Pilot Training - Theory of Flight: Fundamentals

7. Definition.-Officially defined by the National Advisory Committee for Aeronautics, aerodynamics is the branch of dynamics which treats of motion of the air and other gaseous fluids and of the forces on solids in motion relative to such fluids. This science, one, of the branches of physics, offers the basic principles of flight and is the mainstay of aeronautics.

8. Purpose.-a. The purpose of this section is to present the salient principles of flight without recourse to higher mathematics and without exhaustive analysis. This is due to the fact that the time which is and could possibly be allotted to, theory of flight precludes a complete treatment. Many excellent texts are available on aerodynamics. The student who desires to amplify his knowledge beyond the scope of this manual may refer to some of the commercially published volumes, and the reports and technical notes of the National Advisory Committee on Aeronautics.

b. To acquire the basic facts and theories of aerodynamics, initial understanding must embrace pertinent fundamental physical laws and concepts. The air must be appreciated as a fluid having mass and weight and as having motion which is, as all motion, purely relative.

c. The construction and operation of the wind tunnel are dealt with since it has proved invaluable to aerodynamics and to airplane development.

d. With these considerations as a background it is possible to proceed with the salient phases of aerodynamics, including airfoil characteristics, dimensions and combinations, parasite drag, balance, stability, propeller theory, performance, controllability, and maneuverability. No attempt is made at complete treatment. Rather are those principles stressed which account for the airplane in its present form and for its behavior in flight. These are the items of primary interest to operating personnel.

9. Distinction between airplane and other aeronautical vehicles.-Normal surface craft, submarines, and airships, entirely when at rest and principally when underway, depend for sustentation upon the physical fact called the "Principle of Archimedes," namely, that a body submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced. Ships and submarines are said to be supported by hydrostatic forces, and airships by aerostatic. forces. The science dealing with the latter forces is "aerostatics."

a. The aquaplane and the hydroplane, underway, displace much less water than that required to support them and their loads at rest. They are supported principally by hydrodynamic forces. Aircraft, like surface craft, may be supported by static or dynamic forces. Those aircraft supported by static forces such as free balloons, blimps, dirigibles, etc., are called "lighter-than--air"" craft or airships. Those aircraft supported by dynamic forces are called "heavier-than-air" craft.

b. Heavier-than-air craft may be classified as ornithopters, helicopters, autogyros, gliders, and airplanes.

(1) The ornithopter is an alleged flying machine that is supported, birdlike, by the flapping of wings. The ornithopter may be compared with the Australian bird called the Kiwi, which runs around flapping its wings but never takes off.

(2) The helicopter is a flying machine supported by the upward thrust of horizontally rotating propellers. Successful types of the true helicopter do not exist at present.

(3) The autogyro is a flying machine having a conventional and conventionally placed engine-propeller unit and is supported by the upward thrust of a large rotor or windmill resembling a propeller rotating in a substantially horizontal plane, the rotation being effected by the air forces acting upon it. These air forces are caused by the forward motion of the craft through the air. The autogyro is the most successful of the unconventional types and promises to have military value. The autogyro has a number of excellent features in that it can take off with an exceptionally small run, climb very steeply, and cannot be accidentally spun. Its forward speed may be reduced to the point where it will momentarily hover over an object. Its descent may be vertical at a speed as low as a parachute, and it can be landed within "the circle" without rolling out of it. The autogyro does not possess a great deal of speed at present, and maneuverability is practically nil, but its good features are such as to assure its retention and further development as a type.

(4) The glider is simply a lightweight airplane without motor. It is given initial flight by catapulting or towing and once aloft it may glide to the ground or the pilot may ride upward moving air currents. These currents will supply the energy necessary to sustain flight. A glider being supported by updrafts is said to be "soaring" and the machine is often called a "sailplane." The glider is primarily a sporting or training device.

(5) There remains the airplane as the most , common most successful, and most highly developed type of flying machine. It is supported by dynamic forces caused by air being passed over its wings. The relative motion of the air is effected by having the craft forced through the air by an engine-propeller unit.

10. Fundamental conceptions.-The mathematics of aerodynamics is rather deep and involved but for the purpose of this manual it is less pretentious than that of any other branch of engineering.

a. Force is the term applied to the exertion of muscular effort or to any effort which may be employed in lieu thereof. Thus in pushing with the shoulder against a wall a force is exerted.

b. Matter is a term applied to describe objects or substances that occupy space or possess weight and hence muscular effort or some other force is required to support or move them.

c. Weight is the force urging each body toward the ground and exists whether the body be a liquid, gas, or solid. The air itself is matter and hence a given volume will have a definite weight for definite conditions of temperature and atmospheric pressure.

d. Vectors.-Forces, velocities, and other quantities that involve direction as well as magnitude are said to be "vector quantities" or "vectors" Those quantities that involve magnitude alone are called scaler quantities. A vector may be represented by a straight line, the length of which designates the magnitude of the quantity. An arrowhead on one end indicates the direction of the line of action while the other end constitutes the point of application. Thus a force of 4 pounds applied at A and acting toward the right may be represented by the 1-inch line below, where each pound is 1/4 inch.

