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OSMANIA UNiyERSItY LIBRARY
Call No. 62~*?*/3i~~]> // v//s Accession No, Author ^
Title A ^f\J^ , * .
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This book sli I re the date Kst marked below.
AERODYNAMICS
By the Same Author
A COMPLETE COURSE IN ELEMENTARY AERODYNAMICS WITH EXPERIMENTS AND EXAMPLES
AERODYNAMICS
BY
N. A. V. PIERCY
D.Sc., M.Inst.C.E., M.I.Mech.E., F.R. Ae.S.
Reader in Aeronautics in the University of London
Head of the Department of Aeronautics, Queen Mary College
Member of the Association of Consulting Engineers
SECOND EDITION
AA
ML
THE ENGLISH UNIVERSITIES PRESS LTD LONDON
FIBST PRINTED 1937
REPRINTED ...... 1943
SECOND EDITION, REVISED AND ENLAKGED 1947
ALL RIGHTS RESERVED.
Made and Printed in Great Britain by Hagell, Watson 6* Viney Ltd., London and Aylesbury.
PREFACE TO SECOND EDITION
THE present edition is enlarged to provide, in the first place, an introduction to the mathematical and experimental study of com- pressible flow, subsonic and supersonic. This and other matters now becoming prominent are not collected in a supplementary section but incorporated in place as additional articles or short chapters. Following a well-established practice, the numbering of original articles, figures and chapters is left undisturbed as far as possible, interpolations being distinguished by letter-suffixes. It is hoped this procedure will ensure a minimum of inconvenience to readers familiar with the earlier edition. To some extent the unlettered articles indicate a first course of reading, though a modern view of Aerodynamics requires consideration of Mach numbers equally with Reynolds numbers almost from the outset.
Other matters now represented include various theories of thin aerofoils and the reduction of profile drag. The brief account of the laminar-flow wing is in general terms, but the author has drawn for illustrations on the conformal system, in the development of which he has shared more particularly.
The original text is revised to bring it up to date, and also in the following connection. Experience incidental to the use of the book at Cambridge and London Universities isolated certain parts where the treatment was insufficiently detailed for undergraduates ; these are now suitably expanded.
The aim of the book remains unchanged. It does not set out to collect and summarise the researches, test results and current practice of the subject, but rather to provide an adequate and educational introduction to a vast specialist literature in a form that will be serviceable for first and higher degrees, and like purposes, including those of the professional engineer.
N. A. V. PIERCY.
TEMPLE,
October, 1946.
VI PREFACE
PREFACE TO FIRST EDITION
FIRST steps towards formulating the science of Aerodynamics pre- ceded by only a few years the epoch-making flight by the Wright brothers in 1903. Within a decade, many fundamentals had been established, notably by Lanchester, Prandtl, Joukowski, and Bryan. Yet some time elapsed before these essentially mathematical con- ceptions, apart from aircraft stability, were generally adopted. Meanwhile, development proceeded largely by model experiment. To-day, much resulting empiricism has been superseded and the subject is unique among those within the purview of Engineering in its constant appeal to such masters as Helmholtz and Kelvin, Reynolds and Rankine. A complete theory is stiU far out of reach ; experiment, if no longer paramount, remains as important as analysis ; and there is a continual swinging of the pendulum between these two, with progress in aviation marking time.
This book presents the modern science of Aerodynamics and its immediate application to aircraft. The arrangement is based on some eighteen years' organisation of teaching and research in the University of London. The first five chapters, and the simpler parts of Chapters VI-XII, constitute an undergraduate course ; more advanced matters are included to serve especially the Designer and Research Engineer. No attempt has been made to summarise reports from the various Aerodynamic Laboratories, which must be consulted for design data, but the treatment is intended to provide an adequate introduction to the extensive libraries of important original papers that now exist in this country and abroad.
To facilitate reference, symbols have been retained, for the most part, in familiar connections, though duplication results in several instances, as shown in the list of notations. Of the two current systems of force and moment coefficients, the American or Contin- ental, associated with " C," will probably supersede the British, distinguished by " k." No great matter is involved, a C-coefficient being derived merely by doubling the corresponding ^-coefficient. However, so many references will be made in this country to litera- ture using the ' k" notation that the latter has been given some preference.
My thanks are due to Professor W. G. Bickley for reading the proof sheets and making many suggestions ; also to The English Universities Press for unremitting care and consideration.
N. A. V. PIERCY. TEMPLE.
ART.
PREFACE NOTATION
CONTENTS
CHAPTER I
PAGE V
xiii
AIR AT REST, THE ATMOSPHERE AND STATIC LIFT
1—4. Properties of Air ; Density ; Pressure ....
5—7. Hydrostatic Equation. Incompressibility Assumption. Measurement of Small Pressures .....
8-9. Buoyancy of Gas-filled Envelope. Balloons and Airships . 10. Centre of Pressure ........
11—15. Relation between Pressure, Density, and Temperature of a Gas. Isothermal Atmosphere. Troposphere. The In- ternational Standard Atmosphere. Application to Alti- meters .........
16-17. Gas-bag Lift in General. Vertical Stability
18. Atmospheric Stability and Potential Temperature 19-20. Bulk Elasticity. Velocity of Sound
CHAPTER II AIR FLOW AND AERODYNAMIC FORCE
21. Streamlines and Types of Flow .....
22. Absence of Slip at a Material Boundary .... 23-25. Viscosity : Qualitative Theory ; Maxwell's Definition ; Ex- perimental Laws ........
26-28. Relation between Component Stresses in Non-uniform Flow ; Static Pressure. Forces on an Element
29-33. Bernoulli's Equation : Variation of Density and Pressure ; Adiabatic Flow ; Temperature Variation ; the Incompres- sible Flow Assumption ; Pitot Tube ; Basis of Velocity Measurement ........
34-41. Equation of Continuity. Experimental Streamlines. Stream Function. Circulation and Vorticity. Gradient of Pitot Head across Streamlines. Irrotational Flow .
42-43. The Boundary Layer Experimentally Considered
44-46. Constituents of Aerodynamic Force. Integration of Normal Pressure and Skin Friction ......
47-49c. Rayleigh's Formula. Reynolds Number. Simple Dynami- cally Similar Motions. Aerodynamic Scale. Mach Num- ber. Froude Number. Corresponding Speeds
CHAPTER III
WIND-TUNNEL EXPERIMENT
50-53. Nature of Wind-tunnel Work. Atmospheric Tunnels
54. Coefficients of Lift, Drag, and Moment ....
vii
4
6
11
13 18 20 21
23 26
26 32
35
44 52
54
58
68 75
Vlll
CONTENTS
ART. 66-69.
69A. 60. 61-63.
64-66.
Suspension of Models. Double Balance Method. Aero- dynamic Balance. Some Tunnel Corrections
Pitot Traverse Method
Aerofoil Characteristics .......
Application of Complete Model Data ; Examples. Arrange- ment of Single Model Experiment. Compressed-air Tunnel
Practical Aspect of Aerodynamic Scale. Scale Effects. Gauge of Turbulence .......
PAGE
77 86 89
92
97
CHAPTER III A EXPERIMENT AT HIGH SPEEDS
66. Variable-density Tunnel 103
66A. Induced-flow Subsonic Tunnel. Wall Adjustment.
Blockage 106
66B-66C. Supersonic Tunnel. Illustrative Results . . . .109 66D. Pitot Tube at Supersonic Speeds. Plane Shock Wave . 114
CHAPTER IV AIRCRAFT IN STEADY FLIGHT
67-69. Examples of Heavier-than-air Craft. Aeroplanes v. Air- ships. Aeroplane Speed for Minimum Drag .
70-72. Airship in Straight Horizontal Flight and Climb
73-76. Aeroplane in Level Flight. Size of Wings; Landing Con- ditions ; Flaps . . . . . ...
77-79. Power Curves ; Top Speed ; Rate of Climb
80. Climbing, Correction for Speed .....
81-83. Effects of Altitude, Loading, and Partial Engine Failure .
84-85. Gliding; Effects of Wind; Motor-less Gliders
86-89. Downwash. Elevator Angle; Examples; C.G. Location. 90. Nose Dive
91-96. Circling and Helical Flight. Rolling and Autorotation. Handley Page Slot. Dihedral Angle ....
118 126
129 137 140 143 146 149 155
156
CHAPTER V FUNDAMENTALS OF THE IRROTATIONAL FLOW
96-100. 101-106. 106-109.
110-114. 115.
116-117. 118-120.
Boundary Condition. Velocity-potential. Physical Mean- ing of <f>. Potential Flow. Laplace's Equation
Source. Sink. Irrotational Circulation. Combined Source and Sink. Doublet .....
Flow over Faired Nose of Long Board. Oval Cylinder. Circular Cylinder without and with Circulation
Potential Function. Examples. Formulae for Velocity
Circulation round Elliptic Cylinder or Plate. Flow through Hyperbolic Channel .......
Rankine's Method. Elliptic Cylinder or Plate in Motion .
Acceleration from Rest. Impulse and Kinetic Energy of the Flow Generated by a Normal Plate ....
163 167
172
180
184 186
190
CONTENTS
IX
CHAPTER VI TWO-DIMENSIONAL AEROFOILS
ART.
121-124.
PAGE
Conformal Transformation; Singular Points. Flow past Normal Plate by Transformation. Inclined Plate . .194
125-127. Joukowski Symmetrical Sections ; Formulae for Shape. Velocity and Pressure. More General Transformation Formula. Karman-Trefftz Sections .... 203
128-129B. Piercy Symmetrical Sections. Approximate Formulae. Velocity over Profile. Comparison with Experiment and Example 212
130-133A. Circular Arc Aerofoil. Joukowski and Piercy Wing Sections 220
134-139. Joukowski's Hypothesis ; Calculation of Circulation ; Stream- lines with and without Circulation. Investigation of Lift,
Lift Curve Slope and Moment 227
140. Comparison with Experiment ..... 235
CHAPTER VIA THIN AEROFOILS AT ORDINARY SPEEDS
140A-140B. Method and Equations .
140c. Application to Circular Arc 140D-140F. General Case. Aerodynamic Centre.
Example
237 240 241
CHAPTER VIB COMPRESSIBLE INVISCID FLOW
140G-140i. Assumptions. General Equation of Continuity , . 245 140J-140K. Euler's Dynamical Equations. Kelvin's (or Thomson's)
Theorem 247
140L-140M. Irrotational Flow. Integration of Euler's Equations . 260 140N-140O. Steady Irrotational Flow in Two Dimensions. Electrical
and Hydraulic Analogies ...... 252
CHAPTER VIC
THIN AEROFOILS AT HIGH SPEEDS
140P-140Q. Subsonic Speeds. Glauert's Theory. Comparison with
Experiment. Shock Stall 259
140R-140T. Supersonic Speeds. Mach Angle. Ackeret's Theory. Com- parison with Experiment . . . . .262
CHAPTER VII VORTICES AND THEIR RELATION TO DRAG AND LIFT
141-147. Definitions. Rankine's Vortex. General Theorems . 267
148-150. Induced Velocity for Short Straight Vortex and Vortex
Pair. Analogies 272
X CONTENTS
ART. PAGE
151-155. Constraint of Walls. Method of Images ; Vortex and Vor- tex Pair within Circular Tunnel; Other Examples. Ap- plication of Conformal Transformation; Streamlines for Vortex between Parallel Walls . . . . .276
155A-155B.Lift from Wall Pressures. Source and Doublet in Stream
between Walls ........ 283
156-162. Generation of Vortices ; Impulse ; Production and Dis- integration of Vortex Sheets. Karman Trail ; Applica- tion to Circular Cylinder. Form Drag .... 287
163-168. Lanchester's Trailing Vortices. Starting Vortex. Residual Kinetic Energy ; Induced Drag ; Example of Uniform Lift. Variation of Circulation in Free Flight. Example from Experiment ....... 295
CHAPTER VIII
WING THEORY
169-171. General Equations of Monoplane Theory .... 309
172-177. The 'Second Problem.' Distribution of Given Impulse for Minimum Kinetic Energy ; Elliptic Loading. Minimum Drag Reduction Formulae ; Examples . . .312
178-180. Solution of the Arbitrary Wing by Fourier Series. Elliptic Shape Compared with Others. Comparison with Experi- ment ......... 320
181-186. General Theorems Relating to Biplanes. Prandtl's Biplane Factor ; Examples. Equal Wing Biplane — Comparison with Monoplane ; Examples ..... 327
187-188. Tunnel Corrections for Incidence and Induced Drag . . 335 189-192. Approximate Calculation of Downwash at Tail Plane ; Tun- nel Constraint at Tail Plane ; Correction Formulae. Tail Planes of Biplanes 339
CHAPTER IX VISCOUS FLOW AND SKIN DRAG
193-199. Laminar Pipe Flow : Theory and Comparison with Experi- ment. Turbulent Flow in Pipes ; the Seventh-root Law. Flow in Annular Channel. Eccentric and Flat Cores in
Pipes 346
200-204. General Equations for Steady Viscous Flow. Extension of
Skin Friction Formula . . . . . .357
205-207. Viscous Circulation. Stability of Curved Flow . . . 365 208-209. Oseen's and Prandtl's Approximate Equations . . . 369 210-2 1 7 . Flat Plates with Steady Flow : Solutions for Small and Large Scales ; Formation of Boundary Layer ; Method of Suc- cessive Approximation. Karman's Theorem ; Examples 370 2 18-2 ISA. Transition Reynolds Number. Detection of Transition . 384 219-221. Flat Plates with Turbulent Boundary Layers : Power
Formulae. Transitional Friction. Experimental Results 387 221A-221B. Displacement and Momentum Thicknesses. Alternative
Form of Kdrm&n's Equation . . . . .391
CONTENTS
XI
ART.
222-223.
224-230.
PAGE
Note on Laminar Skin Friction of Cylindrical Profiles. Breakaway. Effect of Wake. Frictions of Bodies and Flat Plates Compared 393
Turbulence and Roughness. Reynolds Equations of Mean Motion. Eddy Viscosity. Mixing Length. Similarity Theory. Skin Drag. Application to Aircraft Surfaces. Review of Passage from Model to Full Scale . . . 399
CHAPTER IX A REDUCTION OF PROFILE DRAG
230A-230B. Normal Profile Drag. Dependence of Friction on Transition Point
230c-230F. Laminar Flow Wings. Early Example. Maintenance of Negative Pressure Gradient. Position of Maximum Thickness. Incidence Effect ; Favourable Range. Veloc- ity Diagrams. Examples of Shape Adjustment. Camber and Pitching Moment .......
