Aerodynamics (Lanchester)/Chapter 2

by Frederick William Lanchester

Chapter II.

 

Viscosity and Skin-Friction.

 

§ 31. Viscosity.—Definition.—The fundamental law of viscosity is enunciated in the form of an hypothesis to Section IX., Book II., of Newton's "Principia," as follows:—

The resistance arising from the want of lubricity in the parts of a fluid is, cæteris paribus, proportional to the velocity with which the parts of the fluid are separated from each other.

The subsequent propositions li., lii., and liii., show that the expression "want of lubricity" is synonymous with the modern term "viscosity," and the motion contemplated by Newton in framing the foregoing hypothesis is motion in shear. The Newtonian law has since received ample verification at the hands of Maxwell and others.

Maxwell, in his "Theory of Heat," gives a quantitative definition of viscosity as follows:—

The viscosity of a substance is measured by the tangential force on the unit area of either of two horizontal planes at a unit distance apart one of which is fixed while the other moves with the unit of velocity, the space between being filled with the viscous distance.

Or if (Fig. 26) a stratum of the substance of thickness I be contained between a fixed plane A B and the plane C D, moving from C towards D with a velocity V, then, when a steady state is established, the motion of the intervening fluid will be in the direction C to D, and its velocity at different points will be in proportion to the height above the plane A B, so that the fluid in immediate contact with the plane A B will remain at rest, and that in immediate contact with the plane C D will have the velocity V in common with it. Then, if F be the horizontal force applied to the plane C D per unit area to overcome the resistance of the fluid, we have—

F = µV/l, where µ is a quantity termed the coefficient of viscosity.

This equation is merely the algebraic expression of the law previously stated, for where V and l are unity we have F = µ.

It will be seen that between the planes A B and C D there will exist a velocity gradient. A series of particles situated at points on a straight line a, a, a, a, at one instant of time, will be situated at points b, b, b, b, on another straight line at another instant,

Fig. 26.

the figure thus giving a pictorial idea of the motion in a viscous fluid.

 

§ 33. Viscosity in relation to Shear.—In the foregoing illustration, which is in substance as given by Maxwell, the nature of viscous strain as a shear is sufficiently obvious. There are cases, however, in which viscosity plays a part in which the conditions are not so straightforward. The modern definition of shearing stress is stress that tends to alter the form of a body without tending to alter its volume, and any strain that involves the geometrical form or proportions of a body requires shearing stress for its production. All stresses and strains can be resolved into shear and dilatation (plus or minus); and such stresses as linear tension or compression of a solid involve stress in shear.

We thus see that changes in the shape of a body of a fluid, such as take place in the course of its passage through a "tube of flow" in the vicinity of a streamline body, are resisted by viscosity in proportion to the velocity with which the change of form takes place, and the work done on the fluid in this manner must be supplied by a propulsive force; that is to say, the body will be resisted in its motion through the fluid from the cause stated. We have here one of the causes of viscous resistance.

We cannot state that this form of resistance will increase directly as the velocity, that is, according to the viscous law, for we do not know that the form of the lines of flow is the same at different velocities. It would appear that this must be so for an inviscid fluid. It would also seem evident that the viscous resistance will modify the form of flow materially. It may therefore be deduced that the form of flow will be more modified for low than for high velocities, in which case the form of resistance we are now discussing will not vary exactly in the direct ratio of the velocity.

Bodies other than of streamline form will also be affected by this type of viscous resistance, when it will appear as an added resistance. The only exception is found in the case of a plane moving tangentially, the consideration of which introduces the important subject of skin-friction.

 

§ 33. Skin-friction.—It is well established that there is no slipping of a fluid past the surface of a solid, but that the film adjacent to the surface adheres to it, and the resistance experienced is of the nature of a viscous drag. This fact has already been assumed in the discussion of the law of viscosity, for otherwise there would be no necessity for the fluid to be set in motion by the plane C D at all. To a certain extent, therefore, the term "skin-friction" is misleading. It is, however, a term sanctioned by usage, and it is difficult to find a more suitable expression.

Let us suppose that a plane having no sensible thickness be put in motion tangentially through a fluid, and be maintained in motion until a steady state is reached. Then the advancing edge of the plane is continually engaging with new masses of the fluid, and setting them in motion by virtue of the viscous stress exerted. But the conditions under which any given mass of fluid is acted on are not those of the previous hypothesis; the force resisting the motion of the plane is that of the inertia of the fluid itself; and if we confine our attention to any one portion of the fluid, its condition is not that of a steady state, but one of acceleration. Now it is evident that when the leading edge first enters the undisturbed region the stratum of fluid affected will be quite thin; and as the following portions of the plane successively traverse the same region the thickness of the stratum set in

Fig. 27.

motion continuously increases, and the velocity gradient will correspondingly diminish. This is illustrated in Fig. 27, in which the line a, a, a, represents the original position of a series of particles of the fluid at some given instant, and b, b, b, the position assumed after a short time has elapsed.

 

§ 34. Skin-friction.—Basis of Investigation.—Owing to the considerations dealt with in the preceding section, it is evident that we cannot regard skin-friction as of necessity amenable to the ordinary viscous law, i.e., F ∝ V where l and µ are constant; it is in fact easy to prove that this law will not apply.

Let us suppose for example that the motion of the fluid is strictly homomorphous in respect of changes of V, that is to say, if u, v, w, be the velocity of the fluid particles at any instant of time in the direction of co-ordinates x, y, z, travelling with the body (or plane), we are supposing that any variation in V results in a variation in like ratio of u, v, w, for all values of x, y, z.

