246 .354 .482 .631 .985 1.42 1.93 2.52	.0101 .040 .090 .161 .252 .363 .494 .646 1.01 1.45 1.97 2.59	.0104 .042 .094 .167 .261 .376 .510 .670 1.04 1.50 2.05 2.68	.0107 .043 .096 .171 .268 .385 .525 .686 1.07 1.54 2.10 2.74
$\xi =.015.$	5 10 15 20 25 30 35 30 50 60 70 80	.0075 .0300 .067 .120 .187 .269 .367 .480 .750 1.08 1.47 1.92	.0078 .0313 .070 .125 .195 .282 .384 .501 .783 1.12 1.53 2.00	.0081 .0325 .073 .130 .203 .292 .398 .520 .813 1.17 1.59 2.08	.0084 .0336 .075 .134 .210 .302 .410 .538 .840 1.21 1.65 2.16	.0086 .0344 .077 .137 .214 .309 .420 .549 .858 1.23 1.68 2.20	.0087 .0350 .079 .140 .218 .315 .430 .560 .875 1.26 1.71 2.24	.0090 .0362 .081 .145 .226 .326 .444 .580 .905 1.30 1.77 2.32	.0093 .0372 .083 .149 .232 .335 .456 .596 .930 1.34 1.82 2.38
$\xi =.010.$	5 10 15 20 25 30 35 30 50 60 70 80	.0060 .024 .054 .097 .151 .242 .297 .387 .606 .871 1.18 1.55	.0063 .025 .057 .102 .159 .254 .312 .407 .637 .916 1.24 1.63	.0066 .026 .059 .105 .165 .264 .323 .422 .660 .950 1.29 1.69	.0068 .027 .061 .109 .171 .273 .335 .437 .683 .984 1.34 1.75	.0070 .028 .063 .112 .175 .279 .342 .447 .699 1.00 1.37 1.79	.0071 .028 .064 .114 .179 .286 .350 .457 .714 1.02 1.40 1.83	.0073 .029 .066 .118 .185 .295 .361 .472 .738 1.06 1.44 1.89	.0076 .030 .068 .122 .191 .304 .373 .487 .761 1.09 1.49 1.95

units, for the pressures in kilos per square metre can be obtained with a sufficient degree of approximation by multiplying by five the figures given in Tables X. and XII.

§ 187. Comparison with Actual Measurements.—The portent of the preceding sections may be illustrated by a few examples from Nature.

The herring gull, according to the system of measurement adopted by the author and subsequently explained, carries its load at the rate of about 1.3 to 1.4 pounds per square foot; its $n$ value is 7. Referring to Table X. ($\xi =$ .02)
Fig. 115. we find this load corresponds to 38 feet per second or about 26 miles per hour, which is probably a fair approximation to its actual speed.

The albatros carries about 3 pounds per square foot, and has an $n$ value of 12; referring to the Table we find the corresponding velocity to be about 50 feet per second or slightly over 84 miles per hour, which again is probably not far from the truth.

If now we take the case of a dragon-fly: an example weighed and measured by the author (Fig. 115) gave a result, from a planimeter measurement of the whole wing surface, of .68 grammes on 3.5 square inches, which is .062 pounds per square foot. The “$n$” value may fairly be taken as about = 4.

Referring to the Tables and taking $\xi =$ .02, we have for pterygoid form the corresponding velocity = 9.1 feet per second, or according to Table XII., considering the wings as planes, the velocity should be from 13 to 15.6 feet per second, according as $\xi$ is taken as .02 or .01.

Unfortunately no scientific measurement of the flight of this insect appears to be available, but its velocity is certainly nearer the latter than the former estimate.

In the case of birds such as those above cited, the soaring mode of flight is so extensively employed that without doubt the process of natural selection, or whatever other method Nature may employ, may be relied upon to have approximated the proportions of least resistance proper to the ordinary velocity of flight of the species. In the case of smaller birds or insects, such as the dragon-fly cited, it is an open question to what extent the problem is modified by the exigencies of active flight, and so the evidence, as confirming or otherwise the present theory, is at the best inconclusive.

§ 188. Considerations Relating to the Form of the Aerofoil.—We have so far specified the form of the aerofoil only so far as the angles $\alpha$ and $\beta$ are concerned, and have now not only to discuss the other attributes of the fore and aft section, but also the plan form of the aerofoil and its variation of section from point to point, and in addition the shape it presents when viewed along the axis of flight.