230G-230H. Boundary Layer Control. Cascade Wing
230i. Prediction of Lift with Laminar Boundary Layer .
230j~230K. High Speeds, back
Minimum Maximum Velocity Ratio. Sweep-
409
412 419 421
423
CHAPTER X AIRSCREWS AND THE AUTOGYRO
231-232. The Ideal Propeller ; Ideal Efficiency of Propulsion . . 425
233-238. Airscrews. Definitions. Blade Element Theory. Vortex Theory ; Interference Factors ; Coefficients ; Method of Calculation ; Example ...... 427
239. Variable Pitch. Static Thrust 438
240-241. Tip Losses and Solidity. Compressibility Stall . . 440
242. Preliminary Design : Empirical Formulae for Diameter and
Inflow ; Shape ; Stresses ...... 443
243-245. Helicopter and Autogyro. Approximate Theory of Auto- gyro Rotor. Typical Experimental Results . . , 446
CHAPTER XI PERFORMANCE AND EFFICIENCY
246-260. Preliminary Discussion. Equivalent Monoplane Aspect
Ratio. Induced, Profile and Parasite Drags; Examples 451 251. Struts and Streamline Wires . . . . . .457
252-263. Jones Efficiency ; Streamline Aeroplane. Subdivision of
Parasite Drag ........ 469
254-258. Airscrew Interference ; Example . . . . .461
259-260A. Prediction of Speed and Climb ; Bairstow's and the Lesley- Reid Methods. Method for Isolated Question . . 466
261-262. Take-off and Landing Run. Range and Endurance. , 473
Xll
CONTENTS
ART.
PAGE
262A-262F. Aerodynamic Efficiency ; Charts. Airscrew Effects ; Ap- plication to Prediction ; Wing-loading and High-altitude
Flying ; Laminar Flow Effect 477
263. Autogyro and Helicopter ...... 487
263A. Correction of Flight Observations ..... 489
CHAPTER XII
SAFETY IN FLIGHT
264-265. General Problem. Wind Axes. Damping Factor . .493 266-269. Introduction to Longitudinal Stability : Aerodynamic Dihe- dral ; Short Oscillation ; Examples ; Simplified Phugoid Oscillation ; Example 496
270-273. Classical Equations for Longitudinal Stability. Glauert's Non-dimensional System. Recast Equations. Approxi- mate Factorisation ....... 503
274-278. Engine-off Stability : Force and Moment Derivatives.
Example. Phugoid Oscillation Reconsidered . . 508
279-280. Effects of Stalling on Tail Efficiency and Damping . . 513 281-284. Level Flight ; High Speeds ; Free Elevators ; Climbing . 514
285. Graphical Analysis . . . . . . .516
286. Introduction to Lateral Stability 618
287-289. Asymmetric Equations ; Solution with Wind Axes ; Ap- proximate Factorisation . . . . . .519
290-292. Discussion in Terms of Derivatives. Example. Evalua- tion of Lateral Derivatives . . . . . .521
293-295. Design and Stalling of Controls. Control in Relation to
Stability. Large Disturbances. Flat Spin . . . 523 296. Load Factors in Flight ; Accelerometer Records . . 526
AUTHOR INDEX 629
SUBJECT INDEX
531
NOTATION
(Some of the symbols are also used occasionally in connections other than those stated below.)
A . . . Aerodynamic force ; aspect ratio ; transverse moment of inertia.
A.R.C.R. & M. Aeronautical Research Committee's Reports and Memoranda.
A.S.I. . . Air speed indicator.
a . . . Axial inflow factor of airscrews ; leverage of Aerodynamic force about C.G. of craft ; slope of lift curve of wings ; velocity of sound in air.
a9 . . . Slope of lift curve of tail plane.
a . . . Angle of incidence.
a, . . . Tail-setting angle.
B . . . Gas constant ; longitudinal moment of inertia ; number of blades of an airscrew.
Blt B* . . Stability coefficients.
b . . . Rotational interference factor of airscrews.
fi . . . Transverse dihedral ; twice the mean camber of a wing.
C . . . Directional moment of inertia ; sectional area of tunnel.
C.A.T. . . Compressed air tunnel.
Clf C2 . . Stability coefficients.
CL, CD, etc. . Non-dimensional coefficients of lift, drag, etc., on basis of stagnation pressure.
C.G. . . Centre of gravity.
C.P. . . Centre of pressure.
c . . . Chord of wing or aerofoil ; molecular velocity.
y . . . Ratio of specific heats ; tan'1 (drag/lift).
D . . . Diameter ; drag.
Dlt D2 . . Stability coefficients.
A, 8 . . Thickness of boundary layer; displacement thick- ness.
E . . . Elasticity ; kinetic energy.
Elt Et . . Stability coefficients.
MV NOTATION
e . . . Base of the Napierian logarithms.
e . . . Angle of downwash.
F . . . Skin friction (in Chapter II) ; Froude Number.
~ . . Frequency.
£ . . . Vorticity.
g . . . Acceleration due to gravity.
H . . . Horse power ; the boundary layer ratio 8/0.
h . . . Aerodynamic gap ; height or altitude.
7] . . .A co-ordinate ; efficiency ; elevator angle.
6 . . . Airscrew blade angle ; angle of climb ; angular
co-ordinate ; temperature on the Centigrade
scale ; momentum thickness.
/ . . . Impulse ; second moment of area. i . . . V-l ; as suffix to D : denoting induced drag ;
incidence of autogyro disc. / . . . The advance of an airscrew per revolution in terms
of its diameter. K . . . Circulation. k . . . Radius of gyration ; roughness. ^A» ^B» *c • Inertia coefficients. *L» *D» etc- • Non-dimensional coefficients of lift, drag, etc., on
basis of twice the stagnation pressure.
k99 kx . . British drag and lift coefficients of autogyro rotor. L . . . Lift ; rolling moment.
/ . . . Length ; leverage of tail lift about C.G. of craft. X . . . Damping factor ; mean free path of molecule ;
mean lift per unit span.
M . . Pitching moment ; Mach number.
m . . . Mass ; with suffix : non-dimensional moment
derivative ; Mach angle, ji . . . Coefficient of viscosity ; * relative density of
aeroplane ' ; a co-ordinate. N . . . Yawing moment. N.A.C.A. . National Advisory Committee for Aeronautics,
U.S.A.
N.P.L. . . National Physical Laboratory, Teddington. n . . . Distance along a normal to a surface ; revolutions
per sec.
v . . . Kinematic coefficient of viscosity ; a co-ordinate. 5 • • .A co-ordinate. P . . . Pitch of an airscrew ; pressure gradient ; total
pressure.
NOTATION XV
p . . . Angular velocity of roll ; pressure or stress.
p . . . Density of air in slugs per cu. ft.
Q . . . Torque.
q . . . Angular velocity of pitch ; resultant fluid
velocity.
R . . . Radius ; Reynolds number. Rlt Rt . . Routh's discriminant.
R.A.E. . . Royal Aircraft Establishment, Farnborough. r . . . Angular velocity of yaw ; lift/drag ratio ;
radius.
ra . . . Over-all lift /drag ratio. 5 . . . Area, particularly of wings.
s . . . Distance along contour or streamline ; semi-span, a . . . Density of air relative to sea-level standard ;
Prandtl's biplane factor ; sectional area of
vortex ; solidity of an airscrew. r . . . Thickness ; thrust. t . . . Period of time in sec. ; the complex co-ordinate
5 + «j.
£0 • • • Unit of time in non-dimensional stability equa- tions.
T . . . Absolute temperature ; skin friction in Chapter IX ; tail angle of aerofoil section ; tail volume ratio.
<f> . . . Aerodynamic stagger ; angle of bank ; angle of helical path of airscrew element ; velocity
'n>n ; yaw.
f undisturbed velocity in the direc- tOz. jcity of a body.
lircraft. / components in the directions Ox,
o.
Deity ; mean loading of wing in
at ; potential function $ + i^. it ibrce. )f stability charts.
'dinates ; with suffixes : non-dimen-
derivatives.
NOTATION
z • • • The complex co-ordinate x + iy.
& • - • Angular velocity of an airscrew.
o> . . . Angular velocity.
t& • - . Impulsive pressure.
V2 • . 9'/9*1 +
V4 . . (
Chapter I AIR AT REST, THE ATMOSPHERE AND STATIC LIFT
i. Air at sea-level consists by volume of 78 per cent, nitrogen, 21 per cent, oxygen, and nearly 1 per cent, argon, together with traces of neon, helium, possibly hydrogen, and other gases. Although the constituent gases are of different densities, the mixture is maintained practically constant up to altitudes of about 7 miles in temperate latitudes by circulation due to winds. This lower part of the atmosphere, varying in thickness from 4 miles at the poles to 9 miles at the equator, is known as the troposphere. Above it is the stratosphere, a layer where the heavier gases tend to be left at lower levels until, at great altitudes, such as 50 miles, little but helium or hydrogen remains. Atmospheric air contains water-vapour in varying proportion, sometimes exceeding 1 per cent, by weight.
From the point of view of kinetic theory, air at a temperature of 0° C. and at standard barometric pressure (760 mm. of mercury) may be regarded statistically as composed of discrete molecules, of mean diameter 1-5 X 10 ~5 mil (one-thousandth inch), to the number of 4-4 x 1011 per cu. mil. These molecules are moving rectilinearly in all directions with a mean velocity of 1470 ft. per sec., i.e. one- third faster than sound in air. They come continually into collision with one another, the length of the mean free path being 0-0023 mil.
2. Density
Air is thus not a continuum. If it were, the density at a point would be defined as follows : considering the mass M of a small volume V of air surrounding the point, the density would be the limiting ratio of M/V as V vanishes. But we must suppose that the volume V enclosing the point is contracted only until it is small compared with the scale of variation of density, while it still remains large compared with the mean distance separating the molecules. Clearly, however, V can become very small before the continuous passage of molecules in all directions across its bounding surface can make indefinite the number of molecules enclosed and M or M/V uncertain. Density is thus defined as the ratio of the
A.D.— l 1
2 AERODYNAMICS [CH.
mass of this very small, though finite, volume of air — i.e. of the aggregate mass of the molecules enclosed — to the volume itself.
Density is denoted by p, and has the dimensions M/Z,8. In Aerodynamics it is convenient to use the slug-ft.-sec. system of units.* At 15° C. and standard pressure 1 cu. ft. of dry air weighs 0-0765 Ib. This gives p = 0-0765/g = 0-00238 slug per cu. ft.
It will be necessary to consider in many connections lengths, areas, and volumes that ultimately become very small. We shall tacitly assume a restriction to be imposed on such contraction as discussed above. To take a further example, when physical properties are attached to a ' point ' we shall have in mind a sphere of very small but sufficient radius centred at the geometrical point.
3. Pressure
Consider a small rigid surface suspended in a bulk of air at rest. The molecular motion causes molecules continually to strike, or tend to strike, the immersed surface, so that a rate of change of molecular momentum occurs there. This cannot have a component parallel to the surface, or the condition of rest would be disturbed. Thus, when the gas is apparently at rest, the aggregate rate of change of momentum is normal to the surface ; it can be represented by a force which is everywhere directed at right angles towards the surface. The intensity of the force per unit area is the pressure pt sometimes called the hydrostatic or static pressure.
It is important to note that the lack of a tangential component to p depends upon the condition of stationary equilibrium. The converse statement, that fluids at rest cannot withstand a tangential or shearing force, however small, serves to distinguish liquids from solids. For gases we must add that a given quantity can expand to fill a volume, however great.
It will now be shown that the pressure at a point in a fluid at rest is uniform in all directions. Draw the small tetrahedron ABCO, of
* In this system, the units of length and time are the foot and the second, whilst forces are in pounds weight. It is usual in Engineering, however, to omit the word ' weight/ writing * Ib.' for ' lb.-wt.,' and this convention is followed. The appropriate unit of mass is the 'slug,' viz. the mass of a body weighing g Ib. Velocities are consistently measured in ft. per sec., and so on. This system being understood, specification of units will often be omitted from calculations for brevity. For example, when a particular value of the kinematic viscosity is given as a number, sq. ft. per sec. will be implied. It will be desirable occasionally to introduce special units. Thus the size and speed of aircraft are more easily visualised when weights are expressed in tons and velocities in miles per hour. The special units will be duly indicated in such cases. Non-dimeasional coefficients are employed wherever convenient.
I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 3
which the faces OAB, OBC, OCA are mutually at right angles (Fig.
1). Denote by S the area of the face ABC. With the help of OD
drawn perpendicular to this face, it is
easily verified from the figure that the
area OCA is S . cos a. The pressure pABC
on the S-face gives rise to a force pAEC-S
which acts parallel to DO. From the
pressures on the other faces, forces simi-
larly arise which are wholly perpendicular
to the respective faces.
Resolving in the direction BO, for equilibrium W + pABC . 5 . cos a - pocA . S . cos a = 0, FIG. i.
where W is a force component on the tetrahedron arising from some general field of force in which the bulk of air may be situated ; such might be, for example, the gravitational field, when W would be the weight of the tetrahedron if also OB were vertical. But W is pro- portional to the volume of the tetrahedron, i.e. to the third order of small quantities, and is negligible compared with the other terms of the equation, which are proportional to areas, i.e. to the second order of small quantities. Hence :
Similarly :
4. It will be of interest to have an expression for p in terms of molecular motions.
Considering a rigid plane surface suspended in air, draw Oy, Oz mutually at right angles in its plane and Ox perpendicular to it (Fig. 2). Erect on a unit area S of the plane, and to one side of it a right cylinder of unit length, so that it encloses unit volume of air. If m is the mass of each molecule, the total number N of molecules enclosed is p/w. They are moving in all directions with mean velocity c along straight paths of mean length X.
At a chosen instant the velocities of all the molecules can be resolved parallel to Ox, Oy, Oz. But, since N is very large, it is equivalent to suppose that JV/3 molecules move parallel to each of the co-ordinate axes with velocity c during the short time A* required to describe the mean free path. Molecules moving parallel to Oy, Oz cannot impinge on S ; we need consider only molecules moving
AERODYNAMICS [CH.
parallel to Ox, and of these only one-half must be taken as moving towards 5, i.e. in the specific direction Ox (Fig. 2).
The interval of time corresponding to the free path is given by —
A* = \/c.