In order to simplify the thinking in connection with this problem it is convenient to suppose the body to be a plane travelling in the direction of the axis of x, and confine our attention to the motion of the fluid taking place in like direction. Let y be taken as the axis at right angles to the plane.

The viscous stress at every point will be proportional to, the velocity gradient, that is du/dy, which on the present supposition varies as V for every point x, y, z, in the region, consequently we shall have F ∝ V, which is the viscous law. Now if the prescribed conditions satisfy the dynamic requirements of the problem, we might conclude that the motion is strictly homomorphous, and that the viscous law obtains, but such is not the case. The momentum communicated per second to any given layer of the fluid, and therefore to the whole fluid, is = mass × the velocity per second imparted, that is ∝ u V ∝V²; so that under strictly homomorphous conditions the viscous stress cannot be satisfied for varying speeds by the inertia of the fluid.

If, when the velocity V increases, we suppose that the layer of fluid affected to any given degree becomes thinner, and vice versa, it is clear that the viscous forces will rise in a greater ratio than directly as V, for the velocity gradient will be steeper, also the inertia forces will be less, for the mass of fluid to be set in motion will be less. It is, therefore, evident that we may suppose the thickness of the affected strata varies with the velocity in the degree necessary to preserve a balance between the viscous and inertia forces.

 

§ 35. Law of Skin-friction.—Let us suppose that in any two systems, differing only as to velocity V, the whole region be divided into strata by an imaginary series of equidistant planes, so that the thickness of corresponding strata in the two systems, and their distance from the material plane, shall be in the constant ratio $n$. And let us denote the distance between adjacent planes by the symbol $\triangle y$, and the corresponding velocity difference in the axis of $x$ by $\triangle u$. Then

$\triangle y\propto n$ and $\triangle u\propto V.$

(1)

Let $F=$ the tangential force.

We have $F$ as measured by viscosity varies as the area (which is constant) $\times$ velocity gradient, or—

$F\propto {\frac {du}{dy}}\propto {\frac {\triangle u}{\triangle y}}\propto {\frac {V}{n}}.$

(2)

And $F$ as dependent on dynamic considerations $=$ momentum imparted per second to the fluid. For unit width of any stratum we have mass $=\rho \triangle y\quad V,$ and velocity varies as $V$ or $F=\Sigma \rho \ \triangle y\ V^{2}.$

∴	$F\propto nV^{2}.$	(3)
By (2) and (3) we have—
	$nV^{2}\propto {\frac {V}{n}},$
or	$n\propto V^{-{\frac {1}{2}}},$
substituting in either (2) or (3)—
	$F\propto V^{1.5}.$	(4)

This may be taken as the normal law of skin-friction.^[1]

§ 36. Kinematical Relations.—In dealing with problems relating to fluid resistance it is found to lead to simplification to eliminate the density of the fluid by introducing two new quantities, kinematic resistance and kinematic viscosity.

Kinematic resistance, which we will denote by the symbol $R$, may be defined as the resistance per unit density, or $R=F/\rho$, and is consequently of the dimensions ${\frac {L^{4}}{T^{2}}}$.

Kinematic viscosity, which we will denote by the symbol $\nu ={\frac {\mu }{\rho }},$ and and is consequently of the dimensions ${\frac {L^{2}}{T}}.$

Writing the law of viscous resistance in its kinematic form we have $R=A\ \nu \ {\frac {V}{l}}$ where $A$ is the area of the surface; it will be noted that this expression is dimensional.

If we similarly write the law of skin-friction $R=A\ \nu \ {\frac {V^{1.5}}{l}},$ we find that the dimensions do not harmonise.

Let us examine this expression in a general form, where

$R=c\times A^{p}\times \nu ^{q}\times V^{r};$

dimensionally:—

	${\frac {L^{4}}{T^{2}}}=c\times L^{2p}\times {\frac {L^{2q}}{T^{q}}}\times {\frac {L^{r}}{T^{r}}};$
∴	$2p+2q+r=4,$
and	$q+r=2,$
∴	$q=2\ {\boldsymbol {-}}\ r,$
∴	$2p+4\ {\boldsymbol {-}}\ 2r+r=4,$
∴	$2p\ {\boldsymbol {-}}\ r=0,$
	$p={\frac {r}{2}}.$

The general expression therefore becomes:—

$R=c\ \nu ^{q}\ A^{\frac {r}{2}}\ V^{r}$

(5)

in which

$q+r=2.$

This is the general equation to the kinematic resistance of bodies in viscous fluids, and correlates the variations in respect of viscosity, area, and velocity; the application extends to both normal and inclined planes and bodies of the most diverse form.^[2]

It may be illustrated here in its relation to the law of skin-friction; we have, $R$ varies as $V^{1.5}$ and the full kinematic expression therefore becomes—

$R=c\ \nu ^{\frac {1}{2}}\ A^{.75}\ V^{1.5},$

(6)

and we have the unexpected but experimentally established result that the resistance does not vary with the area, hut according to a fractional power of same.

If, as is customarily assumed, the resistance of a body is taken as proportional to the square of the velocity, then we shall have q = zero, and the pressure is independent of viscosity altogether; this result is due to Allen.^[3] Under these conditions the resistance is directly as the area, and conversely if viscosity have any influence on the resistance, then the resistance cannot vary directly as the area, hence the existence of viscosity may be regarded as giving a definite scale to the fluid.