e. Resultant.-(1) The resultant of two vectors is defined as the single vector which will produce the same effect upon a body as is produced by the joint action of the two vectors. Thus if two parallel forces are acting in the same direction their resultant is equal to the sum of the two forces, whereas if they are acting in opposite directions the resultant is equal to the difference between them and acts in the direction of the greater force. If these two vectors, such as two

forces, are not parallel, the resultant will lie within the angle formed by the two vectors and is determined as follows:

(a) From the point of application A, draw line AB in the direction of the first force and of length representing its magnitude to a definite scale;

(b) From A draw line AC in the direction of the second force and of length representing its magnitude, to the same scale as AB;

(c) With lines AB and AC as two sides of a parallelogram, complete the same in dotted lines;

(d) Draw the diagonal from the point of application A. This diagonal AD represents in magnitude and direction the resultant of the two forces AB and AC.

(2) As a practical example of the use of vector diagrams, take the case of an airplane flying on a course east at airspeed of 65 m. p. h. Wind from southeast, velocity 25 m. p. h. The resultant of the wind and airspeed velocity vectors will give the vector of the ground speed. Let A be the point of departure of the airplane. Draw the wind vector AB to scale 25 m. p. h. in direction 315°, the direction in which the wind is blowing from the point of departure. Draw the airspeed vector AC to scale 65 m. p. h. Complete the parallelogram ABCD. The diagonal AD is the resultant ground speed, 50 miles per hour in the direction 69°.

(3) Vector diagrams enable the student to solve graphically, with ease and simplicity, problems in forces, velocities, etc., which would otherwise require tedious mathematical calculation. It must be emphasized that accurate results depend upon accurate construction and measurement of the vector diagram.

f. Component of a force.-A force vector will obviously produce a maximum effect along its line of action. Nevertheless, in any other

direction an effective force is evidenced, this effective force being termed a "component" for the stipulated direction. To find the component of a force in any given direction

(1) Represent the force by a line such as AB in figure 5

(2) Using AB as a diagonal, construct upon it a rectangle, the sides AC and AD of which are respectively parallel and perpendicular to the direction of the required component;

(3) The length of the side A C which is parallel to the given direction represents the magnitude of the desired component.

11. Moment of a force equilibrium.-The tendency of a force to produce a rotation around a given axis is called the moment of the force with respect to that axis.

a. The amount and direction of the moment of a force depend upon the direction -of the force and upon its distance from the axis. The perpendicular distance from the axis to the line of the force is called the moment arm and the moment is measured by the product of the force and the moment arm. Thus a force of 10 pounds acting at a distance of 2 feet from the axis exerts a turning moment of 20 footpounds,

b. In order to distinguish between moments tending to produce rotation in opposite directions, those tending to produce a clockwise rotation are called positive moments; those in the opposite direction, negative moments.

c. If the algebraic sum of the moments is zero, that is, all the positive or clockwise moments equal the negative or counterclockwise moments, there will be no rotation. It is usual to express this in the form SM=0. S is the Greek letter sigma and SM means the sum of all the moments M both positive and negative.

Taking moments about point A:

.

or using the usual notation SM=0 Since the moments balance about axis A there is no rotation.

d. There are 8 pounds of force acting down, and unless the axis is supported by an upward force of 8 pounds, there will be downward movement but no rotation. If the total vertical forces equal zero there will be neither rotation nor translation and the body would then be said to be in equilibrium.

Vertical forces acting upward are called positive; those acting down, negative. For equilibrium the algebraic sum of the vertical forces must equal zero. This is indicated by the symbols SV=0.

e. A group of horizontal forces have the same relation and for horizontal balance SH=0.

f. Forces at an inclined angle can be resolved into their vertical and horizontal components as shown in the section on vectors. Hence the conditions for equilibrium are

SH=0    SV=0    SM=0

g. It is important for the student of aerodynamics to realize that the above conditions for equilibrium hold true if such a system of forces is imposed upon a body (e. g. an aeroplane) which is moving, since the original motion is in no wise affected by this superimposed system.

12. Velocity.-Velocity of a moving body or particle is the distance traveled divided by the time during any particular period. Velocity may be expressed as miles per hour (m. p. h. or mi./hr.), feet per second (f. p. s. or ft./sec.) or any other ratio employing standard units. It is customary in aerodynamics in the United States to use the ratios mi./hr. and ft./sec. as standard.

13. Acceleration.-a. If a moving body changes its velocity it is said to accelerate. Acceleration is a rate of change of velocity or the average change in velocity in a given time.

b. The most common example of acceleration is that of a falling body. If a body is dropped in a vacuum its velocity at the start will be zero feet per second. At the end of the first second its velocity will be 32.2 ft./sec. and its acceleration is expressed as (32.2 ft./sec.-0 ft./sec.)/sec. or 32.2 ft./sec./sec. At the end of the next second its velocity is 64.4 ft./sec. indicating a change in velocity of 32.2 ft./see. in one second.

c. Accelerations and forces are closely associated, so closely in fact, that it is frequent that in discussing forces reference is made to the acceleration produced rather than to the force itself. This is done for purposes of comparison.