During this interval all those mole- cules moving in the direction Ox FIG. 2. which are distant, at the beginning
of A/, no farther than X from S, will
strike S. Their number is evidently >JV/6. Each is assumed perfectly elastic, and so will have its velocity exactly reversed. Thus the aggregate change of momentum at S in time A2 is 2mc . AIV/6. The pressure p, representing the rate of change of momentum, is thus given by :
6. A*
= *?* 0)
Thus the pressure amounts to two-thirds of the molecular kinetic energy per unit volume.
5. The Hydrostatic Equation
We now approach the problem of the equilibrium of a bulk of air at rest under the external force of gravity, g has the dimensions of an acceleration, L/T*. Its value depends slightly on latitude and altitude, increasing by 0-5 per cent, from the equator to the poles and decreasing by 0-5 per cent, from sea-level to 10 miles altitude, At sea-level and 45° latitude its value is 32»173 in ft.-sec. units. The value 32-2 ft./sec.2 is suffi- ciently accurate for most purposes.
Since no horizontal component of external force acts anywhere on the bulk of air, the pressure in every horizontal plane is constant, as otherwise motion would ensue. Let h represent altitude, so that it increases upward. Consider an element- cylinder of the fluid with axis vertical, of length SA and cross-sectional area A (Fig. 3). The pressure on its curved surface clearly produces Fia. 3.
Sh
K
^fc-^'^ |pA
I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 6
no resultant force or couple. If p is the pressure acting upward on the lower end of the cylinder, the pressure acting downward
on its upper end will be p + J~8h. These pressures give a resultant
downward force : A-~8h. The gravity force acting on the cylinder ah
is pg . A8h. Therefore, for equilibrium—
Thus the pressure decreases with increase of altitude at a rate equal to the local weight of the fluid per unit volume.
6. Incompressibility Assumption in a Static Bulk of Air
Full use of (2) requires a knowledge of the relationship existing between p and p, but the particular case where p is constant is im- portant. We then have
ftp = — pg \dh + const.
or for the change between two levels distinguished by the suffixes 1 and 2 :
~hi)- • • • (3)
This equation is exact for liquids, and explains the specification of a pressure difference by the head of a liquid of known density which the pressure difference will support. In the mercury barometer, for example, if A, > hlt p* = 0 and pl is the atmospheric pressure which supports the otherwise unbalanced cohimn of mercury. At 0° C. the density of mercury relative to that of water is 13-596. When A, — A! = 760 mm., pt is found from (3) to be 2115-6 Ib. per sq. ft. at this temperature.
7. Measurement of Small Pressure Differences
Accurate measurement of small differences of air pressure is often required in experimental aerodynamics. A convenient instrument is the Chattock gauge (Fig. 4). The rigid glasswork AB forms a U-tube, and up to the levels L contains water, which also fills the central tube T. But above L and the open mouth of T the closed vessel surrounding this tube is filled with castor oil. Excess of air pressure in A above that in B tends to transfer water from A to B
6 AERODYNAMICS [CH.
by bubbling through the castor oil. But this is prevented by tilting the heavy frame F, carrying the U-tube, about its pivots P by means of the micrometer screw S, the water-oil meniscus M being observed for accuracy through a microscope attached to F. Thus the excess air pressure in A is compensated by raising the water level in B above that in A, although no fluid passes. The wheel W fixed to S is graduated, and a pressure difference of O0005 in. of water is easily
FIG. 4. — CHATTOCK GAUGE.
detected. By employing wide and accurately made bulbs set close together, constantly removing slight wear, protecting the liquids against appreciable temperature changes and plotting the zero against time to allow for those that remain, the sensitivity * may be increased five or ten times. These gauges are usually constructed for a maximum pressure head of about 1 in. of water. Longer forms extend this range, but other types are used for considerably greater heads.
At 15° C. 1 cu. ft. of water weighs 62-37 Ib. Saturation with air decreases this weight by about 0'05 Ib. The decrease of density from 10° to 20° C. is 0-15 per cent. A 6 or 7 pet cent, saline solution is commonly used instead of pure water in Chattock gauges, however, since the meniscus then remains clean for a longer period.
8. Buoyancy of Gas-filled Envelope
The maximum change of height within a balloon or a gas-bag of an airship is usually sufficiently small for variation of density to be
* Cf. also Cope and Houghton, Jour. Sci. Jnstr., xiii, p. 83, 1936.
I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 7
neglected. Draw a vertical cylinder of small cross-sectional area A completely through the envelope E (Fig. 5), which is filled with a light gas of density p', and is at rest relative to the surrounding atmosphere of density p. Let the cylinder cut the envelope at a lower altitude-level ht and at an upper one Aa, the curves of inter-
FIG. 6.
section enclosing small areas Slt Sa, the normals to which (they are not necessarily in the same plane) make angles oclf a8 with the vertical. On these areas pressures />',, p'2t act outwardly due to the gas, and^>lf p9 act inwardly due to the atmosphere.
There arises at h2 an upward force on the cylinder equal to
(pi — ^a)S2 cos a,.
The similar force arising at h^ may be upward or downward, depend- ing on the position of Sl and whether an airship or a balloon is considered, but in any case its upward value is —
(Pi — P()SI cos «i-
Since Sa cos oc8 = A == St cos al, the resultant upward force on the cylinder due to the pressures is
Substituting from (3), if AL denotes the element of lift — AL = ~
8
AERODYNAMICS
[CH. I
The whole volume of the envelope may be built up of a large number of such cylinders, and its total lift is :
L = (p - p'fcZfA. - hJA
= (P-p')gF' ..... (4)
where V is the volume of gas enclosed. For free equilibrium a weight of this amount, less the weight of the envelope, must be attached.
The above result expresses, of course, the Principle of Archimedes. It will be noted that the lift acts at the centre of gravity of the enclosed gas or of the air displaced, called the centre of buoyancy, so that a resultant couple arises only from a displacement of the centre of buoyancy from the vertical through the centre of gravity of the attached load plus gas. For stability the latter centre of gravity must be below the centre of buoyancy.
If W is the total load supported by the gas and a' the density of the gas relative to that of the surrounding air, (4) gives
W = 9gV'(l-v') (5)
For pure hydrogen, the lightest gas known, a' = 0-0695. But hydrogen is inflammable when mixed with air and is replaced where possible by helium, for which a' = 0-138 in the pure state.
9. Balloons and Airships
In balloons and airships the gas is contained within envelopes of cotton fabric lined with gold-beaters1 skins or rubber impregnated. Diffusion occurs through these comparatively impervious materials,
and, together with leakage, con- taminates the enclosed gas, so that densities greater than those given in the preceding article must be assumed.
Practical values for lift per thousand cubic feet are 68 Ib. for hydrogen and 62 Ib. for helium, at low altitude. Thus the envelope of a balloon weighing 1 ton would, in the taut state at sea-level, have a diameter of 39-8 ft. for hydrogen and 41-1 ft. for helium ; actually it would be made larger, filling only at altitude and being limp at sea-level. FlG. 6. Referring to Fig. 6, OA represents
A.r>. — 1*
10
AERODYNAMICS
[CH.
the variation of atmospheric pressure from the level of the top of the open filling sleeve S to that of the crest of the balloon, OH the corresponding variation of pressure through the bulk of helium filling the envelope. The difference between these external arid internal pressures acts radially outward on the fabric as shown to the right. The upward resultant force and part of the force of expansion are supported by the net N, from which is suspended the basket or gondola B, carrying ballast and the useful load.
Balloons drift with the wind and cannot be steered horizontally. Airships, on the other hand, can maintain relative horizontal velocities by means of engines and airscrews, and are shaped to streamline form for economy of power. Three classes may be distinguished.
The small non-rigid airship, or dirigible balloon (Fig. l(a)} has a faired envelope whose shape is conserved by excess gas pressure maintained by internal ballonets which can be inflated by an air scoop exposed behind the airscrew. Some stiffening is necessary, especially at the nose, which tends to blow in at speed. A gondola, carrying the power unit, fuel, and other loads, is suspended on cables from hand-shaped strengthening patches on the envelope. (Only a few of the wires are shown in the sketch.)
In the semi-rigid type (b) some form of keel is interposed between the envelope and gondola, or gondolas, enabling excess gas pressure to be minimised. Several internal staying systems spread the load carried by the girder over the envelope, the section of which is not as a rule circular.
The modern rigid airship (c) owes its external form entirely to a structural framework covered with fabric. Numerous transverse frames, binding together a skeleton of longitudinals or 'stringers, divide the great length of the hull into cells, each of which accommo- dates a gas-bag, which may be limp. Single gas-bags greatly exceed balloons in size, and are secured to the structure by nets. Some particulars of recent airships are given in Table I.
TABLE I
|
Airship |
Zeppelins Graf Hindenburg |
R101 |
Akron (U.S.A.) |
|
|
Length (ft.) . Max. diam. (ft.) . Gas used Volume (cu. ft.) . Approximate gross lift (tons) . |
776 100 Hydrogen 3-7 x 10« 112 |
800 135 Hydrogen 6-7 X 10* 203 |
777 132 Hydrogen 6-6 X 10a 167 |
785 133 Helium 6-5 X 10« 180 |
The largest single gas-bag in the above has a lift of 25 tons.
I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 11
10. Centre of Pressure
Th£ point on a surface exposed to pressure through which the resultant force acts is called the centre of pressure. The centres of pressure with which we are concerned relate to the pressure differ- ence, often called the gas pressure, unevenly spread over part of an envelope separating gas from the atmosphere. Gas pressures are small at the bottom of an envelope and reach a maximum at the top, as illustrated in Fig. 6, and positions of the centres of pressure are usually high.
The high centres of the total gas pressures exerted on walls which restrain a gas-bag, as in the case of the wire bulkheads or transverse frames of a rigid airship, lead to moments internal to the structure.
BCDE (Fig. 8) is a (full) gas-bag of an airship which is pitched at angle a from a level keel. The longitudinal thrusts P, P' from the
* gas pressure ' are supported by bulkheads EC and DE of areas A, A', assumed plane, B and E being lowest and C and D highest points. The gas is assumed to be at rest, so that pressure is constant over horizontal planes, and its pressure at B, the bottom of the bag, is taken as equal to that of the atmosphere. Let p be the excess pressure at height h above the level of B. Then from (3) p = pigA, where px is the difference in the densities of the gas and the surround- ing air.
Lower Bulkhead BC. — Let 8A be the area of a narrow horizontal strip of BC distant y from a horizontal axis in its plane through B. Then h = y cos a, and the total thrust on BC is given by :
re re
P = p dA = pig cos a y dA
JB JB
= P!# cos a . AyQ . . (i)
12 AERODYNAMICS [CH.
where y0 is the distance of the centroid of BC from the axis through B.
Let the centre of pressure of P be distant y0 + Ay from the B- axis, and take moments about this axis.
P(yQ + Ay) = py dA = ^g cos a y* dA JB JB
= ?£ cos a . 7B . (ii)
where 7B is the second moment of the area about the B-axis. If 70 is this moment about a parallel axis through the centroid, /B = /0 -f Ay <?. Substituting in (ii) :
__ Plg cos a (J0 + Ayl) _ AV-" p - y*
Hence from (i) :
where &0 is the radius of gyration.
The result is independent of pitch. For a circular bulkhead of radius rf for example, /„ = nr*/4: and Ay = r/4. In practice, how- ever, an excess pressure is often introduced, so that pB is not zero, when a correction must be made, as will be clear from the following : Upper Bulkhead DE.— Measuring now y in the plane of ED from a parallel horizontal axis through E, we have :
P = 9iS(y cos a + / sin a), where / is the distance apart of the bulkheads.
ro
P' = Plg (y cos a + / sin &)dA JE
— pig^'^o cos a + / sin a).
TD
P'(yi + A/) = pig (y* cos a + yl sin <x.)dA
J E
= Ptf[(Ji + AW*) cos oi + A'yil sin a]. This gives
A ' - • .«i _ «'
^ ~ X + / tan a -X"
__ __^? M
""yj + /tana i;
The additional term in the denominator is EF. Hence (6) is generalised by (7), since it is always possible to draw a horizontal line BF at which any super pressure would vanish.
AIR AT REST, THE ATMOSPHERE AND STATIC LIFT
13
ii. Relation between Pressure, Density, and Temperature of a Gas
By the experimental laws of Boyle and Charles, for constant temperature the pressure of a gas is proportional to its density ; for constant volume the pressure of a gas is proportional to its absolute temperature. The absolute temperature is denoted by T and, if 6 is the temperature on the centigrade scale, is given by
T = 0 + 273.
Combining these laws, we have, for a given mass of a particular gas:
pV^Bt (8)
where V is the volume, or, if V is the volume of 1 lb.,
P/9=gBi: (9)
B is a constant which is made characteristic of a particular gas by treating 1 lb. of the gas ; it is then evaluated from measurements of pressure and volume at a known temperature. It follows that B will vary from one gas to another in inverse proportion to the density under standard conditions of pressure and temperature.
If N is the ixumber of molecules in V, N will, by Avogadro's law, be the same for all gases at constant p and T. Hence, writing pV/N = B'T, B' is an absolute constant having the same value for all gases. Equation (9) is more convenient, however, and the variation of B is at once determined from a table of molecular weights. Some useful data are given in Table II. It will be noticed that, if p is kept constant, B measures the work done by the volume of gas in expanding in consequence of being heated through unit temperature change. The units of B are thus ft.-lb. per lb, per degree centigrade, or ft. per ° C.
TABLE II
|
At 0° C. |
and 760 mm. |
mercury |
||
|
B |
||||
|
Fluid |
ft /°C |
|||
|
Ib./cu. ft. |
cu. ft./lb. |
P slugs/cu. ft. |
||
|
Dry air ... |
0-0807 |
12-30 |
0-00261 |
06-0 |
|
Hydrogen Helium |
0-00561 0-01113 |
178-3 89-8 |
0-000174 0-000346 |
1381 606 |
|
Water-vapour |
0-0501 |
10-05 |
0-00156 |
155 |
12. Isothermal Atmosphere
We now examine the static equilibrium of a bulk of gas under gravity, taking into account its compressibility. Equation (2)
14 AERODYNAMICS [CH.
applies, but specification is needed of the relationship between p and p. The simple assumption made in the present article is that appropriate to Boyle's law, viz. constant temperature TO, so that />/p remains constant. From (2) :
*.-*.
9g From (9) : 1 _ J5r0
9g P ' Hence : ,.
BiQ = - dh.