 

§ 37. Turbulence.—The steady state of viscous motion depicted in Figs. 26 and 27, on which the laws of viscosity and skin-friction have been based, is found in practice to obtain over a moderate range of velocity only. When a certain critical velocity is exceeded the continuity becomes broken and the phenomenon of turbulence manifests itself. Under conditions involving pure viscosity (in contradistinction to the more complex phenomenon of skin-friction), this critical point has been investigated experimentally by Mr. Osborne Reynolds in the case of liquid flowing through a straight tube. It is found that up to a certain velocity the flow is everywhere parallel to the axis, but when this critical velocity is reached the parallel flow breaks up, and is replaced by an irregular turbulent motion. Up to the critical velocity the law deduced by Poisouille^[4] for viscous flow through a tube holds good; beyond this point the resistance rises more rapidly, and for high velocities approximates to F varies as V², when the energy is mostly expended in generating the turbulent motion.

The method of investigation employed by Osborne Reynolds consisted of observing the behaviour of a coloured filament of liquid introduced in the centre of a tube containing liquid in motion; the result obtained is that steady motion ceases to exist if the mean velocity exceeds ${\frac {1000\ \nu }{\alpha }}$ where $\alpha$ is the radius of the tube^[5] (c.g.s. units).

§ 38. General Expression.—Homomorphous Motion.—Let us examine generally the relations of geometrically similar systems possessed of homomorphous motion—that is, under circumstances when the theory of dimensions is strictly applicable, then the quantities upon which the motion depends are comprised by—velocity $=V,$ kinematic viscosity $\nu$, and a linear (scale) dimension $l.$

Let us write

$l=c\ V^{p}\ \nu ^{q},$

or, in terms of dimensions

$L={\frac {L^{p}}{T^{p}}}\times {\frac {L^{2q}}{T^{q}}},$

and we have the equations

	$p+q=0,$
	$p+2q=1,$
∴	$q=1\quad \quad p={\boldsymbol {-}}1,$
∴	$l=c\ {\frac {\nu }{V}}\quad$ or $\quad V=c\ {\frac {\nu }{l}},$	(7)

which may be taken as the general equation connecting all similar systems of flow in viscous fluids.

In a number of tubes, such as may be supposed employed for experimentally investigating the phenomenon of turbulence, we have a number of such similar systems, and it will be noted that the expression is identical with that arrived at by Mr. Osborne Reynolds, in whose equation we have $c=1000$ as expressing a particular state of motion.

§ 39. Corresponding Speed.—The above expression enables us to formulate at once a law of corresponding speed for motion in any viscous fluid, for, if the physical properties of the fluid do not vary in any way $\nu$ will be constant and we have $V\propto {\frac {1}{l}},$ that is to say, for submerged model experiments, in which the condition of acceleration = constant does not apply, the smaller the model the higher the speed, in the direct proportion of the linear dimension—a rather unexpected result.

The law of corresponding speeds employed in naval architecture is primarily influenced by considerations of wave-making, in which (as shown later in the present work) the dimensional basis is acceleration $\left({\frac {\mbox{L}}{{\mbox{T}}^{2}}}\right)=$ constant; the author has proved from aerodonetic considerations that the same law obtains in connection with aerial flight.^[6] In this law we have $V$ varies as the square root of $l$ so that the two laws are incompatible—that is to say, not capable of simultaneous fulfilment. This fact is well known in connection with model experiments relating to ship resistance, the results of experiment being subject to correction according to certain rules for frictional resistance, and similar correction will be required in the case of aerodrome experiments.

If it were possible, as by employing some different fluid, to alter the value of $\nu$ when experimenting with scale models, the necessity for applying a correction might be obviated; we have:—

By Fronde's law $V^{2}=c_{1}\ l,$ where $c_{1}$ is a constant.

or, the kinematic viscosity is required to vary with the 3/2 power of the linear dimension.

We cannot always obtain fluids with viscosity to order, but if we select two fluids such as air and water, whose kinematic viscosities are, at 15° C., in the approximate ratio of 14:1, and if $l_{1}$ and $l_{2}$ represent the lengths of the two models, and $\nu _{1}$ and $\nu _{2}$ the values of the viscosities respectively, then,—

${\frac {l_{1}}{l_{2}}}=14^{\frac {2}{3}}=5.8.$

That is to say, that a model aerodrome, made to a ${\tfrac {1}{5.8}}$th scale and adapted for motion under water, will, at a velocity proportioned to the square root of its linear dimension, that is ${\tfrac {1}{2.4}}$th the full scale velocity, give rise to a geometrically similar disturbance in the fluid, and will itself undergo geometrically similar disturbance, and density for density the resistance will be proportional to the cube of the linear dimension—that is to say, in the ratio of ${\tfrac {1}{196}}$ of the full scale model; or, taking count of the relative density of air and water, the resistance of the smaller model will be approximately four times that of the greater.

 

§ 40. Energy Relation.—In all cases of purely viscous resistance the law of viscosity requires that the resistance shall vary directly as the velocity; and the whole of the energy expended disappears at once into the thermodynamic system. In cases where the resistance is dynamic—that is to say, where it is due to the continuous setting of new masses of the fluid in motion—the whole of the energy expended remains in the fluid in the kinetic form (being only subsequently frittered away), and the resistance varies as the square of the velocity. Where the resistance is due to both causes combined, as in the case of skin friction, the portions of the total resistance varying directly, and as the square, are respectively proportional to the energy expended in the two directions.