14. Mass.-In dealing with moving bodies it is necessary to use not only the weight of the body but also the acceleration of gravity.

This relation is used as a ratio of, W/G and is called "mass".

15. Specific gravity-density.-a. Specific gravity.-The ratio of the weight of a solid or liquid to the weight of an equal volume of water at some standard temperature is called its specific gravity. The specific gravity of gases is usually referred to hydrogen or air.

b. Density.-(1) The density of a body is its weight per unit volume, example W/V

(2) The mass density is used in discussing moving bodies and is the ratio of mass to volume and is expressed as M/V.

(3) In the case of solids the ratio of mass to volume cannot change. This is not the case with gases, however. If a sounding balloon is inflated with air until it has a volume of 2 cubic feet, a, definite mass of air will be contained therein. By the heating of the balloon it can be expanded to about 4 cubic feet. The volume has been doubled but the amount of air trapped within the envelope remains the same. Hence the density of the air is just half what it was before heating. If instead of raising the temperature of the air and thus causing inflation of the balloon, it were compressed until of only 1 cubic foot volume, the density would be twice the initial value. In general, for a given kind of gas, the product of the pressure and volume, divided by the mass and by the absolute temperature is constant or -

Where P=pressure, V=volume, m=mass, T=absolute temperature and R is a constant depending only on the kind of gas. This is the law of gases and is a combination of two well-known laws of physics, that is, "Boyle's Law" and "Charles' Law".

16. Standard atmosphere.-a. A cubic foot of air will weigh more, the greater the atmospheric pressure, and the lower the, temperature. Since the reaction of the air against an airfoil depends upon the mass of air coming in contact with it, there must be a definite standard for the measurement of mass. The National Advisory Committee for Aeronautics has adopted the following conditions as standard: dry air under a barometric pressure of 29.92 inches of mercury at a temperature of 15° C. A cubic foot of air under these conditions weighs 0.0765 pound. Air containing water vapor is somewhat lighter than dry air.

b. The earth being surrounded by a blanket of air, estimated as more than 200 miles in thickness, it is obvious that with altitude the density must decrease. Increase in altitude leaves less air above to compress that below. Figure 8 shows the relations existing in temperature, relative density, and relative pressure with altitude.

17. Center of gravity.-a. Newton's law of universal gravitation states that-any two bodies in the universe attract each other with a force which is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. This holds whether the bodies are two planets or an airplane and the earth. In fact, every particle of any body at or near the earth's surface will be pulled toward the center of this greater mass, and the total force will be equal to the sum of the pulls on the individual particles. This total force is termed the weight of the body and may be considered as acting at a point in the body called the center of gravity.

b. This center of gravity is the point of application of the resultant of all the pulls on individual particles of the body. It may be determined as follows: The body, an airplane, is suspended from a cable clear of the ground. The cable will assume a vertical position and be subjected to a tension equal to the weight of the airplane. The same will hold regardless of the point of attachment of the cable and consequent attitude of the airplane. Since the line of action of the weight of the airplane will be that of the cable in all cases, the point of intersection of the lines of action for two or more attitudes must represent the point of application of the resultant force, namely, the center of gravity. Figure 9 illustrates one method.

18. Graphs.-a. The position of a point may be fixed by reference to two known straight lines intersecting at right angles in the same plane, as the point P (0X and 0Y of fig. 7). Such lines are known as rectangular coordinate axes. The horizontal line 0X is called

the "axis of abscissae" or 'X axis." The vertical line 0Y is called the "axis of ordinates" or "Y axis." The point of intersection is called the origin. The abscissa of a point is its horizontal distance from 0Y; its ordinate is its vertical distance from 0X. These given, the position of the point is determined. A familiar example of plotting the position of a point by reference to coordinate axes is the latitude and longitude of a point on a Mercator Chart.

b. A succession of points may be plotted with reference to coordinate axes and these points connected by a smooth line, thus forming a curve. Such curves are frequently the most convenient and the clearest way of representing some physical relation. For convenience squared or cross section paper is used for this purpose. A typical example is that shown in figure 8.