P
Integrating between levels Ax and h2, where p = pt and p2 respec- tively,
BTO log (pjpj = h, - h, . . . (10)
The logarithm in this expression is to base e. Throughout this book Napierian logarithms will be intended, unless it is stated otherwise. The result (10) states that the pressure and therefore the density of a bulk of gas which is everywhere at the same tem- perature vary exponentially with altitude.
The result, although accurately true only for a single gas, applies with negligible error to a mass of air under isothermal conditions, provided great altitude changes are excluded. The stratosphere is in conductive equilibrium, the uniform temperature being about — - 55° C. The constitution of the air at its lowest levels is as given in Article 1. As altitude increases, the constitution is subject to Dalton's law : a mixture of gases in isothermal equilibrium may be regarded as the aggregate of a number of atmospheres, one for each constituent gas, the law of density variation in each atmosphere being the same as if it constituted the whole. Hence argon and other heavy gases and subsequently oxygen, nitrogen, and neon will become rarer at higher levels. The value of B for the atmosphere will consequently increase with altitude, although we have assumed it constant in order to obtain (10). The variation of B for several miles into the stratosphere will, however, be small. At greater altitudes still the temperature increases again.
13. The Troposphere
The atmosphere beneath the stratified region is perpetually in process of being mechanically mixed by wind and storm. When a bulk of air is displaced vertically, its temperature, unlike its pressure, has insufficient time for adjustment to the conditions obtaining at the
I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 15
new level before it is moved away again. The properties of this part of the atmosphere, to which most regular flying so far has been restricted, are subject to considerable variations with time and place, excepting that B varies only slightly, depending upon the humidity. There exists a temperature gradient with respect to altitude, and on the average this is linear, until the merge into the stratosphere is approached. It will be found in consequence that the pressure and density at different levels obey the law —
P/9* = k ..... (11)
where k and n are constants. This relationship we begin by assuming.
Substituting for p from (11) in (2) leads to
or
i
nk» »=*
----- m p n = __ gh _j__ const. w ~— 1
Putting p = p0 when h = 0 gives for the constant of integration —
i nk" «-i
Therefore :
M-l n-l n — I
To evaluate k let p0, TO, be the density and absolute temperature atA=0. By (11):
while by (9) :
Hence :
1
PS \ P. and
£ .... /rrT?^ \*A l-»
or
Substituting in (12)
p ( n — l h \£i
. (13)
n tQ v '
16 AERODYNAMICS [CH.
The temperature gradient is found as follows. From (9) and (11):
n-l
=(!)"
Substituting in (13)
— 1 A
or
6 denoting temperature in ° C. This shows that while n remains constant :
dt dQ n — 1
i.e. the temperature variation is linear.
As the stratosphere is approached, the law changes, the gradient becoming less and less steep.
14. The International Standard Atmosphere
It is necessary to correct observations of the performance of air- craft for casual atmospheric variation, and for this purpose the device of a standard atmosphere is introduced. A number of countries have agreed upon the adoption of an international standard, representing average conditions in Western Europe. This is defined by the temperature-altitude relationship :
6 = 15 — 0-00198116& . . . (17)
A being in feet above sea-level (the number of significant figures given is due to h being expressed in the metric system in the original definition). The dry air value of B, viz. 96-0, is also assumed.
This definition leads to the following approximations : From (16) n == 1-235
From (13) p/p, = (1 - 0-00000688A)*'M'
From (14) pfp9 = (T/288)B-*" f • • I18)
Similarly P/PO == (T/288)4-a85
Some numerical results are given in Table III.
I] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT
TABLE III
17
|
h (ft.) |
0 (° C.) |
PIP, |
<r = p/Po |
|
0 |
16-0 |
1-000 |
1-000 |
|
5,000 |
5-1 |
0-832 |
0-862 |
|
10,000 |
— 4-8 |
0-688 |
0-738 |
|
15,000 |
- 14-7 |
0-664 |
0-629 |
|
20,000 |
- 24-6 |
0-459 |
0-534 |
|
25,000 |
- 34-6 |
0-371 |
0-448 |
|
30,000 |
- 44-4 |
0-297 |
0-375 |
|
35,000 |
- 54-3 |
0-235 |
0-310 |
|
40,000 |
- 55 |
0-185 |
0-244 |
|
45,000 |
- 56 |
0-145 |
0-192 |
|
60,000 |
- 55 |
0-116 |
0-161 |
15- Application to Altimeters
A light adaptation of the aneroid barometer is used on aircraft, with the help of a thermometer, to gauge altitude. To graduate the instrument, increasing pressure differences are applied to it, and the dial is marked in intervals of h according either to the isothermal or to the standard atmosphere laws.
In the former case, the uniform temperature requiring to be assumed is usually taken as 10° C. From (10) and Table III the altitudes indicated are then excessive, on the basis of the standard atmosphere, by about 1-6, 5-7, and 10 per cent, at altitudes of 10,000, 20,000 and 30,000 ft. respectively. Correction for decrease of temperature with increase of altitude is made by assigning esti- mated mean temperatures to successive intervals of altitude. Thus, if TM applies to the true increase of altitude A#, corresponding to a decrease of p from pl to p2, while AA is indicated by the altimeter whose calibration temperature is TO, we have from (10)
A# = AA.^ . . . . (19)
Readings of altimeters with a standard atmosphere scale require correction for casual variation of temperature. Let H, TH, and p denote the true altitude, temperature, and pressure respectively, and h the altimeter reading corresponding to p and the graduation temperature T. Use suffix 0 for sea-level and write 5 for the tem- perature lapse rate, so that T — TO == — sh. Then from (14), if s remains constant —
t - 1
(i)
18 AERODYNAMICS [CH.
Hence :
To giving
by (i).
. . . (20)
TO — Sfl v
1 6. Gas-bag Lift in General
The assumption of constant density made in Article 8 to obtain expression (4) for the lift L of a gas-filled envelope may now be examined. Although a balloon of twice the size has been con- structed, 100 ft. may be taken as a usual height of large gas-envelopes. The maximum variation from the mean of the air density then follows from the formulae (18). At sea-level, where it is greatest, it amounts to 0-15 per cent, approximately. Similarly, the maximum variation of the air pressure from the mean is found to be less than 0*2 per cent. Equation (3) shows that the corresponding variations in the gas will be smaller still.
Although the buoyancy depends on differences between atmos- pheric and gas pressures, these are negligible compared with varia- tions caused in both by considerable changes in altitude. Gas-bags should be only partly filled at sea-level, so that the gas can, on ascent, expand to fill an increased volume without loss.
To study the condition of a constant weight W' of gas enclosed, (4) is conveniently written :
L = W'(±-\Y . . . (21)
We also have from (9), always distinguishing the gas by accented symbols :
1 __ p __£V a' ~ p ' ~~ ~B^ at all pressures, and, therefore, altitudes. So (21) becomes
• • • (22)
and it is seen that since B, B' are constants, L remains constant in respect of change of altitude, provided that no gas is lost and that no temperature difference arises between the gas and the surrounding air. The last requirement involves very slow ascent or descent to
l] AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 19
allow sufficient transference of heat through the envelope, or the envelope must be held at a new altitude — as is possible by aero- dynamic means with airships — until such transference has taken place.
Gas-bags are too weak to support a considerable pressure, and safety valves operate when they become full, leading to a loss of gas. Thus the volume held in reserve at sea-level decides the maximum altitude permissible without loss of gas. This is called the static ceiling. A lighter-than-air craft can be forced to still greater alti- tudes by the following means : aerodynamic lift ; heating of the gas by the sun ; entering a cold atmospheric region ; or by discharging ballast. The condition then is that V' remains constant. Exclud- ing the case of variation of weight, we find from (4) that the gas lift will remain constant only if p — p' remain constant or, by (9) if
P \ 5 W' )' i-e- if PI1 — HT7 ) remain constant. Hence, from
\ £>T D T / \ JD T /
(11) the condition is that n — _ Ljmust vary inversely as p*.
In this way it is simple to calculate the excess gas temperature required for static equilibrium at a given altitude in excess of the static ceiling. Gas having been lost, when the temperature differ- ence vanishes ballast must be released for static equilibrium to occur at any altitude.
17. Vertical Stability
The foregoing conditions depend upon the absence of a propulsive or dragging force ; the envelope must move with the wind, otherwise a variation of external pressure, different from that investigated, may contribute to lift. A difference between gas-lift and total weight, brought about by release of gas, for instance, or discharge of ballast, creates vertical acceleration which leads to vertical velocity relative to the surrounding air, equilibrium again being attained by the supervention of an aerodynamic force due to the relative motion. Variation of weight carried or of gas provides con- trol of altitude, but even if, as in the case of airships, vertical control by aerodynamic means is also possible, the practical feasibility of lighter-than-air craft requires further investigation, since their level of riding is not obviously fixed, as is the case with ships only partly immersed in water. The first question is whether, in a stationary atmosphere, a balloon would hunt upwards and downwards, restrict- ing time in the air through rate of loss of gas due to the need for con-
20 AERODYNAMICS [ClL
tinual control. A second question is whether the atmosphere is liable to continual up and down currents. These would have the same effect on the duration of flight of a balloon, but the second question has a wider significance, since such currents, if sufficiently violent, would make flight by heavier-than-air craft also impossible. Consider the rapid ascent of an envelope without loss of gas from altitude hlf where the atmospheric pressure p = pt and the absolute temperature T = TJ, to h2f where p = pt. When the atmosphere is in standard condition we have :
/A\*~T /* \°t1903 T* = #A = (*!\
TX W \pj '
For the gas within the envelope the thermal conductivity is so small that heat transference can be neglected. The gas then expands according to the adiabatic law :
P
r = const.
Distinguishing properties by accented symbols, we have, since Y= 1-405:
/ r,/x 0-288
T, (P*\
3 = (p'J •
Now assume that initially r( = TJ. Very closely, pl = p( and pz = P*- Since p* < plf we then have that -ri < T2. Hence, by Article 16 the gas-bag will sink, the load attached to it being con- stant. Conversely, a rapid descent of a gas-bag results in temporary excessive buoyancy.
Thus a lighter-than-air craft riding below its static ceiling tends to return to its original altitude if displaced, provided displacement is sufficiently rapid for passage of heat through the envelope to be small. It is said to be stable in respect of vertical disturbance. The state of the atmosphere is part and parcel of the question, for a necessary proviso is seen to be that n < y.
If the craft is above its static ceiling, the stability in face of down- ward disturbance is the same, since no further gas is lost. But for upward displacement the stability is greater, since the weight of gas enclosed decreases.
1 8. Atmospheric Stability and Potential Temperature
The foregoing reasoning may be applied to the rapid vertical dis- placement of a bulk of the atmosphere, and we find that if, for the
l] AIR AT REST, THE ATMOSPHERE AND STATIC IIFT 21
atmosphere, n has a value less than 1-405, up and down currents are damped out. If n = 1-405, the stability is neutral ; the atmosphere is then said to be in convective equilibrium. When n > y, a condi- tion that may arise from local or temporary causes, vertical winds occur and make aeronautics dangerous, if not impossible.
The condition for atmospheric stability is discussed alternatively in terms of ' potential temperature/ The potential temperature at a given altitude is defined as the temperature which a given bulk of air at that altitude would attain if displaced to a standard altitude, such as sea-level, the compression taking place without loss or gain of heat. We see that f or n = y the potential temperature would be the same for all altitudes. For n < y the potential temperature increases upward. An atmosphere is stable when the potential tem- perature is greater the greater the altitude. It will be noticed that the stratosphere is more stable than the troposphere.
19. Bulk Elasticity
Fluids at rest possess elasticity in respect of change of volume. The modulus of elasticity E is defined as the ratio of the stress caus- ing a volumetric strain to the strain produced. An increase of pressure from ptop + $p will change a volume V to V — W. The strain is — 8V I V and E is given by
'-*/(-?)•
Since
dV
£ = P ..... (23)
tfp
In the case of liquids the compressibility is very small, and (23) sufficiently defines Et but with gases we must specify the thermal conditions under which the compression is supposed to take place. The interest of E in Aerodynamics is chiefly in respect of changes of pressure, and therefore of density, occurring in air moving at approxi- mately constant altitude. The changes are usually too rapid for appreciable heat to be lost or gained, having regard to the small thermal conductivity of air. In these typical circumstances the adiabatic law is again assumed, viz. p = &pv, so that from (23)
E =
22 AERODYNAMICS [CH. I
20. Velocity of Sound
The condition under which (24) has been derived is ideally realised in the longitudinal contractions and expansions produced in elements of the air by the passage of waves of sound. Newton demon- strated the following law for the velocity a of such waves in a homo- geneous fluid :
a = V(£/p).
Thus for gases, from (24) —
a = Vy^/p .... (25) or, substituting from (9),
a = VjgBi: (26)
The velocity of sound is seen to depend on the nature of the gas and its temperature only.
We shall always employ the symbol a for the velocity of sound in air. With y = 1-405, g = 32-173 and B = 96-0,
a = 65-9-v/r .... (27) nearly. For 15° C., T = 288,
a = 1118 ft. per sec. . . . (28)
A disturbing force or pressure suddenly applied to a part of a solid body is transmitted through it almost instantaneously. From the preceding article we infer that through air such a disturbance is pro- pagated more slowly, but yet at a considerably greater rate than the velocities common in aeronautics. Disturbance of the stationary equilibrium of a bulk of air follows from swift but not instantaneous propagation through it of pressure changes. It may be noted, for example, that a moving airship disturbs the air far in front of it ; a fast bullet, on the other hand, overtakes its propagation of distur- bance and fails to do so. This change assumes great significance in connection with stratospheric flying, for two reasons : a decreases to between 970 and 975 ft. per sec., the flight speeds of low altitude are at least doubled to compensate for the reduced density of the air.
Chapter II AIR FLOW AND AERODYNAMIC FORCE
21. Streamlines and Types of Flow
It is familiar that motions of air vary considerably in character. Means of discriminating with effect between one kind of flow and another will appear as the subject develops, but some preliminary classification is desirable.
Streamlines. — Discussion is facilitated by the conception of the streamline. A streamline is a line drawn in the moving fluid such that the flow across it is everywhere zero at the instant considered.
Uniform Flow. — The simplest form of flow is uniform motion. By this we mean that the velocity of all elements is the same in magni- tude and direction. It follows that the streamlines are all parallel straight lines, although this is not sufficient in itself to distinguish uniform motion.