Now for any particular velocity, the total resistance—that is, the sum of the viscous and dynamic resistances—may be expressed as varying as the $n$ th power of the velocity; it is not necessary that the value of $n$ should be constant over the whole range of the $R\ V$ curve; it may be a quantity varying as a function of $V,$ the form of which is unknown; but, for the particular value of $V$ chosen we have

${\frac {dR}{dV}}=n\ V^{(n-1)},\quad {\mbox{or}}\quad =n\ {\frac {R}{V}}.$

Let $R_{1}$ be the resistance varying as $V,$ and $R_{2}$ be the resistance varying as $V^{2},$ then $R=R_{1}+R_{2},$ and we have

${\frac {dR}{dV}}={\frac {R_{1}+2\ R_{2}}{V}}=n\ {\frac {R}{V}},$

or $R_{1}+2\ R_{2}=n\ (R_{1}+R_{2}),$ from which

If we apply this to the case of a body obeying the normal law of skin-friction we have $n=1.5,$ or ${\frac {R_{2}}{R_{1}}}={\frac {.5}{.5}}=1,$ that is to say, the energy expended dynamically is equal to that expended in viscosity.

When the conditions are such that turbulence supervenes the expenditure of energy dynamically in the fluid disproportionately increases and consequently $R_{1}$ becomes greater than $R_{2},$ and in accordance with (9) the value of $n$ rises, until for very high velocities it approximates more and more closely to 2, when the law becomes more nearly $R$ varies as $V^{2}.$

The foregoing applies not only to the resistance of a plane moving tangentially through a fluid but to all cases of submerged fluid resistance; but at present the changes of the value of the index $n$ have been but imperfectly investigated.

 

§ 41. Resistance-Velocity Curve.—Let us suppose that a curve $a,\ a,\ a,\ a$ (Fig. 28) represents by its ordinates the resistance of a body of some particular geometrical form for different values of $V$ (abscissae), which we may suppose have been determined experimentally; then if $b,\ b,\ b,\ b$ be the curve for some other body of the same geometrical form but of different linear proportions, we shall, by the law of corresponding speeds, have for every given value of $R,\ V\propto {\frac {1}{l}},$ that is to say, the proportion $a\ c/b\ c$ is everywhere constant and the two curves are similar in relation to the axis of $y.$ Also we have the relation $a\ c/b\ c$ in the inverse ratio of the respective linear dimensions, so that a single curve may be employed to represent the velocity-resistance relation of any given geometrical form, the velocity being read to a scale varying according to the linear dimension of the body, so that the diameter or some other definite linear dimension of the body is some definite and constant multiple or submultiple of the scale unit employed. Thus, if the curve be plotted for a one foot diameter circular plane and one foot per second velocity is represented by one inch, then for a two foot circular plane a one foot per second velocity will be represented by two inches, that is to say, a given pressure will be developed at one-half the velocity.

Fig. 28. This result is independent of the value of the index connecting $R$ and $V,$ or generally of the indices relating $R,\ V_{1},\ A_{1}$ and $V,$ of Equation (5); it would appear to be fundamental.

 

§ 42. Resistance - Linear Curve.—We may express the relationship of linear dimension and resistance directly in the form of a curve in which $R$ is given by the ordinates as before, and $l$ is represented by the abscissae, the curve being drawn for any given value of $V.$ Now we have in Equation (5) $R=c\ v^{q}\ A^{\frac {r}{2}}\ V,$ which we may write in the form $R=c\ v^{q}\ l^{r}\ V^{r},$ where $l$ is a linear dimension on which $A$ depends; in this form the expression is symmetrical in respect of $l$ and $V,$ and we have Equation (7) $V=c\ {\frac {\nu }{l}}$ also symmetrical with regard to these quantities, so that the form of the $R\ l$ curve will be identical with the $R\ V$ curve.

Thus let $a,\ a,\ a,$ (Fig. 28) represent the $R\ V$ curve for a body of a certain geometrical form which we will entitle $(F)\ l$ where $l$ is a linear dimension, then, for any value of $l$ assigned to the body, there is for a given value of $R$ a corresponding value of $V$ such that $V\ l$ is constant; this is merely an expression of Equation (7) $V\ l=c\ \nu .$ But this holds good equally when the curve is read as an $R\ l$ curve, the value of the constant product of $V\ l$ being unaffected; consequently in reading the curve either as $R\ V$ or $R\ l$ the units are interchangeable. It may be noted that $l$ may be the length, i.e., the axial dimension of the body, or the transverse or some other dimension, without affecting the result, provided it is in all cases the same, and thus truly represents the linear size of the body.

§ 43. Other Relations.—In considering the relations of the curve of resistance we have hitherto taken the kinematic viscosity $\nu$ as constant; we will now study the consequences of taking this as a variable. So far the treatment has covered the case of variations of the velocity and linear dimensions of bodies in a fluid of constant physical properties; in supposing the viscosity to vary-we are introducing the condition of a change of fluid, or at any rate such a change in the physical state of the fluid as is equivalent thereto.

Now, (7) $V=c\ {\frac {\nu }{l}}$ is the equation to similar systems, so that the similar system when $\nu$ varies is found when $V$ or $l$ or their product $V\ l$ varies in like ratio, that is the scale value of the axis of $x$ varies with the kinematic viscosity. But by (5) $R=c\ \nu ^{q}\ (l\ V)^{r},$ or for similar systems where $l\ V\propto \nu ,$ we have $R\propto \nu ^{q}\ \nu ^{r},$ where $q+r=2,$ that is, $R\propto \nu ^{2},$ or the scale value of the axis of $y$ varies as the square of the kinematic viscosity.

The conclusion may therefore be stated that:—The resistance of a body of any definite geometrical form, in a stated aspect, may he represented as a function of its linear dimension (that is its size) and its velocity, by means of a single curve which may be termed its characteristic curve of resistance, the form of which is constant whether the abscissae represent linear dimension or velocity, and whatever the value of the kinematic viscosity may be. And further, in the interpretation of the curve the linear and velocity quantities alternatively represented by abscissae are interchangeable, and the scale value of the axis of $x$ is proportional to the magnitude of the kinematic viscosity, and the scale value of the axis of $y$ to the square of the kinematic viscosity.