19. Relative motion.-a. If a body changes its position it is said to be in motion. The position of the body is fixed by its distance from surrounding objects. Hence, a body which has moved has changed its position with respect to some other object regarded as fixed. In other words, there has been relative motion. A person sitting quietly in a Pullman seat may be said to be not moving though the train is traveling at 60 miles per hour. Here the individual is at rest with respect to the Pullman in which he is riding though the train itself is rushing rapidly over the earth's surface which is considered as fixed in stating the velocity over the ground.

b. So it is with objects sustained in the air. On a calm day a balloon will maintain a fixed position over its point of departure from the ground. With a wind blowing it is carried away from the point of departure at a velocity equal to that of the wind. There is no relative motion between the balloon and the wind if it maintains its altitude but there is between the balloon and the ground. The airplane, on the other hand, depends for its sustentation on relative motion between itself and the surrounding air. This relative motion is expressed as the airspeed. On a calm day an airplane requiring an airspeed of 60 in. p. b. to maintain itself in flight will leave the point of departure at this velocity. If headed into a 60 in. p. h. wind, the airplane will remain over its point of departure or will have zero ground speed, that is, no relative motion between the airplane and the ground.

c. Since all motion is relative, velocity which fixes the rate of motion is also relative, and a body may have at the same time different velocities (either in amount or direction) with respect to different bodies. Thus, an airplane flying at 60 m. p. h. airspeed against a 20 m. p. h. breeze will have a velocity of 60 in. p. h. with respect to the air and a velocity of 40 m. p. h. with respect to the ground.

20. Momentum.-A body moving at uniform velocity by reason of its inertia (resistance to change of motion) tends to continue at uniform velocity. This tendency is called momentum and is measured by the product of the mass and its velocity. This momentum can only be changed by the application of some external force to the body. When the force is applied to the body, its mass will not change, hence the velocity must change.

21. Newton's laws of motion.-a. Newton has given as three laws of motion:

(1) Every body tends to remain in a state of rest or of uniform motion in a straight line except insofar as it is acted upon by an impressed force.

(2) Change of momentum is proportional to the impressed force and to the time during which it acts and takes place in the line of action of that force.

(3) To every action there is an equal and opposite reaction.

A clear understanding of these laws is absolutely essential to the study of aerodynamics.

b. The first law is generally spoken of as the law of inertia. It simply means that a body at rest will not move unless force is applied to it. If it is moving at uniform speed in a straight line, force must be applied to increase or decrease that speed.

c. The second law means that if a body moving at uniform speed is acted upon by an outside force, the change of motion will be proportional to the magnitude of the force acting, and the new direction of motion will be the resultant of two components, one the original motion the other that produced in the line of action of the force. Mathematically the law may be expressed by

kF= Va        (4)

(4) where F, a force acting on a mass W, produces a rate of change of motion (acceleration) a, and k is a constant depending on the units used to measure F, W, and a. By experiment it may be determined that k=g, the acceleration of gravity, and the equation therefore becomes

Fg=Wa        (5)

Many persons are irritated by the presence of the multiplier g and get rid of it by using an artificial unit of force, the poundal. In this system the poundal=1/g pounds and the equation is written

F (poundals) = W (pounds) x a  (6)

Another way of concealing the multiplier g is to measure the forces in the common pound unit, but to adopt an artificial unit of mass, the slug. In this system, the slug= g pounds, and the equation is written

The National Advisory Committee on Aeronautics customarily uses the slug as the unit of mass and the pound as the unit of force. This system of units will be used in this manual, but the student must always be aware that g lurks in the background and will appear and disappear in a most confusing manner.

d. The third law is well exemplified by the action of a swimmer's hands. He pushes water aft and thereby propels himself forward, since the water resists the action of the hands. The action of the 16-inch gun is likewise typical. On being fired the mass of the gun times the velocity of recoil will equal the mass of the shell times its velocity. In general, when one body acquires momentum in one direction another body will acquire an equal and opposite momentum.

22. Dynamic reaction of airstream.-a. Since air possesses mass and inertia, a stream of air moving in a certain direction at a certain velocity will, according to Newton's first law of motion, continue to move in the same direction at the same velocity until some outside force is exerted against it. If a flat plate is held normal to an airstream, the air impinging upon the plate must change both its immediate direction and velocity to pass around the plate. The plate exerts a force against the airstream and the latter, in accordance with Newton's third law, exerts an equal and opposite force against the plate. The equilibrium of these forces, as indicated in figure 10, will be maintained as long as the relative motion is maintained constant; in this case as long as the plate is held in position.

b. If the plate, on the other hand, were moved through still air at a rate sufficient to produce the same relative motion the force required to move the plate through the air would equal that previously required to hold it in position, while the resistance offered by the air to the moving plate would equal the force previously exerted by the impinging airstream.

c. Should the plate be held at an acute angle to the relative wind. a similar condition must hold, that is, the plate will produce a dynamic reaction. The air is pushed downward and its velocity reduced by the interference of the plate. Hence, the plate must exert a force on the air downward and forward. The reaction of the air must be an equal pressure upward and backward as shown by the vector AR. The vectors AL and AD are respectively the vertical and horizontal components of the pressure of the airstream against the plate. In order to maintain the relative motion between air and plate there must be a, downward force AL. equal to the upward force AL, and a forward force AD, equal to the backward force AD,.

23. Streamline flow and turbulence.-a. If a thin flat plate is held edgewise to an airstream, as shown in figure 12, the air will part at the leading edge and flow smoothly over the upper and lower surfaces, reuniting abaft the trailing edge.