Many of the influences at work to affect the form of the aerofoil do not belong to the province of aerodynamics. The question of form, as viewed along the axis of flight, is governed almost entirely by aerodonetic considerations, and the discussion of this point will therefore be reserved.

The present subject has already been examined in Chap. IV., § 120 ; it remains for us now to continue the discussion in the light of the present theory.

If we suppose, provisionally, that the aerofoil section is of the form of the arc of a circle, then such a form would manifestly carry out the requirements of hypothesis with a uniform distribution of pressure on its surface, for we are supposing that the “fluid” consists of a limited layer composed of a number of strata whose individual continuity is preserved, after the manner of Fig. 108. If we suppose such an aerofoil to be gliding in a frictionless fluid, then its trailing and leading edges will be at the same level, for owing to considerations of symmetry it is then that the reaction is vertical. Under these conditions we see (Fig. 116) that the gliding angle will be $={\frac {\beta -\alpha }{2}}.$
Fig. 116.

This result may be deinonstrated more generally for, $x$ resistances being absent,—

$\gamma ={\frac {y}{W}}$ $={\frac {\beta ^{2}-\alpha ^{2}}{2\ (\beta +\alpha )}}={\frac {\beta -\alpha }{2}}.$ Compare §§ 174, 176.

Now the ratio of the angles $\alpha$ and $\beta$ does not depend upon the leading and trailing angles given to the aerofoil, but upon the aspect ratio, so that the design of the aerofoil requires to conform to the ratio so imposed. If we take an aerofoil of arc section there is a particular direction in which it must be propelled in order that it should fulfil the necessary condition, and this direction is in practice determined by a directive organ which usually takes the form of a tail plane.

Let us examine the effect of an incorrect adjustment of the directive organ; that is to say, we will examine the effect of incorrectly designing the aerofoil in respect to the value of $\epsilon \ (=\alpha /\beta ).$ Firstly, suppose it be adjusted so that the “dip” of the front edge is insufficient, then the up-current will no longer strike the edge conformably, and, in the case of a real fluid, a discontinuity will result, as illustrated in Fig. 117; such a discontinuity may be a trivial matter involving only a small pocket of "dead water" (a), or it may be more serious so that the form of flow resembles that generated by an aeroplane (b); in either case we

Fig. 117a.

know, from the great efficiency obtainable from the aeroplane, that the effects are not disastrous.

If, secondly, we suppose that the leading edge has too much

Fig. 117b.

"dip," the want of conformity is in the opposite direction, and the surface of discontinuity springs from beneath the leading edge as depicted in Fig. 118; the result of this is destructive to the whole peripteral system of flow, for the moment the pressure region commences to occupy the upper surface of the aerofoil a condition of instability arises and a new system of flow is inaugurated which produces a downward instead of an upward reaction. This is a fact easily demonstrated experimentally: a model in which the adjustment has been carried to its limits will behave in a most capricious manner, sometimes gliding perfectly and at others dropping suddenly in the midst of a flight like a bird when shot.

§ 189. The Hydrodynamic Standpoint.—Let us revert to the Hydrodynamic aspect of the subject as expounded in Chap. IV. We have seen that the supporting reaction is due to a cyclic motion in the fluid which is maintained by, and is in equilibrium with, the load on the aerofoil; it is of course understood that there is a superposed motion of translation, i.e., the motion of flight. Now in the case of an aerofoil of infinite lateral breadth we have seen that this equilibrium is permanent, and we have in

Fig. 118.

several instances plotted the resulting field of flow. When we have to deal with a case of finite lateral extent we have seen that there must be a continual dissipation of the cyclic motion, which vanishes in the manufacture of the trailing vortex filaments which the aerofoil is continually shedding on either hand.

In the Eulerian fluid there is no reaction on the aerofoil possible except that due to the cyclic motion, but in a real fluid this is not the case; a reaction may always be generated and always is generated when the motion gives rise to discontinuity, whether kinetic or physical.

Now the cyclic motion is in equilibrium with the reaction to which it gives rise, so that if we suppose an aerofoil supported entirely by the cyclic reaction and in equilibrium at any instant with the cyclic reaction, then if, firstly, it be supposed of infinite extent it will be in equilibrium at every other instant of time; if, secondly, it be supposed of finite extent, then, since the cyclic motion is decaying, its equilibrium must vanish. In the case of the finite aerofoil we must consequently have the load in part supported by a reaction due to discontinuity, and in part only to the cyclic motion; the part of the reaction sustained by the discontinuity of motion may be regarded as that required to augment the cyclic motion at the same constant rate as its rate of decay. Hence: an aerofoil of finite lateral extent cannot be so designed that it shall be everywhere conformable to the lines of flow, and any such aerofoil must give rise to discontinuity in the motion of the fluid, involving surfaces of discontinuity, and presumably dead water regions.