 

§ 44. Form of Characteristic Curve.—The form of the characteristic curve of resistance for different forms of body can, in all probability, only be determined experimentally. There are three ways by which the curve could be plotted, (a) by experimentally determining $R$ for different values of $V$ (or vice versa), for any given body; (b) by the determination of the resistances of a number of bodies of geometrically similar form but of different scale dimensions, at any standard velocity; and, (c)  by employing fluids of different viscosity and plotting indirectly, using a standard body at a standard velocity. The same curve should result in every case.

Of the three methods the last (c) may be dismissed as impracticable; the two former, (a) and (b), are, however, well suited to experimental conditions, and would furnish a complete check on the foregoing investigation. At present the experimental data are fragmentary and the evidence inconclusive.

The general properties of the curve, common to all forms of body, may be gathered from the circumstances of the problem. For very small values of $V$ we allow that quantities varying as $V^{1.5}$ and $V^{2}$ become negligible, and the curve will be of the form $R\propto V$ and leave the origin as an inclined straight line. When the velocity is very great resistances that vary as the lower powers of the velocity will be negligible in comparison to those that vary as $V^{2},$ and consequently the curve will approach asymptotically to the form $R\propto V^{2}.$ It is questionable whether the $R\propto V$ stage can exist when the viscous reaction of the fluid is due wholly to its own inertia; in the demonstration of the "normal law of skin-friction" it was shown that this condition results in the "1.5 power" law, and it would appear probable that this law is capable of more general demonstration, in which case the form of the curve (in an infinite region) will be limited to a minimum index of 1.5 and the straight line stage will disappear.

It is to be expected in any case that in actual experiment the ordinary viscous law will be found to apply for very low velocities on account of the fact that the size of the vessel containing fluid in which the experiment is performed cannot be made infinite, and for very low velocities the viscous stress or part thereof will be carried across to the walls of the vessel. Under these circumstances the condition that the reaction of the viscous substance shall be borne by its own inertia will not apply; it is consequently of importance that experiments should be conducted in as large a tank as can be conveniently employed.

§ 45. Consequences of interchangeability of V and l.—It is evident that the general results relating to the form of the curve which have been deduced from the obvious relations of resistance and velocity apply to the less obvious relationship of resistance and linear dimension, owing to the interchangeability of V and l previously demonstrated (§ 42); we therefore see that—

For small similar bodies, obeying the viscous law, the resistance varies with the linear dimension, that is as the square root of their area.

For bodies of larger size, the resistance may be found to vary as the 1.5 power of the linear dimension, that is as the .75 power of the area.

For bodies of very large size, the resistance will approach to vary as the linear dimension squared, that is directly as the area.

For planes moving tangentially it would appear possible that the latter condition is never attained but that some lower power may prove to be the limiting condition.

 

§ 46. Comparison of Theory with Experiment.—The foregoing theory receives substantial support from the experimental work of Froude, Dines, Allen and others.

In a series of experiments to determine the skin-friction of surfaces moving tangentially in sea water. Froude found that an increase in area is not accompanied by a proportionate increase in resistance; he also found that the index connecting resistance and velocity is in general less than 2, the mean result of several experiments giving 1.92. Colonel Beaufoy, also experimenting in sea water, gives the value 1.7 to 1.8.

Dines, experimenting in the open air, obtained results that have some interest from the present standpoint. In spite of some conflicting evidence, it would, in the main, appear that, under the conditions of experiment, the V² law is a very close approximation to the truth. In this Dines agrees with the previous experiments of Newton. Hutton, and others, and with the contemporary work of Langley.

It is to be inferred that in cases of direct resistance the Stokes (R ∝ V) and Allen (R ∝V^1.5) stages are confined to bodies of very small size and very low velocity. The bodies employed by Dines varied from some few square inches to some few square feet area.

Allen's work in connection with the present subject is of the greatest moment. The present application of dimensional theory is largely due to him as also its experimental verification. His investigations principally relate to spherical bodies of very small dimensions, and demonstrate positively that which has been already inferred negatively, i.e., the small size and low velocities belonging to the Stokes and Allen stages of the characteristic curve.

§ 47. Froude's Experiments.—Owing to the condition of constant geometrical form not having been complied with in these experiments, some doubt exists as to the exactitude of the theory in its application. The planes employed differed in length alone, and it is evident that the skin-friction on a long narrow plane moving endwise will be less proportionately than one of more nearly square proportions, and consequently the effect of departing from the conditions will be to show a fictitiously low co-efficient for the longer of the planes employed.

If the width of the planes in proportion to their fore and aft length were sufficiently great, this effect would be negligible, as under these circumstances the sectional area of the fluid affected would vary substantially with the width itself; we will provisionally assume this to have been the case, and treat the fore and aft length as the l of the dimensional equation, at the same time bearing in mind the direction in which error is to be expected.

The first three series of experiments are as follows, the figure quoted being in each case the mean resistance per square foot taken over the whole area at a velocity of 10 feet per second:—

Nature of surface.	Length of Surface (Fore and Aft Dimension).
	2 feet.	8 feet.	20 feet.
Varnish	.41	.325	.278
Paraffin	.38	.314	.271
Tinfoil	.30	.278	.262

The values of the index calculated from the above observations, on the basis of the dimensional equation, are given in columns 1 and 2 of the Table that follows.