Laminar Parallel Flow. — There are other motions whose stream- lines are parallel straight lines, In which the velocity of the element, although uniform in direction, depends upon distance from some fixed parallel axis or plane. Such motions are properly called laminar, although the name laminar is nowadays frequently used in a wider sense, strictly laminar motions being characterised as ' parallel/
Both uniform and laminar motions are steady, i.e. the velocity at any chosen position in the field of flow does not vary in magnitude or direction with time. They are more than this, however, for the velocity of any chosen element of fluid does not vary with time as it proceeds along its path. (It is specifically in this respect that wider use is commonly made of the name laminar.)
General Steady Flow. — We may have a steady motion which is neither uniform nor strictly laminar. The streamlines then form a picture of the flow which does not vary with time, but the velocity along a streamline varies from one position to another. Thus the elements of fluid have accelerations. The streamlines are riot parallel and in general are not straight. It is this more general kind of flow that is usually intended by the term ' steady motion ' used without qualification.
23
24 AERODYNAMICS [Cfl.
Unsteadiness and Path-lines. — Steady motions are often called ' streamline/ All steady motions have one feature in common : the streamlines coincide with the paths of elements, called path- lines.
Unsteady motions are common in Aerodynamics, and in these the path-lines and streamlines are not the same. The velocity varies with both space and time. At a chosen instant streamlines may be drawn, but each streamline changes in shape before an element has time to move more than a short distance along it. An unsteady motion may be such that an instantaneous picture of streamlines recurs at equal intervals of time ; it is then said to be periodic or eddying, though use of the latter term is less restricted.
Turbulence. — When unsteadiness of any kind prevails, the motion is often called turbulent. In addition to periodic we may have irregular fluctuations. These may occur on such a scale that transient streamlines might conceivably be determined. But in other cases the fluctuations are much more finely grained, conveying the impression of a chaotic intermingling of very small masses of the fluid accompanied by modifications of momentum. This last type of unsteadiness is, unfortunately, at once the most difficult to under- stand and the most important in practical Aerodynamics. It has come to be the form usually intended by the name turbulence.
Stream-tube. — A conception of occasional use in discussing steady flow is the stream-tube. This may be defined as an imaginary tube drawn in the fluid, of small but not necessarily constant section, whose walls are formed of streamlines. Clearly, no fluid can enter or leave the tube through the walls except in respect of molecular agitation.
Two-dimensional Flow. — Another conception, of which we shall make very frequent use, is that of two-dimensional flow. Consider fixed co-ordinate axes Ox, Oy, Oz drawn mutually at right angles in the fluid. Let the velocity components of any element in the direc- tions of these axes be u, vt and w9 respectively. Two of the direc- tions, say Ox and Oy, are open to selection, the third then following. If the motion is such that we can select Ox, Oy in such a way that w = 0 for all elements at all times, and also if neither u nor v then vanishes, the motion is of general two-dimensional form. The streamlines drawn in all planes parallel to a selected #y-plane will be the same. It is then sufficient to study the motion in the ^y-plane, tacitly assuming that we are dealing with a slice of the fluid in motion of unit thickness perpendicular to this plane.
If besides w = 0 we have another velocity component, say vt
II] AIR FLOW AND AERODYNAMIC FORCE 25
everywhere vanishing, the motion is strictly laminar, or parallel, and we may have u depending either upon distance from the plane xOz or upon distance from the axis Ox. In the former case, where u is a function of y only, the motion is two-dimensional ; in the latter, the flow is of the kind that occurs in certain circumstances along straight pipes of uniform section, when it is sufficient to consider unit length of the pipe because the distribution of flow will be the same through all cross-sections.
22. Absence of Slip at a Boundary
The theorem of Article 3 holds equally for a fluid in uniform motion if the rigid surface exposed in the fluid moves exactly with it. The pressure in uniform motion is thus constant and equal in all directions. Unless the whole motion is uniform, however, the theorem fails, and considerable investigation is necessary to establish precisely what we then mean by ' pressure/
Imagine a small, rigid, and very thin material plate to be im- mersed and held stationary in the midst of a bulk of air in motion ; let its plane be parallel to the oncoming air, considered for simplicity to be in uniform motion. The disturbance caused by the plate might, on account of its extreme thinness, be expected to be neglig- ible. This would, however, be completely at variance with experi- mental fact. Experiment clearly shows that the fluid coming into contact with the tangential surfaces of the plate is brought to rest, whilst fluid that passes close by has its velocity substantially reduced.
To explain this phenomenon in molecular terms we may suppose the plate to be initially chemically clean, each surface being a lattice-work of atoms of the substance of which the plate is made. As such it exposes a close distribution of centres of adhesive force. The force of adhesion is very intense at distances from the surface comparable with the size of a molecule, and a molecule of gas impinging on the surface is held there for a time. Considering the whole lattice-work, we may say that the air is condensed on it, since the molecules no longer possess a free path. But the layer of con- densed gas receives energy, partly from the body of the plate and partly from bombardment by free gas molecules, and where the energy attains to the latent heat of evaporation the molecules free themselves and return to the bulk of the gas— thereby only giving place, however, to others. Thus the film of condensed gas molecules is in circulation with the external free gas.
Regarding the action of the plate on the stream of air, we must
26 AERODYNAMICS [CH.
suppose, therefore, two effects to result from molecular constitu- tion : (a) impinging air molecules are brought to rest relative to the bulk or mass motion — just as they are, for a time, in regard to the molecular motion ; (b) air molecules released from the plate are de- prived of mass motion, and, requiring to be accelerated by the other molecules, retard the general flow to an appreciable depth. Thus the rate of change of molecular momentum at the plate is no longer normal to its surface, but has a tangential component ; in other words, the ' pressure ' on the plate is oblique. Further, the retarda- tion occurring at some distance into the fluid shows that the pressure in this affected region away from the plate cannot be equal in all directions. It will be noted that the mass flow is no longer uniform ; its initial uniformity has been destroyed by introducing the plate which has a relative velocity.
The phenomenon of absence of slip at the surface of separation of a material body from a surrounding fluid occurs quite generally and is of fundamental importance in Aerodynamics. It is known as the boundary condition for a real fluid. No matter how fast a fluid, gaseous or liquid, is forced to rush through a pipe, for example, the velocity at the wall is zero. The velocity of the air immediately adjacent to the skin of an aeroplane at any instant is equal to that of the aeroplane itself.
Thus a uniform fluid motion cannot persist in the presence of a material boundary which is not moving with the same velocity (although the motion may remain steady). The * pressure ' at a point in the unevenly moving fluid will depend upon the direction con- sidered. The matter is further investigated in the following articles.
VISCOSITY 23. Nature of Viscosity
If air is moving in other than uniform motion, a further physical property is brought into play in consequence of the molecular structure of the fluid. Its nature will be discussed with reference to laminar (or parallel) two-dimensional flow. Let this flow be in the direction Q%% and draw Oy so that u, the mass velocity, is a function of y only.
Consider an imaginary plane, say y =y't perpendicular to Oy (Fig, 9). This plane is formed of streamlines, but owing to mole- cular motion, molecules are continually darting across it in all direc- tions. Density remains uniformly distributed, and this condition entails that the same number of molecules crosses a chosen area of
n]
AIR FLOW AND AERODYNAMIC FORCE
27
the plane in unit time from either side. The molecules possess, in addition to their molecular velocity, a superposed mass velocity ut which by supposition is different on one side of the plane from on the other. Hence molecules crossing in one direction carry away, per unit area of the plane and in unit time, a different quantity of mass momentum from that which those crossing in the opposite direction bring with them. Hence momentum is being transported across the direction of flow. This phenomenon is called the viscous effect. Clearly, it exists only in the presence of a velocity gradient, which it tends to destroy in course of time.
Qualitative Theory of Viscosity
Consider an imaginary right cylinder (Fig. 9) of unit length and unit cross-section, whose ends are parallel to and, say, equidistant from the imaginary plane y = yf. Denote by 5 the unit area of the plane which the cylinder encloses. If p is the density of the air and m the mass of each molecule, the number of molecules within the cylindrical space is p/w and is constant. These molecules are moving in all directions with a mean molecular velocity c FIG. 9.
along straight paths of mean
length X. They have in addition a superposed mass velocity u whose magnitude depends upon their values of y at the instant considered, subject to the consideration that the u of any particular molecule cannot be modified while it is in process of describing a free path, for changes can come only from collisions.
The molecular velocities of all the p/w molecules can be resolved at any instant parallel to Ox, Oy, Oz, but, as in Article 4, the number being very large, this procedure may be replaced statistically by imagining that p/6w molecules move at a velocity c in each of the two directions which are parallel to each of the three co-ordinate axes. This equivalent motion must be supposed to extend through the interval of time At which is required for a displacement of the molecules through a distance X. Thus A/ = X/c. At the end of this interval collision occurs generally.
We are concerned only with molecules which cross S, and so ignore
28 AERODYNAMICS [CH.
all moving parallel to Ox, Oz. Of molecules moving parallel to Oyt only those within a distance X of y = y' can cross during Atf. Thus, S being unity and there being no displacement of mass, Xp/6w molecules cross in each direction during this time. For clarity we shall speak of y increasing as ' upward ' and assume u to increase upward. There is also no loss of generality in supposing that all molecules penetrating S from above or below y9 start at distance X from that plane, the velocity at y =y' being u.
Consider a single exchange by the fluid above y'. It loses on ac-
/ du count of the downward-moving molecule momentum =mlu + ^-
whilst it receives by the upward-moving molecule momentum = mu,
du
a loss on this account of wX— . But in addition it must, by
Ay
collision at the end of A£, add momentum to the incoming molecule
du
to the amount wX— . Thus the total change in the momentum of Ay
the fluid above y' in respect of a single molecule exchanged with one
du from below is a loss amounting to 2wXT-. Summing for all pairs, the
dy
aggregate loss is :
du
- . dy
The rate of this loss is :
1 pX2 du
AV 3 dy or, since A£ = X/c, the rate is
du
The rate of change of mass momentum being parallel to Ox, it may be represented by a force in the fluid at^ =y' acting tangentially on the fluid above. If the intensity of this traction is F, we have, since
The direction of F is such as to oppose the motion of the fluid above. Similarly, we find that the fluid below / gains momentum at the ;ame rate. We note the passage downward of momentum and that i traction F acts at S in the opposite direction on the fluid below, irging it forward.
29
II] AIR FLOW AND AERODYNAMIC FORCE
The coefficient by which du/dy is to be multiplied in order to determine F is called the coefficient of viscosity, and is denoted by JA. Its dimensions are (Af/Z8) . (LIT) . L = M/LT.
24. Maxwell's Definition of Viscosity
The following example is instructive from several points of view. A number of layers of air, each of thickness ht are separated from one another by a series of infinite horizontal plates. Alternate plates are fixed, while the others are given a common velocity U in their own planes. The resulting conditions in all layers will be the same except fcfor a question of sign, and we shall investigate one layer only.
Draw Ox (Fig. 10) in the fixed plate (taken to be the lower one)
y
u=U
u
tr
U-O
FIG. 10.
and in the direction of motion of the other, and Oy vertically up- ward. By Article 22 air touching the fixed plate has a velocity u = 0, while for air touching the moving plate u = U, and the fluid between is urged forward from above, but the ensuing motion is retarded from below.
Now it is assumed that, after sufficient time has elapsed, the motion in the layer becomes steady. In these circumstances con- sider a stratum of air of thickness 8y between the plates and parallel to them. If the velocity at the lower face distant y from Ox is u,
du that on the upper face is u + ^~8y. The intensity of traction F on
ay
the lower face is equal in magnitude to u,—~, and retards the stratum ;
ay
that on the upper face is jx ~ ( u + -- 8y Y and tends to accelerate the
dy\ dy /
stratum. The resultant traction on the stratum in the direction Ox
. j d ( du 1S ^_ + _
du\ d*u ^ — j- r = pi j -8y. But as the motion is steady.
30 AERODYNAMICS [CH.
there cannot be a resultant force on the stratum. Hence :
Integrating twice,
u = Ay + B
where A and B are constants of integration. Now insert in this equation for u the special values which are known, viz. u = 0 when y = o, w = U when y = h. Two equations result, viz. :
0 =o + B
U =Ah + B
which are sufficient to determine A and B. We find :
5 =0
A = Z7/A. Inserting these values in the original equation for u,
u = j,y ..... (30)
Thus the fluid velocity between the plates is proportional to y . The distribution of velocity is plotted in the figure.
Let F be, as before, the intensity of traction, and reckon it positive in the direction Ox. The traction exerted on the fluid adjacent to the lower plate by the fluid above is given by
This traction is transmitted to the lower plate and a force of equal intensity must be applied in the opposite direction to prevent it from being dragged in the Ox direction. Similarly, it is found that a force
F= ^
must be applied to the upper plate to maintain the motion. But from (30) —
/du\ __ E7 _ /du
)y^ ~~ h ~~~ \
Hence the forces on the plates are equal and opposite, as is otherwise obvious. F is, in fact, uniform throughout the fluid. Hence this case of motion is known as uniform rate of shearing.
If U = 1 = A, the intensity of either force is equal to (z. Hence, Maxwell's definition of the coefficient of viscosity as the tangential
U] AIR FLOW AND AERODYNAMIC FORCE 31
force per unit area on either of two parallel plates at unit distance apart, the one being fixed while the other moves with unit velocity, fMfiuid filling the space between them being in steady motion. , ' In the general case the moving plate does work on the layer of fluid at the rate fjtC7a/A per unit area of the plate. The result is a gradual rise in temperature of the fluid unless the heat generated is conducted away.
25. Laws of Viscosity
The traction on a bounding surface past which a fluid is flowing is called the skin friction. It differs in nature from the rubbing friction between two dry surfaces, but is essentially the same as the friction of a lubricated surface, such as that of a shaft in a bearing.
In certain cases of laminar flow, as will be seen in Chapter IX, the boundary value of the velocity gradient can be calculated in terms of a total rate of flow which can be measured experimentally, while the skin friction can also be measured. Hence the value of \i can be deduced without reference to the theory of Article 23. By varying the density, pressure, and temperature of the fluid in a series of experiments, empirical laws expressing the variation of y. can be built up.