We thus see that a perfectly conformable motion, such as we

Fig. 119.

have tacitly supposed possible, is not possible when dealing with a real fluid, and at some point or points along its length the aerofoil must give rise to a discontinuity. This does not affect the validity of the foregoing theory, which has been founded on a hypothesis that admittedly does not fully represent the actual conditions; but it may be found that the matter now under discussion renders this hypothesis less valid than would otherwise be the case, especially where we are concerned with the quantitative estimation of the work done, i.e., the computation of the gliding angle.

§ 190. Discontinuous Motion in the Periptery.— We may take as a simple example of the phenomenon under discussion the case of an aeroplane where, as we have seen, we have a system of flow of the Rayleigh-Kirchhoff type (Fig. 98). Let Fig. 119 represent such a plane in front elevation, then surfaces of discontinuity will spring from the ends in the manner shown, and these surfaces, constituting at first a Helmholtz vortex sheet, will break up into a number of vortex filaments which conceivably become the vortex cores discussed in Chap. IV, with reference to Figs. 83—86.

This view must at present be regarded as tentative, and is not altogether in agreement with the explanation put forward in the chapter to which reference has been made; it is highly suggestive, however, and on that account requires to be recorded; thus if we examine the wing of a bird we find that the middle portion is of a very characteristic arched section, but towards the extremities the arching is very much less pronounced; in fact, the form becomes such as might easily become the seat of discontinuity. This observation applies more particularly to the soaring birds, which in all respects constitute the best criterion.

If the view put forward in § 106 is correct, as to the cause of the noise made by bodies in rapid motion, then the " whirring " noise made by the wing of a bird in flight is a direct proof of the existence of discontinuity.

Of the two explanations of the genesis of the vortex continuations offered here and in Chap. IV., it is not necessary that either should be in error. The previous explanation also opens out a possibility that must not be lost sight of as bearing on the phenomenon now under discussion.

Let us suppose that the two air streams passing under and over the aerofoil find themselves when they meet at the trailing edge possessed of different velocities. Then their common surface would constitute a surface of discontinuity which might in itself fulfil the requirements of theory. But such a condition is impossible in an irrotational mass of fluid, for where there is a difference of velocity there is also a difference of pressure; and the fluid in the periptery is certainly irrotational in the sense of the argument.^[8] But let us modify the supposition and take it that the two streams, although moving with the same velocity, are moving in different directions along the surface of separation; there would appear to be nothing contrary to hydrodynamic principles in this supposition; and the result would be a surface of discontinuity which might conceivably satisfy the condition.

The subject is one of very great difficulty, and it is impossible to do more than point out the more probable interpretations.

§ 191. Sectional Form.—The simple arc form of section employed as an illustration in § 188 is, qualitatively speaking, representative of that which may be considered essential, although the actual section more commonly resembles that given in Fig. 57, which may be regarded as typical.

We have seen that the consequence of an excessive "dip" on

Fig. 120.

the front edge is a loss of sustaining reaction, and it would appear that the trouble is not so much due to the excessive angle of dip but to the fact that the leading edge comes down too low; it is evident, therefore, that the leading edge, after being allotted the theoretical angle and position, as in Fig. 120 (indicated by the dotted line), should be curtailed somewhat in the manner shown.

In the wings of birds the elastic nature of the trailing portion probably acts as a considerable safeguard, for should the pressure reaction show any sign of falling off, the elasticity of the plumage will immediately rectify matters; it is at least impossible to get any sudden reversal as may happen when the aerofoil is a rigid structure.

In § 120 it was suggested that the form of section might be uniform throughout the length of the aerofoil, but of changing scale, i.e., that it should tail off to a point at each extremity. Such a form is not generally found in Nature; the section nearly always becomes flatter and the angles of dip and trail become less as the extremities are approached. It is not known whether this fact is due to the reasons suggested in the preceding section, or whether it is attributable to aerodromic considerations alone, or whether there is some further subtle reason that has hitherto escaped detection. It is certain that the feature in question is valuable from the aerodonetic standpoint, and that is all that can be said with certainty at present.