The observed index, that is to say the index calculated from observations made at different velocities is (taking the mean of all observations) given in column 3.

	Observation at 2 feet in relation to Observation
	At 8 feet.	At 20 feet.
	(1)	(2)	(3)
Varnish	1.825	1.z832	1.88
Paraffin	1.86	1.854	1.94
Tinfoil	1.94	1.940	1.97

We here find the theory receives confirmation, inasmuch as, firstly, the order in which the indices arrange themselves is the same whether the computation is made, as by Froude, on the basis of experiments at different velocities, or as done here on the basis of change of linear dimension; and secondly, the quantitative discrepancy between columns 1, 2 and 3 is in the direction anticipated from the nature of the provisional assumption.

§ 48. Fronde's Experiments (continued)—Roughened Surfaces.—When we examine the cases of roughened surface which form part of the series of experiments quoted, we find results that are not capable of such ready interpretation. In the case, for instance, of a surface coated with coarse sand, the index determined by Froude from experiments at different velocities was found not to differ sensibly from the maximum possible; that is, the index value is given as = 2. The constant velocity data in this case are:—

Surface.	2 feet.	8 feet.	20 feet.
Coarse sand . . .	1.10	.714	.588

Calculating as before, we obtain the index values 1.69 and 1.728 respectively, which have no apparent resemblance to Froude's value. It is not possible to attribute this failure to the dissimilarity of geometrical proportion, for the previous calculations give an indication of the maximum value of the error introduced on this account; it is evident, therefore, that the cause must be sought elsewhere.

In the first three examples the nature of the surface is physically speaking smooth, that is to say, the roughness, such as it is, may be considered as molecular. Now we know in such a case that the drag produced on the fluid arises from the viscous connection between the film of the fluid actually contiguous to the surface and the strata more remote, and this connection—viscosity—is one fully taken account of in the equation; and even if the molecular roughness of one substance differs from that of another, the application of the theory will not be affected. When, however, we have to deal with a physical roughness, the conditions are altered, and in order that the theory should apply, the scale of the roughness, i.e., the coarseness of the sand, must be increased as the length of the plane is increased; that is to say, the contour of the protuberances that constitute the roughness of surface becomes part of the geometrical form of the body. Thus, in the example quoted, the roughness, and so the resistance, is less on the 8 feet and 20 feet planes than it should be, and so the results are not comparable. In all probability the difference between the values of the resistance for varnish, paraffin, and tinfoil is due to some difference in the physical roughness of these bodies, and so we shall expect to find the best agreement with theory in the case of tinfoil (which shows the smallest co-efficient); this is actually the case.

§ 49. Dines' Experiments.—The most suggestive experiments of Dines are those in which wind planes of different area are balanced about a vertical axis and the relative pressure so determined. Mr. Dines found that the pressure on normal planes does not increase in proportion to their area, but is proportionately greater on small than on large planes. The actual results obtained by observations on planes 6 ft. by 7 ft., 3 ft. by 3 ft., and 1 ft. 6 in. by 1 ft. 6 in., were that the pressure per square foot on a plane 6 ft. by 7 ft. is only 78 per cent. of that on one 3 ft. square, and that on the plane 3 ft. square is 89 per cent. of that on the 1 ft. 6 in. square plane. The actual velocity of the wind in which these experiments were made is not stated.

On the other hand, Mr. Dines specifically states that he finds the wind pressure on the normal plane and on bodies generally varies strictly as the square of the velocity, a result which it is difficult, in view of dimensional theory, to harmonise with the above experiments.

It is probable that the departure from the V² law is less than is indicated by the balanced plane experiment, owing to the smaller plane being unduly affected in each case by its proximity to the larger one. It is conceivable that the smaller plane being situated in the counterwake of the larger, will in effect be surrounded by air moving with above the normal wind velocity, and so show a fictitiously high pressure value. Mr. Dines' elegant method of determining the V² law, by balancing against centrifugal force, would appear to be quite above suspicion, although it may not be sufficiently sensitive to demonstrate the departure from the law, which for the normal plane is certainly very small indeed. In any case the results, without some such explanation as given, are not altogether consistent, and a repetition of these experiments ought to be made.

§ 50. Allen's Experiments.^[7]—Mr. H. S. Allen, experimenting with bubbles and small solid spheres in liquids, found that for very small velocities the viscous law holds good, whereas for very great velocities the V² law prevails ; he also shows that there is an intermediate well-defined range, over which the V^1.5 law applies. His results are summarised as follows:—

"Three distinct stages have been recognised:

"(1) When the velocity is sufficiently small the motion agrees with that deduced theoretically by Stokes for non-sinuous motion, on the assumption that no slipping occurs at the boundary ; in such motion the resistance is proportional to the velocity.

"(2) When the velocity is greater than a definite critical value, the terminal velocity of small bubbles and solid spheres is proportional to the radius, less a small constant; it may be expressed by the formula given.

"(3) For velocities considerably greater than those just considered, the law of resistance is that which Sir Isaac Newton deduced from his experiments, namely, that the resistance is proportional to the square of the velocity."