The experimental value of y. for air at 0° C. is given by
N = 3-58 x 1(T7 slug/ft, sec. . . (31)
It is interesting to compare this with a numerical value obtainable from the qualitative theory. Equations (1) and (9) of Chapter I together give
c* = 3gJ3T ..... (32)
The value of c calculated from this expression for 0° C., viz. 1591 ft. per sec., is greater than the mean molecular velocity given in Article 1 for this temperature, because it is a root-mean-square value. Hence, according to (29) —
(i)
giving, for 0° C., pt = 2-58 x 1G~7. The error, amounting to 28 per cent., is removed by more elaborate analysis, which shows that the mean free path must effectively be increased in the viscosity formula. But this mathematical development is not required in Aerodynamics, since JJL can be measured accurately.
The first law of viscosity is that the value of the coefficient is independent of density variation at constant temperature. This
32
AERODYNAMICS
[CH.
surprising law is expressed in (i), because it is obvious that X must be inversely proportional, approximately, to p. After predicting the law, Maxwell showed it experimentally to hold down to pressures of 0-02 atmosphere. It tends to fail at very high pressures.
According to (i) a second law would be [i oc VT, but experiment shows [i to vary more rapidly with the temperature. An empirical law for air is
'4
• • ' • (33>
(Rayleigh), where (JLO is the value of the coefficient at 0° C.
PRESSURE IN AIR FLOW 26. Relation between Component Stresses
We now prove a relationship that exists between the stresses on an element (in the sense of Article 2) of a fluid in any form of two- dimensional motion. In the general case we have four com- ponent stresses to deal with, and a certain nomenclature is adopted, as follows. For a face drawn perpendicular to Ox, the normal pressure on it in the direction Ox is denoted by pxx and the tangential com- ponent in the direction Oy by pxr The corresponding normal and tangential pressures on a face perpendicular to Oy are
x
FIG. 11.
,and^.
It will always be possible to find two axes at right angles to one another, moving with the element, such that, at the instant considered, the pressures in these directions tend to produce either simple compression or simple dilatation in the element. These axes are called principal axes and the pressures in their directions principal stresses.
Let G be the centre of the element which is moving in any manner in the plane xOy. Let Gxf, Gy' be the principal axes at any instant, inclined at some angle a to the fixed axes of reference Oxt Oy. Denote by p^ (Fig. 1 1) the principal pressure parallel to Gx' and by p^ that parallel to Gy' , and take a negative sign to indicate that the
II] AIR FLOW AND AERODYNAMIC FORCE 33
pressure is tending to compress the element and a positive sign that it is tending to dilate it.
Adjacent to G draw X'Y' perpendicular to Oy and X'Y " of equal length perpendicular to Ox, forming with the principal axes the element-triangles GY'X' and GX"Y" (Fig. 11). These triangles are to be regarded as the cross-sections of prisms A and B, respectively, which have the same motion as that of G and whose faces are per- pendicular to the Ay-plane. Let X'Y' = X"Y" = A, then A is equal to the area of each of these two particular faces of the prisms per unit length perpendicular to the #jy-plane. Similarly the area per unit length of the GX' face = that of the GY" face = A cos a, etc.
The prisms of fluid form part of the general motion and have accelerations. The forces arising from these are proportional, however, to mass, i.e. to A8, and, as A is supposed very small, are negligible compared with forces arising from the stresses, which are proportional to A2. Hence the stresses are related by the condi- tion for static equilibrium.
For the equilibrium of prism A we have, first, resolving in the direction Ox —
y or
pyx .A — p1 . A sin a . cos a + P* • A cos a . sin a = 0,
Pyx = (Pi — ^2) sin a cos a. ;cti(
pxy . A — />! . A cos a . sin a + pz • A sin a . cos a = 0,
Resolving in the direction Oy, we have, in regard to the equilibrium of B—
or
Pxy = (Pi — P*) sin a COS a«
Hence :
Ay = Py* = i(#i — #«) sin 2a. . . (34)
The pressure pxy is identical with the tractional stress F of equation (29) and involves an equal tractional stress at right angles. This conversely is the condition for principal axes to exist.
With regard again to the equilibrium of A, but resolving now in the direction Oy —
pyy . A — pl . A sin2 a — p* . A cos1 a = 0, while resolving parallel to Ox with regard to B —
pxx . A — p1 . A cos* a — pt . A sin2 a = 0. These two equations together give :
A.D.— 2
34 AERODYNAMICS [CH.
This equation is independent of a. Hence the arithmetic mean of the normal components of pressure on any pair of perpendicular faces through G is the same.
27. The Static Pressure in a Flow Let us write :
-#=*(#! + PI = *(#« + Pyy) • • (36)
where, it has been found possible to say, x and y are any directions at right angles to one another. Then p does not depend upon direction and is the compressive pressure we shall have in mind when referring to the ' static pressure/ or simply the pressure, at a point of a fluid in motion. (The system of signs adopted in the last article will be found convenient in a later chapter.) It will be noted that, if the fluid were devoid of viscosity, p would be the pressure acting equally in all directions at a chosen point, although not necessarily equally at all points.
The basis of the experimental measurement of p is as follows. The mouth of the short arm of an L-shaped tube is sealed, and a ring of small holes is drilled through the tube wall a certain distance from the closed mouth. The long arm is connected to a pressure gauge, so that the outer air communicates with the gauge through the ring of holes. The other side of the gauge is open to the atmosphere. The tube is then set in motion in the direction of its short arm through approximately stationary air. It is apparent, from Article 22, that the pressure acting through the ring of holes will not in general be the same as with the tube stationary. Nevertheless, a design for the short arm can be arrived at by experiment, such that the gauge shows no pressure difference when the tube is given any velocity, large or small. Adding to the whole system of tube and air a velocity equal and opposite to that of the tube converts the case of motion to that of a stationary tube immersed in an initially uniform air-stream. The pressure communicated is then the same when the tube is immersed in uniform flow or in stationary air. Thus the tube correctly transmits the static pressure of a uniform motion. To cope with motions in which the static pressure varies from point to point, the tube may be reduced to suitably small dimensions ; even 0«5 mm. diameter is practicable, the ring of holes then degenerating to one or two small perforations.
28. Forces on an Element of Moving Fluid
The forces on the three-dimensional element fcSyftr are con- veniently grouped as due to (a) external causes, such as gravity,
nj AIK FLOW AND AERODYNAMIC FORCE 35
(b) variation of the static pressure p through the field of flow, (c) tractions on the faces.
In regard to (a) it may be remarked generally that, although air- craft traverse large changes of altitude, the air motions to which they give rise are conveniently considered with the aircraft assumed at constant altitude and generalised subsequently. The air will be deflected upward or downward, but its changes of altitude are then sufficiently small for variations of density or pressure on this account to be neglected. An element of air may be regarded as in neutral equilibrium so far as concerns the gravitational field, its weight being supposed always exactly balanced by its buoyancy.
(b) We shall require very frequently to write down the force on an element due to space variation of p. Choose Ox in the direction in which p is varying and consider the forces due to p only on the faces of the element 8x8y8z. The forces on all the 8x8y and 8x8z faces cancel, because p is varying only in the ^-direction. On the 8y8z face that is nearer the origin, the force is p . 8y8zf while on that
farther from the origin the force is (p + ~ 8x j 8y8z. The resultant force in the direction Ox is thus
p . 8y8z -(P + '-r- 8*)
= — -~- X the volume of the element . . (37) dx * '
This result should be remembered.
(c) The tractions have already been discussed to some extent. They are proportional to p,, which is small for air. Close to the sur- faces of wings and other bodies studied in Aerodynamics, the velocity gradients are steep and the tractions large. Away from these boundaries, however, the velocity gradients are usually sufficiently small for the modification of the motion of the element due to the tractions to be neglected.
BERNOULLI'S EQUATION 29. Derivation of Bernoulli's Equation
The following five articles treat of flow away from the vicinity of material boundaries, and such that the tractions on the element can
36 AERODYNAMICS [CH.
be neglected, i.e. the pressure p is assumed to act equally in all direc- tions at any point. It is also assumed that the flow is steady. Consider steady flow of air at velocity q within a stream-tube
(Article 21) of cross-sectional area A (Fig. 12). Denote by s distance measured along the curved axis of the tube in the direction of flow. The condition of steady motion means that p, q, p, A may vary with s, but not, at any chosen posi- FlG 12> tion, with time. Since fluid does not
collect anywhere — pqA = constant . . . . (38)
The volume of a small element 8s of the air filling the stream-tube is A 8s, and the force on it in the direction of flow due to the pressure
dp variation is — - . ,48s, by (37). The mass of the element is
and its acceleration is dq/di. By Newton's second law of motion—
But
Jt = ds ' It = qds* Hence :
I dp , dq
~~~ + q-~^=Q (39)
p ds ds v '
Integrating along the stream-tube, which may now be regarded as a streamline —
h l?a = constant. . . . (40)
J p v '
This is the important equation of Bernoulli. Evaluation of the remaining integral requires a knowledge of the relationship between p and p. The constant appertains, unless proved otherwise, only to the particular streamline chosen ; it must be regarded in general as varying from one streamline to another of the same flow. Another form is obtained by integrating (39) between any two values of s, where the conditions are denoted by 1 and 2 :
{41)
II] AIR FLOW AND AERODYNAMIC FORCE 37
30. Variation of Density and Temperature
1°. — Let us first assume the flow to be isothermal, so that p and p obey Boyle's law : dp /dp = constant. The integral remaining in (40) may then be written :
dp _
* P Y P
from (25), a being the velocity of sound and y the ratio of the specific heats = 1-405. This reduction is possible because (Article 20) a remains constant under the isothermal condition. Hence (41) becomes, on evaluating the integral —
C
J <
T Pi or
Pi Expanding in an exponential series —
Pl «
under isothermal conditions.
2°. — Density variations actually occur so rapidly in, most aero- dynamical motions that the isothermal assumption is inappropriate, and, in fact, the condition is closely approached that no heat is lost or gained. The adiabatic law then relates the pressure to the den- sity, viz. :
p = kf . . . . (44)
The absolute temperature T now varies from point to point according to
. (45)
From (44) :
dp — Y*pY - l dp. Thus :
f*dp (• Y
— = Y* P
J i P J i
38 AERODYNAMICS
or, eliminating k by (44),
1 pi l\pl>
'-l} • . . (46)
where, it will be noted, the velocity of sound introduced from (25) refers to the position slt where T = Tt. Substitution in (41) leads to—
— 1) — ~- > . . (47)
Finally, expanding by the Binomial Theorem—
— = 1 2 — i- J- -_ I[±l iL | __ (A0\
pi 2dj* \2_ \ 2a^ / * '
Comparison with (42) and (43) shows density variation now to be less, also that the convenient expression —
P~ = * Ul • . . (49)
applies closely to adiabatic flow, provided the velocity change is not great. If qj — q* amounts to \a? the error in (49) is only 1-3 per cent. ; this would occur, for example, if <72 = 2^ = 912 ft. per sec., or if q* = 3#x = 838 ft. per sec.
There is an important limit to the application of (47) ; q2 cannot exceed a2, because #a gives the limiting velocity with which pressure waves can be propagated. It will be noted that, since the tempera- ture is reduced on expansion, 0a < ax. When q2 = a2 and ql = 0, we find the minimum value of the density ratio :
But
/-i
Hence :
Minimum ?? = (-^y "... (50)
If TJ = 288, i.e. Oj = 15° C., this gives 0-634 and £2 max. = a9 =
II] AIR FLOW AND AERODYNAMIC FORCE 39
1019 ft. per sec. The final temperature is — 33*7° C., a drop of 48-7° C.
The examples worked out in Table IV further illustrate adiabatic flow. Two cases of common occurrence are studied : (a) a stream brought to rest (q^ = 0), (b) the velocity doubled (qn = 2^). In all cases the initial conditions assumed are : pl = 760 mm. mercury, Oj = 15° C. The values of Ai\A± are obtained from (38), by which, since q, = 2ql9 AJA^ = J(p!/p»).
TABLE IV EXAMPLES OF ADIABATIC FLOW
|
A |
ft (ft. per sec.) |
100 |
200 |
300 |
400 |
621 |
|
0 |
— ™ (per cent.) |
0-4 |
1-8 |
3-6 |
6-5 |
11-1 |
|
0 |
M°C.) . |
15-4 |
16-8 |
19-2 |
22-5 |
27-7 |
|
2?, |
Pi - Pi / ccnt % |
- 1-2 |
-4-7 |
- 10-5 |
- 18-2 |
-30 |
|
Pi v±~* """"' |
||||||
|
2?, |
<?a (° C.) . |
13-6 |
9-4 |
2-4 |
- 7-4 |
- 23-1 |
|
2ft |
^^t • • • |
0-50(5 |
0-525 |
0-56 |
0-61 |
0-71 |
The variation of temperature affects such questions as the trouble- some formation of ice on wings and the location of convective radiators. Otherwise it is ignored.
31. Variation of Pressure — Comparison with Incompressible Flow
Equations for pressures corresponding to those of the preceding article for densities are obtained in a similar way. They follow immediately, however, by use of the relations : pi/p* = pi/pt for isothermal and pi/p* = (pi/p2)Y for adiabatic flow. Thus Ber- noulli's equation for adiabatic flow, which alone will now be con- sidered, is found, with the help of (46), to be —
•r-}
Y-l This gives, corresponding to (47) —
= 0.
• (51)
(52)
Now an outstanding result of the investigation of density variation is that it is small provided velocities do not approach that of sound.
40 AERODYNAMICS [CH.
The condition p = constant is then a first approximation. Making this assumption gives at once, from (40) —
P + ip?a = constant . . . (53)
for incompressible flow along a particular streamline, provided always that tractions can be neglected. (41) becomes —
of which convenient non-dimensional forms are
. . . (55)
or
^=A'-*UrV-i|- • • • <56)
These alternative expressions of Bernoulli's theorem for an incom- pressible fluid are of great importance.
We now determine the error involved in applying (53) to a gas which is flowing adiabatically. Expanding (52) by the binomial theorem —
fr = 1 ~ 2 ~~^7~~ + sv «7/ ""••••
or
Since Y^>I/«IS == p! by (25), this reduces, with r written for q^qit to
A similar expression is readily obtained to compare with (56).
The above series is rapidly convergent, and the equation indicates that the error involved in applying (55) to a gas in adiabatic flow is small, provided that q? is small compared with a,2. Since a± is only approached by ql in particular cases, as for example at the tips of airscrews, it follows that air in motion may usually be treated as an incompressible fluid, such as water.