§ 192. A Standard of Form.—In 1894 the author, with a view to embarking on some experiments in flight, took measurements of the plan-form of several of the soaring birds, including an albatros, a herring gull, and a kittiwake gull, with the result that an elliptical form was adopted as being a simple geometrical form whose ordinates approximate very closely to the average of those adopted by Nature. No attempt was made to imitate the "sinuosity" of the bird wing plan-form, this being regarded as an anatomical accident.

The form of section adopted has been given in Fig. 58; the aerofoil being made of timber, it was necessary to adopt a form easy to produce; on this account the hollow in the underside was very soon abandoned owing to the results not justifying the additional labour.

The "grading" is segmental or parabolic^[9], that is to say, the maximum thickness of the section at different points along the length of the aerofoil is given by the ordinates of the segment of a circle whose cord is constituted by the flat face. This method of grading, independently of the plan-form to which it is applied, ensures the proper tailing off of the load towards the extremities as set forth in § 120. It is evident that if the thickness of the stratum at different points along the aerofoil is in constant relation to its width at each point, the mass dealt with per unit length will be everywhere as the width. But for a given height of the arched section the values of $\alpha$ and $\beta$ will be inversely as the width, and consequently the load sustained for any particular element of the length will be constant in respect of width; that is to say, the sustaining power of the aerofoil for a given midsection and a given grading is constant, no matter what the plan-form, both as to total and as to each element of length.
Fig. 121.

This is a very convenient rule to remember, but one which, from the nature of the assumptions made, is more or less approximate; it can be applied legitimately to all ordinary modifications of plan-form.
Fig. 122.

When an aerofoil is designed according to the foregoing specification, whether as a solid as in the case in point, or as a lamina of the same mean section, the equivalent area for uniform values of $\alpha$ and $\beta$ will evidently be that of a plane whose plan ordinates are those of a segment, that is, proportional to the thickness ordinates. Such a form may be taken as having two-thirds the area of the circumscribed rectangle; that is, if $L$ be the length of the aerofoil the equivalent area will be:—

${\frac {2}{3}}\times {\frac {L^{2}}{n}}.$

By adopting and adhering to some standard such as that above defined, the experimental information obtainable becomes of greater value than when a variety of forms are employed. It is only desirable to modify the design when the data for some defined form is fully established and available as a standard of reference.

In experimenting with mica models the author has frequently adopted the form of natural flexion of an elliptical mica plate secured to a central bolster, to give the desired mid-section form; the grading is found to take the form given approximately in Fig. 121. In other cases the mica plate has been artificially graded to approximate to the standard given above, by fitting additional ribs as shown in Fig. 122.

Fig. 123.

§ 193. On the Measurement of "Sail Area."—The appropriate measurement of the sail area or wing area of birds of various species is a rather vexed question. Some writers have regarded the wings as the sole organs of support, and the actual wing area alone has been reckoned as effective. Others (notably M. Moulliard) have assumed that the whole plan area (or ombre) of the bird contributes to the support, and have made pressure computations on this basis.

The author's view is that the influence of the body as a supporting member cannot be ignored, but that probably its effect can be best estimated as equivalent to an imaginary band of appropriate width forming a junction between the two extended wings as represented in Fig. 123. This view is based on the knowledge that the cyclic system must be continuous from wing to wing, and, on the whole, will produce a reaction on the body just as if the wing were continued in the manner shown; also on the improbability that there is any augmentation due to discontinuity, for the body is of a rounded form little likely to produce motion of a discontinuous type, and the tail is essentially a directive organ.^[10]

We have seen that the different portions of a bird's wing are of different sectional form and adapted to carry different pressure values at different points. It sometimes happens that, for the purposes of comparison, in place of the actual area the equivalent area is required, as for example when comparing the results of theory, as in § 185, with the proportions adopted by Nature; this area being the area that would be required on an assumption of uniform pressure distribution, or constant values of $\alpha$ and $\beta$ throughout the length.

In order to rightly assess this equivalent area we require to know the grading of the aerofoil, a knowledge which we do not possess, and which, owing to the flexibility of the wing structure, it is almost impossible to obtain.

The probability is that for birds of similar habits the grading will be found to be similar; that is to say, the distribution of the load along the length of the outstretched wings will be identical. In the absence of more definite knowledge, a rough assumption has been made for the purposes of the present work, i.e., that the grading is substantially that of the standard form adopted by the author, and consequently the effective area is given by the expression ${\frac {2\ L^{2}}{3\ n}}.$

 

§ 194. The Weight of the Aerofoil as influencing the Conditions of Least Resistance.—The subject of the influence of aerofoil weight as affecting the conditions of least resistance has been discussed in a previous chapter (§ 171), and a general equation has been deduced from the conditions.