Of the above three stages, (2) corresponds approximately to the law, resistance varies as the 1.5th power of the velocity, for, where the force overcoming the resistance is supplied by the difference of specific gravity of the fluid and the sphere, we have—

	$R\propto l^{3},$
and	$R\propto V^{r}\ l^{r}\ \nu ^{2-r},$
∴	$l^{3}\propto V^{r}\ l^{r}\ \nu ^{2-r},$
∴	$l^{3-r}\propto V^{r}\ \nu ^{2-r},$
that is,	$V\propto {\frac {l^{\frac {3-r}{r}}}{\nu ^{\frac {2-r}{r}}}}.$
But for a given fluid $\nu$ is constant, and we have—
	$V\propto {\frac {l^{\frac {3-r}{r}}}{,}}$
or if	$V\propto l^{n}\quad n={\frac {3-r}{r}},$  or $\ r={\frac {3}{n+1}}.$

Now, if in stage (2) (Fig. 29) we ignore the small constant, we have $V\propto l$ or $n=1,$ and $r={\frac {3}{2}},$ that is to say, the general expression in this case becomes:—

$R=c\ \nu ^{\frac {1}{2}}\ l^{1{\frac {1}{2}}}\ V^{1{\frac {1}{2}}},$

that is, during this stage the resistance follows the normal law of skin-friction, or Allen's law.

 

§ 51. Characteristic Curve, Spherical Body.—The form of the experimental curve, as plotted by Mr. Allen, is given in Fig. 29, in which ordinates $=V,$ and abscissae = values of linear dimension, i.e., radius of sphere. The first stage or Stokes portion of this curve is a parabola, $V\propto l^{2};$ this corresponds to an $r$ value = unity; the second stage is approximately a straight line, the value of $r$ here being as shown 1.5; the third (or Newtonian) stage of the curve, not shown on this plotting, has an $r$ value equal 2, that is $n=.5$ or $V\propto l^{.5}.$ This form of plotting is the outcome of the method of experiment, i.e., measuring the limiting velocity acquired under the influence of gravity; if we re-plot as a resistance-velocity diagram (Fig. 30), the size of the body being supposed constant, we are able to obtain a general idea of the “characteristic curve of resistance” for a spherical body. In Figs. 29 and 30 the curve does not extend to the $V^{2}$ stage; in Fig. 31 we have the three stages represented diagrammatically, firstly, by a straight line departure from the origin where $R\propto V$—this is the Stokes stage; next we have a section of the curve following the $R\propto V^{1.5}$ law, the Allen stage, and lastly the curve will approximate to a parabola where $R\propto V^{2}.$ The latter stage is that investigated both theoretically and experimentally by Newton ("Principia," Book II., Section VII.), determinations being made both in water and air;
Fig. 29. also in the year 1719 by Dr. Desaguliers, who employed spherical bladders let fall from the cupola of St. Paul's. Newton's theoretical investigations were based on the hypothetical medium of discrete particles, but the experimental verification was sufficiently close, qualitatively, to establish the velocity squared law as substantially correct, so far at least as the sizes of sphere and velocities employed in his own and Desaguliers' experiments are concerned.

§ 52. Physical Meaning of Change of Index.—The nature of the alteration in the system of disturbance that accompanies each change of "law" is a matter of considerable interest. The Stokes law is based on a system of motion of the fluid that has been mathematically investigated and the lines of flow plotted from an equation.^[8] It may be remarked that this system of motion can never exist in its entirety, for it involves an infinite

Fig. 30.

quantity of momentum and an infinite quantity of energy^[9]; in other words, the steady state involves a force applied for an infinite time through an infinite distance; it also constitutes a violation of the principle of no momentum of § 5.

If we suppose that the stress, due to the propulsion or to the resistance of the body, be transmitted by viscosity to the walls of the vessel, as when the body is moving quite slowly, or when the thickness of intervening fluid is small; then the resistance will

Fig. 31.

evidently follow the ordinary viscous law. When, however, the viscous drag is resisted by the inertia of the fluid, that is to say, there is no continuity of viscous stress from the body to the walls of the vessel, then it would appear probable that the law of skin-friction applies. If this view is correct, the extent of the stage where R varies as V will depend, as already stated, upon the size of the containing vessel.

§ 53. Change in Index Value (continued).—We have so far confined ourselves to the discussion of the first change of index, that which takes place when the curve passes from the Stokes to the Allen stage.

The second change of index value evidently takes place when the motion of the fluid becomes turbulent, for it is then that the conditions leading to the normal law of skin friction are violated, and the energy relation becomes disturbed. In all probability also the V² law, in cases involving other than pure skin-friction, is closely associated with the phenomenon of discontinuity. A system of flow of the discontinuous type is almost certainly accompanied by resistance following the V² law.

The conclusions of this and § 52 at present lack experimental demonstration. There would appear to be some evidence to show that the Stokes stage may exist independently of the size of the vessel; this at least is a conclusion reached by Allen. If this should prove to be the case the explanation here given will need modification.

§ 54. The Transition Stages of the Characteristic Curve.—The junction or transition portions of the curve connecting the various stages are not angular as shown diagrammatically in Fig. 31, but pass gradually from the one to the other. The transition stages, however, are not such as to mask the distinct individuality of each portion of the curve, but merely enough to render uncertain the precise point at which the change of "law" takes place.

It appears that there is a small departure from the exact expression given, both in the second stage (as found by Mr. Allen) and in the third stage. In the latter case we may suppose that when the velocity becomes very great the geometrical form of the lines of flow becomes sensibly constant, and such resistance as is due to viscosity will then vary as the velocity, the net resistance curve being thus the sum of the ordinates of a parabola

Fig. 32.

and an inclined straight line. It can be shown geometrically that this results in the curve of resistance approximating to a parallel to the true parabola as shown in Fig. 32. In the second stage we have the experimental result of Mr. Allen as a guide. We know that if the second stage law were to hold good down to zero velocity, we should have a certain small residuary resistance (see Fig. 29). This means on the velocity-resistance diagram that the origin for the 1.5 index curve will be situated a short distance up the axis of y; we may conveniently construe this as an approximate parallel to the curve struck from the true origin, when Fig. 32 will represent the manner in which the resistance curve may be supposed to be built up.