As an example, consider the case q2 = 2qt. The error involved in employing (55) instead of (57) is as follows :
ql (ft. per sec.) : 100 200 300 400
y, (ft. per sec.) : 200 400 600 800
error (per cent.) : 0-6 2-4 5-5 10
II] AIR FLOW AND AERODYNAMIC FORCE 41
32. The Pitot Tube and the Stagnation Point
Consider an L-shaped tube immersed and held stationary in a stream, one arm being parallel to it with open mouth direetly facing the oncoming air ; suppose the other end to be connected to a pres- sure gauge so that no air can flow through. There must exist an axial streamline about which fluid approaching the mouth divides in order to flow past. Air following that streamline will arrive at some point within the mouth of the tube, where the time-average of the velocity is zero ; we refrain from saying that the velocity will be zero, because some unsteadiness may possibly exist in the mouth of the tube ; but we can assert that the time-average of the square of the velocity will be negligible in ordinary circumstances, compared with the square of the velocity of the oncoming stream. Denote by p, q, p, a the pressure, velocity, density, and the velocity of sound at a point of the streamline far upstream, and use suffix 0 for the mouth of the tube. Ignoring the small unsteadiness that may arise, and also, for the moment, variation of density, the pressure p0 in the mouth of the tube is given from Article 31 by :
.... (58)
Such a tube is called a pitot tube (after its eighteenth-century inventor), and p0 the pitot head, or total head, for air flow whose changes of pressure due to variation of altitude can be neglected. Comparing with (53), we note that the constant of that equation is measured by a pitot tube. Variation of p0 from one streamline to another is readily determined by a pitot tube in experiment, its diameter being made very small where the space-variation of total head is rapid. For accurate work the tube must be oriented to lie parallel to the local streamlines of the flow.
Variations of pQ are small compared with^>, and it is convenient to deal with the quantity p0 — p, sometimes called the dynamic head. For incompressible flow, to which Bernoulli's equation applies, we have pQ — p = Jpy2. For the corresponding flow of a gas we find, in the same way as for (57) :
s^\ ft 1
. . (59)
Putting a = 1118 ft. per sec., the value appropriate to 15° C., gives, for example :
q (ft. per sec.) : 100 200 400
(po — p)/foq*: 1*002 1*008 1-032
A.D.— -2*
42 AERODYNAMICS [CH.
Thus the correction on (58) due to compressibility remains small for moderately large velocities.
In the above case of motion the dividing streamline is obviously straight, and collinear with the axis of the tube. Imagine a solid of revolution, of the shape of an airship envelope, for instance, having this same axis and situated with its nose at the mouth of the tube. The pressure in the tube remains unchanged, and indicates a pressure increase of %pqz occurring at the nose of the body. An airship nose requires special strengthening to withstand this pressure (cf. Fig. 7). If the body and tube are tilted with respect to the oncoming stream, i.e. are given an ' angle of incidence/ the pressure in the tube de- creases. But it must then be possible to find a new position for the tube, in the neighbourhood of the nose of the body, such that the pressure difference 'in the tube is again Jp#2, for there must still exist a dividing streamline, although now it may be curved (Fig. 13). Experiment confirms this con- clusion.
The point at which the dividing streamline meets the nose of an im- fftersed body is called the front
stagnation point. ^The increase of pressure there is known as the stagnation pressure. ^ The fact that a stagnation point must exist is of considerable help in constructing curves of pressure variation round the contour of a body from meagre experimental data.
33. Basis of Velocity Measurement
The undisturbed static pressure of a stream is measured as de- scribed in Article 27. A combination of a pitot tube and a static pressure tube, called a pitot-static tube, enables local velocity to be measured if p is known. For from (58)
FIG. 13. — FRONT STAGNATION POINT.
? =
(60)
The velocity thus obtained may be corrected, if need be, for com- pressibility by (59). A concentric form of pitot-static tube is shown in Fig. 14 ; other designs exist.
Other methods of measuring velocity are readily devised, although none is so convenient. The present method has a theoretical advan- tage in determining directly not q but pq2. It is usually the latter
AIR FLOW AND AERODYNAMIC FORCE
43
quantity that is required to be known with accuracy in Aerody- namics ; often a comparatively rough knowledge of q itself is sufficient.
Various problems in connection with the use of pitot-static tubes are described later ; but a certain limitation may be referred to here.
1=3
FIG. 14. — N.P.L. PITOT-STATIC PRESSURE TUBE.
Putting q = 10 ft. per sec. gives for standard conditions at sea-level pQ — p =0-119 Ib. per sq. ft. This pressure difference balances a head of water —0-023. in. only. Gauges (compare, for instance, Article 7) can be devised to measure such a pressure with high accuracy, but the required sensitivity makes simple forms unsuited to rapid laboratory use, owing to various small disturbing factors, which are usually negligible, beginning to become important. Varia- tion of temperature, vibration, and slight wear are instances. 1 or 2 per cent, of the above head is a convenient limit to sensitivity. It follows that the pitot-static tube becomes unsuitable for smaller velocities, and other means of measurement are then substituted. Of these, the change in electrical resistance of a fine heated wire due to forced convection in a stream has proved most convenient.
A pitot-static tube, usually of divided type, is employed on air- craft to indicate speed. Wherever located within practical limita- tions, it is subject to disturbance from near parts of the craft to an extent depending on speed. Especially if fitted to an aeroplane, the tube can only be tangential to the local stream at one speed. Errors due to an inclination of 10° amount to 2-3 per cent., depending upon the type of tube. A mean alignment is adopted, but for the several reasons stated calibration in place is necessary for accurate readings.
The tube is connected with a pressure gauge of aneroid barometer type, deflection of a diaphragm of thin corrugated metal moving a
44 AERODYNAMICS [CH.
needle over a scale calibrated in miles per hour. The reading is termed indicated air speed (A.S.I.), and gives the true speed of the craft relative to the air at low altitude only. If true speed is required at considerable altitudes, readings must be increased in the ratio VT/cr, where cr is the relative air density.
Special forms of pressure tube also exist for aircraft, designed to permit use of a more robust gauge. Since increase of pressure cannot exceed Jp#2, except on account of compressibility, the static tube is replaced by a device giving less than static pressure. This some- times consists of a single or double venturi tube. Particulars of Venturis for this and other purposes are given in the paper cited * (as exposed on aircraft they are not constrained to 'run full'). A formula for the pitot pressure at speeds exceeding the velocity of sound is given later.
SUBDIVISION OF FLOW PAST BODIES
34. Taking advantage of the outstanding result of Article 30, it will now be assumed, except where stated otherwise, that the fluid is sensibly homogeneous and flows incompressibly. From Article 3 1 , maximum velocities must not ap- proach that of sound in air. A very useful ex- pression of the assump- tion is obtained as follows.
Consider part of the field of a two-dimensional
pv
o
x
FIG. 15.
flow enclosed within any
small rectangle ABCD
(Fig. 15), of sides $x, Xy, u,
v being the components parallel to Ox, Oy of the resultant velocity.
The rate at which fluid mass tends to be exhausted from the rectangle
owing to difference in velocities and densities at BC and DA is
^ } Sjy — p«8y = ~~ 8#8y . Comparing similarly the mass- es / ox
flow across the sides AB and CD, the rate at which matter is ex-
o
hausted from the rectangle on this account is -~- S#8y. Now density
* Piercy and Mines, A.R.C.R. & M. 664, 1919.
II] AIR FLOW AND AERODYNAMIC FORCE 45
is assumed to remain constant. Hence :
du dv
This expression is known as the equation of continuity for an incom- pressible fluid.
35. When a wind divides to flow past an obstacle, such as an air- ship, held stationary within it, the inertia of the air tends to localise to the vicinity of the body the large deflections that must occur in the stream, so that laterally distant parts are little affected. Imagine a hoop of diameter several times as great as the maximum trans- verse dimension of the body to be held across the stream, enclosing the body. The volume of air flowing through the hoop per sec. is little diminished by the presence of the body, the air flowing faster to make up for the obstructed area. As the diameter of the hoop is decreased, this statement becomes less true, but at first only slowly. In other words, the increase of speed increases as the body is ap- proached. If there were no friction at the surface of the body, the speed would reach a maximum there. But (Article 22) the air is stationary on the surface, and is retarded for some distance into the fluid. We are concerned with the manner in which such retardation consorts with the more distant, though still close, increases of speed, which are often large.
36. Experimental Streamlines
It is always possible to plot the streamlines for a steady motion from experimental knowledge of the velocity distribution. Fig. 16 has been prepared from actual measurements of the approximately two-dimensional motion in the median plane of a scale model of an aeroplane wing of the section shown. The model wing, or aerofoil, was immersed in a stream whose velocity U and pressure pQ were initially uniform. Explorations of the magnitude and direction of the disturbed velocity q were made along several normals to the wing surface ; values of q sin a/C7 are plotted for the two shown, viz. SlNl and 52]V2, distance from the surface along either normal being denoted by n and the angle between q and the normal by a.
The flow across any part of a normal is given by the value of the integral
\q sin a dn
over that part. Choose a point A± on SlNl through which it is desired that a streamline shall pass. Evaluate graphically
46 AERODYNAMICS [CH.
ql sin a dn = k, say. For n small, sin a = 1-0. Now find a point A* on S2NZ such that
I-
O2 O3 O4 0-5 0-6
FIG. 16.
i.e. find the line n = A* (Fig. 16), such that the area OA^A* equals the area OA ^A^ Similarly, determine points A, A . . . along other normals. Now there is no flow across the aerofoil contour. There- fore there is no flow across the curve AAtA2A . . , Hence this curve is a streamline.
Successive streamlines follow by changing k to k', k* . . . It is
II] AIR FLOW AND AERODYNAMIC FORCE 47
convenient to make k = k' — Tc = k" — fc' = . . ., for then, if the intervals are sufficiently small, the velocity is inversely proportional to the distance apart of successive streamlines. A second streamline is, of course, most conveniently constructed from the first, a third from the second, and so on.
37. The Stream Function
It would be possible to fit to an experimental streamline a formula / (x> y) = constant. The fit would not be so close, however, nor the original measurements so accurate as to ensure obtaining another stream- line by equating the same function of x and y to another constant. In a motion that is known analytically, on the other hand, these difficulties
B
B'
DC
FIG. 17,
disappear. We then have * a function of x and yt fy (#> y)> which, on equat- ing to any constant, gives corresponding values of x and y for points lying on one of the streamlines of the motion, fy is called the stream function of the motion.
Consider a steady two-dimensional motion in the #y-plane. Let A and B be two points, not on the same streamline ; join them by any curve (Fig. 17), and let q make an angle a with an element 8s of the curve. Define the flow across this curve by fy3 — t];A, i.e.
fB . YB — YA ==\ ? sin a as.
J A
This value is unique, for the flow across AB is independent of the shape of the curve, being the same as that across any other curve, such as ACB, joining the points, since otherwise fluid would be com- pressed within, or exhausted from, the area ACBA.
With A fixed let B move in such a manner that the above flow remains constant. Then B traces out a streamline, because there is no flow across its path. If the value of fy (x, y) at A = k, for all points on the streamline BB',
It follows that the equation to all streamlines is
ty = constant .... (62)
48 AERODYNAMICS {CH.
the constant changing from one to another. A definite value is assigned to the constant of a particular streamline by agreeing
to denote some chosen streamline by fy (x, y) = 0. A question of sign is involved ; the increment of <|j is taken as positive if the flow is in a clockwise direction about the origin, but sign is determined generally by (63) below.
38. Let A and D be adjacent points on two streamlines : = k and
O
FIG. 18.
<Jj = fc + Jty. The co-or- dinates of A are x and y, those of D x + %%> y + Sy. From Fig. 18, the flow across AD = that across ED less that across AE. Hence if u, v are the components parallel to Ox, Oy, respectively, of the velocity q,
Now 8^ is the total variation of a function of the two independent variables x and y. It is assumed that the partial derivatives dfy/dx and 3^/3^ are also continuous functions of x and y. It is shown in text-books on Calculus that then
Hence :
-s ty
Sx
(63)
As an example, suppose <]* = Uy, where U is a constant. From (63) u = [7, v = 0, and the flow evidently consists of uniform motion at constant velocity U in the direction Ox. Putting ty/U = 0, 1, 2 . . . gives a series of streamlines all parallel to Ox and spaced equally apart. Again, consider the flow fy = Qy8, where C is a constant. Putting <J*/C = 0, 1, 2 . . . again gives streamlines parallel to 0#, but at a decreasing distance apart (Fig. 19). From (63) u = 2Cy, v = 0, and we recognise the flow as including that of
AIR FLOW AND AERODYNAMIC FORCE
49
y/c*6
y/os
Wo*
y/c=3
n]
Article 24, where 2C = U/h, and there are certain restric- tions on the area occupied by the flow.
39. Circulation and Vorticity
So far we have dealt with the line integral of the normal velocity component across a curve drawn in the field of flow. The line integral of the tangential velocity component once round any closed curve is called the circulation round
that circuit and is denoted by K. If Ss is an element of length of the closed curve of the circuit, q the velocity, and a the angle which q makes with Ss,
FIG. 19.— STREAMLINES FOR UNIFORM SHEARING.
K = Jc q cos a ds .
(64)
There is again a question of sign, and this is taken as positive if K has a counter-clockwise sense.
Let us calculate the circulation $K round the small rectangle ABCD (Fig. 20), of sides 8x, Sy. The sides AD and CB together contribute to counter-clockwise circulation an amount
(du \ Su
u + Y 8y\ 8% = - — S#Sy. Similarly DC and BA together
^i contribute — Sx Sy.
Hence :
B
8x8y dx
!** 9y
VA
Sx
ts
fy
^vtl^Sx
dX
TN
X
The finite limit to which the left-hand side tends as the area decreases is called the vorticity at the point and has the symbol £. Thus :
£=4!-!:? - (66)
FIG. 20.
In words, the vorticity of an element is the ratio to its area of the
50 AERODYNAMICS [CH.
circulation round its contour. Choose the element as circular, of radius r, and so small that its angular velocity G> can be considered constant. Then K is due to G> alone. Writing S for area,
SK = 2-nr . <or ; dK/dS = 2to or
Thus the vorticity of an element is twice its angular velocity.
In the first example of Article 38, where u = £7, a constant, and v = 0, we now have from (65) £ == 0 everywhere ; the elements of fluid are devoid of spin. For the second example, u = 2Cy, v = 0 and (65) gives £ = — 2C, a constant, or there is a uniform distribu- tion of vorticity. Applying the latter result to the motion of Article 24, £ = — C7/A. Imagine the moving plate in Article 24 to be started from rest. Initially u = v = 0 and the fluid is devoid of vorticity, but after a sufficient time a uniform distribution of vorti- city is generated, arising from the boundary condition of zero slip and the action of viscosity. We are thus able to trace the generation of the vorticity to viscosity. If the pressure had acted equally in all directions, it would have exerted no couple on any element of fluid which, being originally devoid of vorticity, would have remained so.