The resistance offered will be small, consisting largely of skin friction caused by the air tending to cling to the surfaces of the plate owing to the viscosity of the fluid and roughness of the surface. Some turbulence, however, will result, for the velocity of streamlines adjacent to the plate will be cut down, and, in consequence, a given volume of such air will have a greater cross sectional area causing a spread in contiguous streamlines or layers. These diverging streamlines must converge behind the trailing edge even though the, plate thickness is infinitesimal. This bowing of streamlines due to setting up a velocity gradient will set up eddies in the flow of air or produce turbulence. The resulting eddies are so infinitesimal, however, that they constitute little drag or resistance, and the flow is said to be

Streamline. This is the case with certain forms other than the thin flat plate edgewise to the wind. Such forms are called "streamline" forms since they offer a minimum disturbance to otherwise parallel or streamline flow. A. typical streamline form is shown in figure 13.

6. If a flat plate, instead of being held edgewise to an airstream, is held at right angles to it as shown in figure 14, a maximum disturbance to linear flow will result. Here the skin friction becomes negligible while the eddy making resistance or turbulence reaches an excessive value.

Both the velocity and direction of the streamlines are abruptly varied with consequent eddying or "burbling."

24. Airfoil.-a. In dynamic reaction, streamline flow, and turbulence (par. 22 and 23) lie the elements vital to the airfoil. An airfoil is a surfaced body which responds to relative motion between itself and the air with a useful dynamic reaction. While a flat plate perpendicular to an airstream will produce dynamic reaction it is in no wise useful owing to the turbulence created and complete absence of a component of force offering sustentation. When such a plate is at an acute angle to the airstream a sustaining component of force is exhibited. Consequently, it constitutes an airfoil when so placed, for a downward momentum is given to the airstream and, therefore, an upward reaction must exist. The vertical component of this reaction opposes the downward pull of gravity. Nevertheless, the plate is inefficient since streamline flow is disrupted and the resulting turbulence offers excessive resistance for the sustentation afforded.

b. Curvature and thickness are paramount requisites for an efficient airfoil, that is, one in which the sustaining force is high for the price paid in skin friction and turbulence. Curvature is of marked assistance toward improving downward momentum and cutting down turbulence at the leading edge at high angles. At very small angles to the airstream such curvature brings about eddies beneath the leading edge as indicated. Thickness then becomes desirable to eliminate this turbulence. It permits, moreover, the housing of the structural members of the wing beneath the surfaces. A typical airfoil section is shown in figure 17.

25. Reaction of air upon airfoil.-a. The resultant dynamic reaction upon an airfoil being a force accompanying a change of momentum of the air affected, Newton's second law of motion is directly applicable to the determination of its magnitude. This force depends upon the mass of air deflected by the airfoil and the acceleration imparted to that mass of air.

b. Referring to figure 18, an airstream of cross section area S sweeps across a line PQ and in time t reaches line P₁ Q₁, a distance 1, behind PQ. The volume of air passing PQ in this time is S x 1. Since the velocity of the airstream, V=1/t, the volume may be expressed as

Vol. = S x V x t                                                (8)

The density of the air, r=mass/volume, from which

        m (mass) =r vol.=r SVt         (9)

 If an airfoil be placed at line PQ deflecting this mass, an acceleration a will be imparted. In accordance with Newton's second law,

        F (force) = kma

The resultant reaction: F = kr SVt x  V/t = Kr SV²         (10)

If a single coefficient is employed to account for the formula dimensions, the airfoil shape, and its attitude with respect to the airstream a simplified equation results:

F=KSV²                                   (11)

2.6. Lift and drag.-a. The resultant reaction on an airfoil placed in an airstream is designated by the vector OR in figure 19. The point of application 0 is the center of pressure or the point at which the aerodynamic forces may be considered as concentrated, for purposes of computation. Its abbreviation is c. p. The c. p. is assumed to be on the chord, a. straight line, extended if necessary, brought into contact with the lower surface of the section at two points. In the case of an airfoil of double convex curvature the chord is a straight line between the leading and trailing edges.

b. The direction and point of application of OR depend upon the shape of the airfoil section and the angle at which it is set to the airstream. The acute angle between the chord of the, airfoil and the relative wind is called the "angle of attack."

c. The magnitude of OR has been determined to be KSV². Consequently, the components of the resultant reaction parallel and perpendicular to the relative wind will have values differing only in the coefficients. The component perpendicular to the relative wind,

designated by OL, is termed "lift," L. The component parallel to or in the line of action of the relative wind, designated by OD, is termed the "drag" D. Hence:

L=K_ySV²                      (12)

D=K_aSV²                          (13)

where K_y and K_a are constants such that if S be expressed in square feet and V in miles per hour, L and D will be respectively the lift and drag in pounds. The lift provides the necessary sustentation overcoming the force of gravity or weight of the airplane. The thrust of the propeller is utilized to overcome the drag of the wings and other parts of the airplane and to maintain the relative motion needed for lift or sustentation.