The most important application of the foregoing mathematical theory is found in the case where $V$ is fixed by external conditions and where the area ($A$) is the variable; this application gives rise to pressure values greater than those given in Table X., the difference being dependent upon the extent to which aerofoil area is “penalised” by the direct effect of its weight in causing additional resistance.

We start the present continuation with Equation (3) of § 171:—

Substituting the constants in full we have—

$V^{4}={\frac {C_{1}\ W_{1}^{2}}{C_{3}\ \xi \ L^{4}}}+$ ${\frac {(2-q)\ C_{1}\ W_{1}\ k}{C_{3}\ \xi \ L^{4-q}}}+{\frac {C_{1}\ k^{2}\ (1-q)}{C_{3}\ \xi \ L^{4-q}}}.$

whence

We now require to substitute for $C_{1}$ and $C_{3}.$ These values were not investigated in § 171; they are obtained as follows:—

We know that—

$W=\rho \ \kappa \ A\ V^{2}\ (\epsilon +1)\ \beta$

and

$y={\frac {\rho \ \kappa \ A\ V^{2}\ (1-\epsilon ^{2})\ \beta ^{2}}{2}}$

we require $y$ in terms of $W$ eliminating $\beta ,$ hence—

${\frac {y}{W^{2}}}={\frac {1-2\ \epsilon }{2\ \rho \ \kappa \ A\ V^{2}\ (1+\epsilon )^{2}}}$ $={\frac {1-\epsilon }{2\ \rho \ \kappa \ A\ V^{2}\ (1+\epsilon )}}$

or

$y={\frac {1-\epsilon }{2\ \rho \ \kappa \ (1+\epsilon )}}\times {\frac {W^{2}}{A\ V^{2}}}$

and if we define $L$ as being $={\sqrt {A}},\quad C_{1}$ is defined by expression $y=C_{1}\ {\frac {W^{2}}{L^{2}\ V^{2}}}.\quad$ (§ 171.)

∴

$C_{1}={\frac {(1-\epsilon )}{2\ \rho \ \kappa \ (1+\epsilon )}}.$

Now $C_{3}$ (see § 171) is defined as such that the quantity $C_{1}\ \xi \ V^{2}\ L^{2}$ is the skin friction on the aerofoil, but we know that this is given in full by the expression—$\epsilon \ C\ \rho \ V^{2}\ A$ and $L^{2}=A,$ therefore $C_{3}=C_{\rho }.$

Substituting for $C_{1}$ and $C_{3}$ in Equation (6) we have—

$L^{4}={\frac {1-\epsilon }{V^{4}\times 2\ \rho ^{2}\ \kappa \ (1+\epsilon )\ \xi \ C}}\times$ $(W_{1}^{2}+\kappa \ W_{1}\ (2-q)\ L^{q}+\kappa ^{2}\ (1-q)\ L^{2q}).$

§ 195. A Numerical Example.—The employment of this equation may be illustrated by a numerical example. Let us take the case of an aerodrome of 1,000 poundals essential weight (31 lbs. approximately), to be designed for a velocity of 50 feet per second. The remaining data are as follows:—

	n taken as 12
therefore—	$C=$	.75
	$\epsilon =$	.75
	$\kappa =$	1.2.
It is found by trial design or by calculation that in the expression $W_{2}=k\ L^{q}$ that—
	$k=$	50
and	$q=$	1.5
We take	$\xi =$	.03
and	$\rho =$	.78.

Substituting values in Equation (7) we obtain—

$L^{4}=$ ${\frac {.25}{6,250,000\times 2.4\times .0061\times 1.75\times .03\times .75}}$ $\times (1,000,000+25,000\ L^{1.5}-1,250\ L^{3})$ $=70+1.75\ L^{1.5}-.0875\ L^{3}$ (approx.).

This form of equation can only be solved by plotting or by guessing^[11]; the solution gives $L=2.96,$ that is to say, the area $A\ (=L^{2})$ is $8.76.$

Now the total weight sustained by this area is $W_{1}+W_{2}$ and $W_{2}=50\ L^{1.5}=254$ (poundals), or $W_{1}+W_{2}=1,254,{\mbox{or}}\ 39$ lbs. almost exactly. We therefore have pressure per square foot for least resistance $={\frac {39}{8.76}}=4.45$ as against $3.70$ as given in Table X.