§ 55. Some Difficulties of Theory.—In all cases of skin-friction where the index exceeds 1.5 the motion is accompanied by turbulence, and if the value of the index rises to 2, as it would appear to do approximately in the case of the roughened surface, then the dimensional equation (as pointed out by Allen) shows that the resistance is independent of viscosity, and the whole of the energy is expended dynamically in producing fluid motion. Under these circumstances we must regard viscosity as merely acting as a gearing by which rotational motion is imparted to the fluid, although it is difficult to understand how such a gearing can be continually imparting rotation to new masses of fluid without a certain amount of slip; and such slip would betoken an expenditure of energy in viscous motion and necessitate the value of the index being less than 2.

Beyond this it is difficult to conceive of the resistance being independent of the value of viscosity without being independent of the existence of viscosity, which appears to be absurd. Consequently it is probable that, so long as the effect of elasticity of the fluid is not felt, the value of the index, connecting resistance and velocity or resistance and linear dimension, can never reach its limiting value, 2, but must always fall short of it by some small quantity.

It is known, from experiments on resistance in the flight of projectiles, that for velocities approaching the velocity of sound, the index may rise considerably above the limiting value given in the foregoing theory ; and therefore we may expect to find in general that the experimental determinations, except for velocities quite small in comparison with that of wave motion, will be in excess of those indicated by the theory. It is evident that in elasticity we have a factor foreign to the dimensional theory as given, and the existence of such a factor invalidates the hypothesis upon which the theory is founded.

An apparent discrepancy occurs in the case of some experiments made by Newton (Book II., Section VI.), who found, from the motion of a pendulum whose spherical "bob" was immersed in water, that the resistance was augmented in more than the duplicate ratio of the velocity.

Newton supposed this to be an error due to the narrowness of the trough employed, but this in the light of dimensional theory is insufficient.

The probable explanation is that for a large arc the discontinuous type of motion has time to establish itself on each swing, whereas for small arcs of motion the flow has not had time to fully develop discontinuity; for very small arcs the flow will approximate to the Eulerian form (compare Chap. III.). Consequently the resistance for small amplitude is far less than is the case for continuous motion, and thus factors are introduced outside the dimensional hypothesis, which presumes a steady state.

§ 56. General Conclusions.—The importance of the results attained in the present chapter, in relation to aerial flight, is to some extent an unknown quantity.

It is evident that under ordinary conditions the law of viscosity does not apply, and it would appear further that the tangential resistance does not follow the normal law of skin friction, but that the conditions commonly involve turbulence, and $R$ varies as some higher power of $V$ between $V^{1.5}$ and $V^{2}.$ It is highly probable that the conditions may be different in the case of the smaller flying insects, such as flies, mosquitoes, etc., and it may be the relatively greater importance of viscosity in such cases that is primarily responsible for the peculiarities of insect flight.

For large birds and flying machines the "$R$ varies as $V^{2}$" law is probably accurate enough for ordinary computations of resistance, whether frictional or direct. The $V^{2}$ law is generally assumed in the present work as sufficiently near the truth; the assumption of one law for both classes of resistance results in a simplification of method which fully justifies its employment, even at the expense of some small degree of accuracy.

The meaning of the statement (§ 36) that viscosity gives a scale to the fluid, may be illustrated by supposing a "blue-bottle" to find itself transformed into a common fly (supposing the two to be strictly proportional in their parts): it would find that the apparent viscosity of the air had increased; in other words, the air would appear to be more "sticky" than usual. The same fact is familiar in other cases—for example, the difference in character of a large and a small flame, etc. Other physical properties are capable of giving a scale to a fluid: thus elasticity as demonstrated by the length traversed by a wave in unit time; surface tension as demonstrated by the velocity of slowest surface wave.

One of the least satisfactory results of dimensional theory, so far as revealed by a comparison with conclusions that would be naturally formed from experience, is the inverse relation that exists for homomorphous motion between $V$ and $l.$ It would appear that for bodies of similar form any state of motion—say the state when discontinuity sets in—is reached when their respective velocities are in the inverse ratio of the linear dimension. Thus, if a salmon and a herring were geometrically proportional, the herring would be capable of a higher velocity, without ceasing to be of streamline form (by definition), than the salmon in the inverse proportion of their respective lengths. Now this seems very unsatisfactory, for a whale would be scarcely capable of locomotion without carrying a dead water region in its wake—a most improbable conclusion. Certain explanations are possible, but the author has been unable up to the present to find any conclusive solution to the difficulty.

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1. The foregoing demonstration is here presented for the first time by the author; the experimental fact was discovered by Mr. H. S. Allen (compare § 50). The relation $F/V^{1.5}=$ const. may appropriately be termed Allen's law. It is evident in the above investigation that the balance of viscous and dynamic forces is demonstrated for all corresponding layers of the region each to each, for any number of cases of $V$ variation, and consequently the method is comprehensive, and includes both the plus and minus momentum of the wake and counterwake currents.
2. The method of dimensions presumes geometrical similarity of figure.
3. Phil. Mag., September and November, 1900.
4. An erratum published in Volume 2 has been applied: "P. 53, line 9 from foot, for 'Poissuille' read 'Poisouille.' " (Wikisource contributor note)
5. Poynting and Thomson, “Properties of Matter,” Chap. XVIII.
6. Aerodonetics.
7. Phil. Mag., September and November, 1900.
8. Stokes' Scientific Papers (t. iii.).
9. "Hydrodynamics," II. Lamb, 1906 ed., p. 553.