27. Units and dimensional relations.-a. All mechanical and nearly all physical quantities may be defined in terms of three arbitrarily selected units, not dependent on any other units. These are called fundamental units, and the others, defined with reference to them, derived units. It is customary to choose as fundamental the units of length, force, and time.

b. An analysis of derived units with reference to the fundamental units is given in the following table.

c. To the student of aerodynamics, a knowledge of the theory of dimensions is of the greatest value as a help to a clear understanding of the relations between physical quantities. Whenever an equation is written between physical quantities the terms of the two sides of the equality must be equal not only in magnitude but in quality, that is, in dimensions. Dissimilar things are never equal and can not be added to or subtracted from one another, though they may be multiplied as in the case in obtaining momentum from mass and velocity, or divided as in the case of density. By substituting the dimensional relations in physical equations and testing the result for equality, it is often possible to detect careless errors in analysis.

28. Absolute system.-The absolute system of units and dimensions used by the N. A. C. A. has the advantage that the force and moment coefficients are dimensionless and do not change with the density of the air. The formulas for lift and drag in the absolute system, as used by the N. A. C. A., are as follows:

Lift = coefficient (CL) x 1/2 mass density of air x area x Velocity ²        (14)

Drag= coefficient (CD) x 1/2 mass density of air x area x Velocity ²         (15)

Lift and drag are in pounds, density of air is in slugs per cubic foot, area is in square feet, and velocity is in feet per second.

(The term "slugs per cubic foot" is expressed as the weight of a cubic foot of air divided by the acceleration of gravity=32.2 ft./sec.² For example, a cubic foot of air under standard conditions weighs 0.0765 pound and has the value of 0.0765/32.2 or 0.00238 slug).

The quantity (1/2 mass density of air x velocity ²) is known as the "dynamic" or "impact" pressure per unit of area. The lift and drag could be written

Lift (or drag) = coefficient X area X impact pressure                     (16)

29. Engineering system.-In the engineering system, the lift and drag formulas are written as follows:

Lift =  K_y x area x Velocity ²          (17)

Drag= K_a  x area x Velocity ²          (18)

Lift and drag are in pounds, area in square feet, and velocity in miles per hour. This system results in the use of units that are more familiar to the student through everyday experience, but the practice of the N. A. C. A. will be followed in this manual in the use of the absolute system.

30. Nature of lift.-It has been demonstrated that lift is the result of downward momentum given to an airstream by an airfoil and the amount of lift is determined by the equation

It remains to examine further the flow of air about the airfoil and to account for the distribution of pressure about the section.

a. When an airstream encounters an airfoil it is bound to separate regardless of the angle of attack. The plane along which the split occurs depends on the angle of attack, for this determines the part of the leading edge which initially influences the airstream, and on which an impact pressure is impressed due to its being at right angles to the flow. At this point on the nose, or leading edge of the airfoil, the relative velocity is reduced to zero and a dead airspace results. For the particular angle of attack, the streamlines will diverge well ahead of the nose much as the bow wave of a ship spreads in advance of the hull. There is this marked difference,

however in the latter case the lines of flow are symmetrical, in the case of the airfoil they are not.

b. The streamlines following the lower surface of the airfoil are deflected an amount dependent on the shape and angular setting of this surface. If essentially flat and at an appreciable angle to the relative wind, the particles impinging nearest the point of impact will hug the surface. Through interference, those adjacent are deflected in advance of where they would otherwise strike, and the net result is that the streamlines are deflected downward for a considerable distance from the airfoil. As the angle of attack is decreased, the deflection becomes less and consequently the augmented pressure decreases. Furthermore, the curvature of the streamlines will be most pronounced further forward, indicating that the center of pressure on the surface moves forward. With the section at a small negative angle of attack, downward deflection obviously no longer obtains. On the other hand, at excessively high angles of attack, there will be pronounced downward deflection but the airstream will break at the trailing edge and curl forward over the upper surface in a decided eddy.

c. A different situation exists with regard to streamlines following the upper surface. The streamline adjacent to the point of impact is deflected upward by encountering a portion of the leading edge. Once deflected, it would naturally follow a straight line since the airfoil section no longer exerts a force on it. That it does not follow such a line is due to its encountering the covering blanket of air. This blanket exerts a downward force on the streamline

causing it to follow the upper surface. But an equal and opposite reaction must be exerted by the streamline in accordance with Newton's third law. This force is away from the surface and hence produces a suction with respect to it. The adjacent streamlines are initially deflected as in the case of the under surface with consequent building up of a positive pressure on the nose but they likewise are forced to follow the upper contour and hence contribute to the suction, though to a smaller and smaller extent the more remote they are. With decrease in angle of attack the deflection of the streamlines is initially not as marked and, consequently, the magnitude of the suction is not so great. Furthermore, the resultant will be further aft since here will be found the most pronounced curvature of the streamlines. At very high angles of attack the air blanket exerts insufficient force to have the streamlines follow the surface of the airfoil. In breaking away from the surface, pronounced burbling results which is augmented by the eddies curling up over the trailing edge. As a result the lift is greatly impaired.