We can calculate the corresponding value of $\beta$ from the expression ${\frac {P_{2}}{V^{2}}}=\rho \ \kappa \ (\epsilon +1)\ \beta$ (§ 185), or $\beta ={\frac {P_{2}}{\rho \ \kappa \ (\epsilon +1)\ V^{2}}}$ which in the present example gives $\beta ={\frac {.0585}{\rho \ \kappa \ (\epsilon +1)}}={\frac {.0585}{.164}}=.356,$ or $20.4$ degrees against $16.8$ degrees according to Table IV.

It would thus appear that unless the aerofoil weight in the above example has been greatly exaggerated, its influence on the conditions of least resistance is a fact that should certainly be taken into account. At present the accuracy with which the fundamental data have been ascertained would scarcely justify the preparation of Tables to include the influence of this factor.

§ 196. The Relative Importance of Aerofoil Weight.—The importance of the present branch of the subject evidently becomes greater with any increase in the size of the aerodrome, for the necessary proportionate weight of the aerofoil will be greater on a large aerodrome than on a small one; this fact is almost self-evident, but is in any case easy of proof.

Let the weight of the aerofoil, as in the preceding section, be represented by $W_{2},$ and as before let $W_{1}$ be the essential load; then we have seen that we can represent W2 by the approximate expression—$W_{2}=k\ L^{q}.$ We assume that the weight of the aerofoil itself does not materially add to its stresses, being supported directly.

Now let us suppose, as is the case for similar bodies, that $W_{1}$ varies as $L^{3},$ then $W_{2}$ will vary as $W_{1}\ k\ L^{q},$ that is, as $k\ L^{3+q},$ or the relation of $W_{2}/W_{1}$ will be represented by ${\frac {L^{3+q}}{L^{3}}}\times$ constant, that is, $L^{q}\times$ constant. Now, if the index $q$ were as low as 1 (and it is improbable that it is lower) the relative weight of the aerofoil $W_{2}/W_{1}$ will increase as the linear dimension. If $q$ be greater the increase will be appropriately more rapid.

The relation $W_{2}/W_{1}$ could only remain constant if the value of $q$ were to sink to zero, a condition which is manifestly impossible.

It is probable that for an aerodrome the size of that chosen as an example in the preceding section the value of $W_{2}/W_{1}$, is in excess of that necessary, but it is questionable whether it would be possible to construct a machine of moderate velocity capable of bearing a man, without the aerofoil considerably exceeding one-fifth of the total weight.

The question of aerofoil weight as affecting the phenomenon of flight as presented by Nature takes its place in the later portion of the work. It is only necessary here to point out that the larger birds must, in general, be more influenced than the smaller ones, just as in the case of other forms of aerodrome; and in the matter under discussion we have one of the causes, if not the most important cause, that constitute determining factors of that critical point at which Nature finds it advantageous to change from the insect mode to the avian mode of flight: that is, from the aeroplane to the pterygoid form.

Note.—Tables, etc., in present chapter relate strictly to incompressible fluid. For method of taking compressibility of air into account, see Appendix I.

---

1. See footnote, § 128.
2. The values of $c$ for the aeroplane are probably not the same as for a pterygoid aerofoil of the same aspect ratio. Neither value has yet been determined with any degree of accuracy; the values given in Figs. 105 and 106 and in tabular form in § 177 are probably more nearly correct for the pterygoid form.
   It will be shown subsequently that the effective value of $\xi$ in the case of an inclined plane may be less than its true value.
3. Compare § 189.
4. See footnote on § 172 as to $c$ values.
5. The results so far obtained by the author on the value of $\xi$ by different methods are not altogether in harmony. Since writing this and the following sections (§§ 182, 183, and 184), certain experiments made with a new instrument, the aerodynamic balance (Ch. X.), seem to indicate that the conclusions here formulated may require qualifying; the results at present available, however, are not conclusive, and it has been thought best to present the argument in its original form.
6. This is on the basis of ignoring the cyclic reaction; if this be taken into account the aeroplane is at a still greater disadvantage.
7. An erratum published in Volume 2 has been applied: "P. 269, line 12 from foot, for 'k' read '$\kappa .$'" (Wikisource contributor note)
8. The continuity of the system of flow cut by a path taken round the aerofoil at some distance away, from one side to the other of the supposed surface of discontinuity, points to this conclusion; the motion in the region traversed by such a path would be sensibly irrotational.
9. For small amplitude the two curves do not sensibly differ.
10. This point will be better appreciated when the aerodromic aspect of the subject has been discussed.
11. The particular case of q = 1.5 is an exception.