31. Pressure distribution.-In paragraph 30 it has been shown that a relatively high pressure to that of the uninterrupted airstream

exists on the under side of the airfoil, whereas a, relatively low one, or suction, exists above it. Just how the pressure is distributed may be determined either by model or by free flight test in a manner shown schematically in figure 23.

a. Parallel rows of small static pressure tubes are installed at right angles to the leading edge of the wing and flush with the upper and lower surfaces. These are connected by means of rubber tubing to manometers located outside the experimental chamber of the wind tunnel or in the cockpit of the airplane as the case may be. For any angle of attack, readings of the manometers will indicate the pressure at the respective points, and plots of distribution both along the chord and from centerline to wing tip can be made. Typical plots of distribution along the chord for different angles of attack are shown in figure 24.

b. Over the bulk of the flight range it should be noted that the suction or relatively low pressure above the airfoil constitutes from 60 to 80 percent of the lifting force. As the angle of zero lift is approached, that is, the attitude in the vertical dive with angle of attack negative, a down load is exerted on the leading edge while the remaining distribution of pressure is such that the resultants for the upper and lower surfaces constitute a pure couple imposing a down load on the front spar and an up load on the rear. Of interest also is the region of high suction at large angles of attack, for it is this suction which is responsible for the opening of the automatic wing slots which are later described.

32. Hydrodynamic theory of lift.-a. The best explanation of the formation of high and low pressure areas about an airfoil is in the application of some of the theories of hydraulics.

b. Bernoulli's theorem would indicate that the air close to the upper surface of the wing, where the pressure is less than that in the air at a greater distance from the wing, has a velocity greater than that in the airstream at a distance from the wing. Similarly the velocity close to the lower surface is less than that at a distance from the lower surface. The theorem of Bernoulli may be stated as follows:

At any point in the path of flow of an incompressible fluid in steady motion, the sum of the "potential head," the "pressure head," and the "velocity head" is constant provided the frictional resistance, is negligible. The term "head" is an old millwright's expression meaning "the height through which a mass of water descends in actuating a hydraulic machine". It designates the energy of the fluid. Mathematically the theorem is

where H if constant, h the potential head,  the pressure head, and  the velocity head.

(1) For the purpose at hand, justification for high and low pressures about an airfoil, it is sufficient to say "where the pressure is high the velocity is low, and vice versa".

(2) Such being the case, there will be an augmented velocity in the streamlines above the airfoil over that of the uninfluenced airstream

and a decreased velocity in those below. This velocity variation is easily accounted for if a circulatory airflow about the airfoil is conceived which may be superimposed on that of the natural flow of the airstream as shown in figure 25.

(3) The potential head will not vary materially for any flight condition and hence may be eliminated from the equation for total head The equation then becomes

(4) -That such a circulation is not farfetched and both can and probably does exist is evidenced by wind tunnel tests. As the airstream initially encounters the airfoil a succession of small vortices are formed

at the trailing edge which curl upward and toward the leading edge. These are assumed to set up the desired counter circulation as indicated. The circulation builds up rapidly and when steady conditions are obtained the trailing edge vortices are no longer perceptible. A velocity gradient is thus assured, with the instantaneous velocity at any point in a streamline the resultant of V_o and V_c at that point. A corresponding pressure gradient will be evidenced with distribution such as depicted in paragraph 31.

33. Drag.-a. The drag of an airfoil or the resistance it offers to passage through the air for range of angle of attack between zero lift and angle at which the lift ceases to increase in direct proportion to the angle of attack is considered as composed of two parts: profile drag and induced drag.

(1) profile drag is the resistance encountered by virtue of forcing an object through a viscous fluid tending to stick to any object immersed in it. Structural drag, form drag, and the various components of parasite drag are all of the same nature as profile drag.

(2) Induced drag is the necessary evil encountered in producing lift. The reaction upon an airfoil, being at an angle to the relative wind, has necessarily a component parallel to the relative wind. This component is called "induced drag."

b. The drag of an airfoil as commonly discussed is the sum of the induced drag and the profile drag.

34. Wind tunnel.-a. General.-(1) A wind tunnel is essentially a venturi tube or meter, used originally for the measurement of the flow of water in pipes. The device consists of a conical nozzlelike reducer AB, shown in figure 26, at the small end of which is a short cylinder BO, called the "throat." To the other end of this throat is attached a conical enlargement CD which attains the same size as the inlet A, but more gradually.

(2) The underlying principle is based on the, theorem of Bernoulli. The quantity of fluid, water or air, which is drawn through inlet A will be discharged through the same sized opening D. The velocity of the fluid must, therefore, increase as it is drawn through the inlet cone, attain a maximum value in the throat and thereafter gradually slow down to its initial value at the outlet. The static pressure in the throat is consequently less than that at the entrance.

Applying the theorem.:

where the subscripts 1 and 2 designate conditions at the inlet and throat respectively. h₁=h₂ in the case where the center of flow is horizontal. Consequently

Since the difference in pressure heads is measurable directly by manometer and since the cross section areas at the inlet and throat are known, the manometer may be calibrated directly to read the throat velocity. This is the unknown which it is desired to measure in the case of the wind tunnel.

b. Types.-(I) The simplest type of wind tunnel is a venturi tube having refinements aimed to reduce friction losses and insure straight line flow through the experimental chamber or throat. Figure 27 illustrates the essential part of a simple wind tunnel and shows the desirable variations from the venturi meter described above. Air is sucked through the wind tunnel by the propeller fan at the outlet at a definite velocity. A system of balance is employed to measure the forces required to maintain the model in equilibrium. All wind tunnels have the following in common:

(a) A large venturi tube to give a uniform airflow with high velocity in the throat.

(b) A power driven propeller.

(c) A honeycomb grill to insure a uniform flow of air with parallel streamlines in the throat.

(d) An experimental chamber or throat fitted to hold the object to be tested in the desired attitude.

(e) A system of linkages for transmitting the forces to measuring balances.

(2) There are three general types of wind tunnel. Both of the types in (a) and (b) below may have a return channel for the air discharged, for the power required to maintain the proper velocity of airflow is thus reduced.

(a) Closed chamber type.-The experiment chamber is closed and the, balances are outside the tunnel. The model is visible through a glass pane in the chamber. The diagram in figure 27 illustrates the type.

(b) Open chamber type.-For the testing of full scale objects such as propellers, fuselages, etc., it is often essential to break the wind tunnel at the throat and place the body to be tested in this opening between the inlet and outlet cones.

(c) Variable density type.-This is a tunnel of the closed chamber, closed return type, differing therefrom in that the air in circulation can be compressed to over 20 atmospheres. Such compression is desirable to maintain dynamic similitude between the action of the air on the model and on the full scale object, thus eliminating the "scale effect".

The larger the wind tunnel the larger the object which can be tested and consequently the more accurate will be the results in the case of a model since its scale more closely approaches that of actual conditions. The size of the tunnel, however, is limited by the cost. A small one would have a 3-foot experimental chamber, and there are several now in use in which an entire airplane can be placed.

(c. Balances.-(1) The balances used for the measurement of forces acting on a model in a wind tunnel are of several types. One type is shown in figure 30.

(a) The model is mounted on a spindle which is attached to the balance. The whole apparatus is supported on a hardened steel or

jeweled point. The model is mounted vertically, since, as far as the air forces are concerned it makes no difference whether the lift is measured vertically or horizontally.

(b) The drag acts horizontally in any case. When the set up is made, the whole apparatus is balanced with the counterweights before air is drawn through. It will be seen that the lift and drag arms of the balance, equipped with scale pans, can be made to balance the lift and drag forces and thus measure them directly. The model is attached to a spindle which is in turn mounted on a movable plate. This plate is calibrated in degrees and by turning it, the angle of the model to the wind can be accurately set without touching the model.

(c) Figure 30 shows the apparatus for measuring moments. In this case the spindle is rigidly attached to a torsion rod, which has been calibrated so that a known number of turns of the micrometer screw is equivalent to a known number of inch-pounds. When the model is set up the reference line on the torsion rod is brought into line with the cross hair in the microscope. When the wind is turned on, the air forces cause a certain moment on the airfoil, in turn causing a rotational movement in the torsion rod.

(d) To measure the moment, the micrometer screw is rotated until the reference line is brought back into coincidence. Reference to a torsion curve will give the moment. Some balances have a third weight pan to bring the torsion rod back to its original position, but the principle is the same.

(2) Figure 32 shows a typical "wire type" balance which is used for large models and high throat velocities. The airfoil is mounted upside down. It will be seen from the figure that the pull in the two drag wires a is exactly equal to the drag through the action of the wires running to the floor at an angle of 45°. The two lift wires b carry part of the lift; the remainder is carried through a single wire c which is attached to a "sting" or spindle extending out from the trailing edge of the airfoil. The center of pressure can be determined without any other measurements being made. Take the moment of the load in wire c about the leading edge, the distance between the wires b and c being accurately known, and divide this moment by the sum of the loads in wires b and c.

d. Uses-(1) Most of the laws of aerodynamics and all of the formulas have been developed or checked through wind tunnel experiments. In no other manner can conditions simulating flight be obtained, and even in flight it would be very expensive or almost impossible to make observations of certain of the forces and airflows existing. The wind tunnel, then, is of incalculable value in aerodynamic, research. Among the qualitative and quantitative results to be expected with adequate engineering accuracy from the tunnel are

(a) The resistance offered by any sort of object to an airstream.

(b) The comparative value of various types of fairings and stream lining.

(c) The lift and drag of airfoils.

(d) The efficiencies of all types of propellers.

(e) The effectiveness of new "gadgets", such as slotted wings, etc.

(f) The air resistance of vessels.

(2) One of the most important uses of the wind tunnel is that of testing complete models. The purpose of these tests is to check the predicted performance, stability, and control of the airplane as laid out on paper prior to undertaking actual construction of the machine, or even the detail design of parts and assemblies. The advantages ocuring include savings in time, labor, and cost, for quick and definite indications are assured as to the adequacy of the design or the need for modifications.