40. Extension of Bernoulli's Equation
We are now in a position to prove a theorem of practical impor-
tance in connection with flow that is sufficiently distant from bodies
and other boundaries. A distribu- tion of vorticity is assumed to exist, but tangential components of stress are neglected.
Consider the fluid element ABCD (Fig. 21), bounded by two adjacent streamlines and the normals thereto. Let the radius of curvature, assumed large, of the streamline AB be R. Let s denote length measured along AB, DC, and n denote length measured along either of the nor- mals towards the centre of curva- ture. Let q be the velocity along
AB. Tangential components being neglected, the pressures act
normally to the faces of the element, The element exerts a centrifugal force p$s8nq*/R which, the flow
RADR
II] AIR FLOW AND AERODYNAMIC FORCE 61
being steady, is balanced by a force due to the difference of the pressures on the faces AB, CD, i.e. by the force — 8s8n(Sp/Sn) (Article 28). Hence :
Now calculate the circulation 8K round the element. There is no flow along the normals ; hence :
. . (i)
From the figure —
CZ> __ CD _ __ 8n
AB ~~&T ~~l ~~~R' Substituting in (i)
-sT .-
dn R dn
The last term is evidently negligible compared with the others. Hence, finally :
- q dq
Multiply both sides of (67) by — pj and substitute from (66) for IjR. (67) becomes
Now p + Jpy* is the pitot head (Article 32). Thus, across the streamlines the pitot head has a gradient proportional to the product of the velocity and the vorticity, provided tractions can be neglected. If, on traversing a pitot tube across a field of flow, the pitot head remains constant, then the flow is devoid of vorticity so far as it is explored.
41. Irrotational Flow
An irrotational motion is one in which it is everywhere true that % = 0. Where velocity gradients exist, this condition usually
52
AERODYNAMICS
[CH.
appears as an ideal which is not exactly attained by a real fluid, but many motions of great Aerodynamic interest approximate closely to the irrotational state. These are discussed theoretically in later chapters. Meanwhile, we note that the theorem of the preceding article leads, as described, to a convenient method of investigating experimentally whether a given flow, or what part of it, is approxi- mately irrotational.
42. Subdivision of Flow Past Bodies
It will now be shown, on experimental grounds, as will be proved theoretically in a later chapter, that Aerodynamic types of flow can
0-0
O
O Ol O2 O-3
l-O
05
be separated into two parts : an outer irrotational motion and an inner flow characterised by the presence of vorticity. For this pur- pose a particular, but typical, case will be described in some detail. The flow selected is that above the aerofoil of Article 36. This aero- foil was small, having a chord c (length of section) of l£ in. It was set at an incidence (angle made with the oncoming stream by the common tangent to its lower surface) of 9-5° in an initially uniform air-stream of velocity U = 41 ft. per sec. and pressure p0. The undisturbed stream was verified to be sensibly irrotational by track- ing across it a pitof 'tube, the pitot head being found to be constant.
II] AIR FLOW AND AERODYNAMIC FORCE 53
The same two normals, S1N1 and S2N2) distant c/3 and 2c/3, respec- tively, from the leading edge, were selected for study as those for which the variation of velocity (q) has been given in Article 36. What will now be described is the variation that was found along them of pitot head and static pressure (p). In Fig. 22 the first of these is given in the form :
; to
ht then, being the loss of pitot head caused by the model ; the second is conveniently expressed as (p — p0)/pU*, both quantities being non-dimensional .
It is seen from the figure that the aerofoil causes negligible change of pitot head beyond n = 0-04c for the upstream and n = 0-2c for the downstream normal. Beyond these limits the stream is con- cluded to be irrotational, approximately ; just within them the velocity gradients are not large and the tractions may be expected to be small, so that, from (68), we infer vorticity to be present. By traversing a fine pitot tube along a number of other normals, or lines across the stream, a number of similarly critical points for pitot head can be found. A line drawn through all such points forms a loop which wraps itself very closely round the nose of the model (where a special form of pitot tube is necessary for detection), widens as the trailing edge is approached, and finally marks out a wake behind the aerofoil. Fig. 23 shows the wake located in this way behind another aerofoil set at smaller incidence. The complete loop may be called, for short, the pit< boundary * and is one way of marking out an internal limitation to irrotational flow.
The pressure decrease pQ — p builds up along the normals as the aerofoil is approached to maxima at the pitot boundary. Actually there was no reason to measure the pressure and velocity separately in the outer irrotational region except as a check, for here the one can be calculated from the other by Bernoulli's theorem. The maximum pressure changes generated at the pitot boundary are transmitted without further variation along normals to the aerofoil surface. This important point is clearly seen from the pressure
* For further illustrations see Piercy, Jour, Roy. Afro, Soc., October 1923.
54 AERODYNAMICS [CH.
curve for the more downstream normal, the three readings nearest the surface being
n\c = 0-047 0-127 0-173
£^~ = 0-310 0-308 0-312
A pressure change of 0-47 pC72 is transmitted to the surface along the more upstream normal. Within the pitot boundary adjacent to the aerofoil the velocity (Fig. 16) and the pitot head (Fig. 22) fall away rapidly. The first, as already seen, vanishes on the surface ; the second decreases from p0 + ipt^2 to p', the value of the static pres- sure on the pitot boundary opposite the position round the contour of the aerofoil considered.
The chain-line curve (Fig. 22) gives a wider view of the manner in which the static pressure drop is built up.
An aeroplane was fitted with wings of the shape of the aerofoil, and some measurements were made in flight. These showed the velocities and pressures, non-dimensionally expressed, to be different from those observed with the model but not greatly so. The pitot boundary was found to be much closer to the full-scale wing surface than it was to the model surface when expressed as a fraction of the chord.
43. The Boundary Layer
From the many experiments which have been made on lines simi- lar to the foregoing, we draw the following preliminary conclusions regarding motions of Aerodynamic interest past bodies :
1. There exists an outer irrotational flow.
2. This is separated from the body by a sheath of fluid infected with vorticity arising from the boundary condition of no slip and the action of viscosity. This sheath or film of fluid increases in thickness from the nose to the tail of the body, but is nowhere thick and is called the Boundary Layer. It merges into the wake.
3. Changes in static pressure are built up in the outer flow, related to the velocity changes there by Bernoulli's equation, and are trans- mitted to the surface of the body through the boundary layer.
AERODYNAMIC FORCE AND SCALE
44. The Aerodynamic force on a body is that resultant force on it which is due solely to motion relative to the fluid in which it is
II] AIR FLOW AND AERODYNAMIC FORCE 55
immersed. Thus forces acting on the body due to gravity, buoyancy, etc., are excluded. Aerodynamic force arises on the body in two ways : (a) from the static pressures over the surface, sometimes called the normal pressures ; (b) from a distribution of skin friction over the surface.
Consider, for example, an aeroplane wing of uniform section. Let 8s denote the area, per unit of span, of an element of the contour of
IVeesui increase
Pressure decrease
FIG. 24.-
-EXPERIMENTAL PRESSURE DISTRIBUTION ROUND SECTION OF AEROFOIL, SHOWING INTEGRATION OF PRESSURE DRAG AND LIFT.
the section at S (Fig. 24), and 0 the angle which the normal SN at 8s makes with SL, the perpendicular to the direction of the relative undisturbed wind. For convenience, subtract from the pressure acting on the surface the static pressure of the oncoming stream, and let p be the normal component of the remainder and F the tangential component. The variation of p is shown by the dotted line in the figure. The force on the element is compounded of p.8s, outwardly directed along SN and F.Ss, perpendicular thereto. For simplicity we shall assume the flow to be two-dimensional, so that F has no component parallel to the span, p and F vary from point to point over the wing, leading to a variation of force from one element to another in both magnitude and direction. To obtain the resultant force we require to effect a summation of the forces on all elements.
56 AERODYNAMICS [CH.
Evaluation is usefully simplified in the following way : compon- ents of the resultant force are determined parallel to SL drawn perpendicular to the relative wind and to the aerofoil span, and SD in the direction of the relative wind. The first component is the lift, the second the drag. It will especially be noted that the Aerodynamic lift of a wing, unlike the static lift of a gas-bag, is not constrained to be vertical, nor even does its direction necessarily lie in a vertical plane ; it is perpendicular to the span of the wing and also to the relative wind, and is taken as positive if it is directed upward when the wing is the right way up.
Denoting lift by L and drag by Z), we have for 8s
8L = (p cos 0 + F sin 0) 8s SD = (- p sin 6 + F cos 6) Ss
Now Ss sin 0 and Ss cos 0 are the projections of Ss perpendicular and parallel to SL. Hence, if the aerofoil section is drawn accurately to scale and all points on the contour at which p is known are projected upon a line perpendicular to SL (i.e. upon a line parallel to the un- disturbed relative wind) and p is set up normally to this line, the area enclosed by the curve obtained by joining the points, completed so as to represent the whole contour, is proportional to that part of the lift which is due to p. If similar projections are made along a line parallel to SL, and p is set up normally to this line and a closed curve is obtained by joining all points and completing so as to include all positions round the contour, the net area enclosed by the curve is pioportional to the contribution to drag by p. As regards drag, the simplest curve found for an aerofoil is of figure-of-eight form, one loop of which is positive and the other negative ; the net area is con- veniently obtained by tracing the point of a planimeter round the diagram in a direction corresponding to one complete circuit of the aerofoil contour. The contributions of skin friction to lift and drag are similarly determined, but the directions of projection are inter- changed. The sense of F depends upon that of the velocity gradient with which it is associated. However, the correct sense is easily decided by inspection.
45. An example of the variation of p round the median section of an aerofoil at a certain angle of incidence, experimentally determined at a certain speed, is given in Fig. 24. Curves are also shown obtained by projection perpendicular to and in the direction of the oncoming stream, the areas under which are proportional to the lift and drag per foot run of the span at the median section, the area ABC giving negative contributions to drag. Apart from scientific
II] AIR FLOW AND AERODYNAMIC FORCE 57
interest, investigations of distribution of force are of technical im- portance, especially in the case of aerofoils, providing data essential to the design of sufficiently strong structural members of minimum weight for the corresponding aeroplane wing. Such analysis is usually required at several angles of incidence. Since the pressure will most conveniently be found at the same points round the contour for all incidences, labour is saved by projecting along and normal to the chord of the wing, resolving subsequently in the wind direction and perpendicular thereto. Graphical processes of integration con- venient for bodies other than wings will be left for the reader to devise.
46. Some limiting cases may be mentioned. When a fluid flows through a straight pipe or past a thin flat plate at zero incidence — i.e. parallel to the oncoming stream — the drag must be wholly fric- tional. Such drag is small with air as fluid. At the other extreme, the drag of a thin flat plate set normal to the undisturbed stream must arise wholly from unequal distribution of pressure. This drag is comparatively large, but is less than that of a cup-shaped body with the concavity facing the direction of flow, as instanced by a parachute. Referring to aerofoils, the contribution of skin friction to lift is negligible. The area enclosed by the negative drag loop of the projected pressure curve (e.g. ABC of Fig. 24) may approach that of the positive loop, when the contribution of the pressures to drag will be small. For 'this condition to be realised, the flow must envelop the back of the body closely, i.e. without ' breaking away ' from the profile. Negative drag loops are absent from the normal plate and very small for the circular cylinder.
A quantity of significance descriptive of an aerofoil is the ratio of the lift L to the drag D, i.e. L/D. Since D = L ~- L/D, for a given lift the drag is smaller the greater L/D. Considering a flat plate at any incidence a and neglecting skin friction, and writing P for the total force due to variations of pressure over the two surfaces, we have L = P cos a, D = P sin a. However P varies with a, L/D = cot a. Values given by this formula must always be excessive, greatly so at small incidences when the neglected skin friction becomes relatively important. Nevertheless, at flying incidences the L/D of a wing, skin friction included, greatly exceeds cot a. A wing having also essentially more lifting power than a flat plate, this comparison is often 'given as illustrating the superiority of the aerofoil over the flat plate for aeroplane wings. The advantage is seen to arise from the pressure distribution round the forward part of the upper surface of the aerofoil, providing positive lift and nega- tive drag.
58 AERODYNAMICS [CH.
As a matter of experiment it is found that the pressure drag of a carefully shaped airship envelope almost vanishes, although the pressure varies considerably from nose to tail, and the drag is almost wholly frictional ; it may amount to less than 2 per cent, of the drag of a normal disc of diameter equal to the maximum diameter of the envelope. The example illustrates the great economy in drag which can be achieved by careful shaping, a process known as fairing or streamlining. So exacting is this process that it pays to shape the contour of a wing, strut section, engine egg, or other exposed part of an aircraft by some suitable formula, instead of using french curves, so as to avoid sharp changes of curvature which, although scarcely apparent to the eye, may increase drag considerably.
47. Rayleigh's Formula
Further investigation of how Aerodynamic force depends upon shape is left to subsequent chapters. The knowledge required for practical use will result partly from theory and partly from experi- ment. For both lines of enquiry we need to establish a proper scale in terms of which Aerodynamic force may be measured. For this purpose we keep the geometrical shape of the body and its attitude to the wind constant, but allow its size to vary ; in other words, we consider a series of bodies of different sizes made from a single drawing, immersed, one after another, at the same incidence in a uniform stream of air. Geometrical similarity must include rough- ness of surface, unless effects of variation are known in a given case to be negligible ; a caution is also necessary against tolerating any lack of uniformity in the oncoming stream. But the velocity of the stream may vary and also the physical condition of the air ; in fact, the bodies may be supposed immersed in uniform streams of different fluids, liquid or gaseous. But it is assumed for simplicity, and as representing a common condition in Aerodynamics, that maximum velocities attained are small compared with the velocity of sound in the fluid concerned, so that compressibility may be neglected.
Preceding articles have shown that the Aerodynamic force A arises from pressure variation and skin friction. The pressure will depend upon the density p and the undisturbed velocity U. The skin friction has been seen to depend upon U and the viscosity jju Comparing different fluids, or air in different states, the general effect of viscosity depends on the ratio of the internal tractions to the inertia, which is proportional to p. Hence it is convenient to sub- stitute for IJL the quantity
v-jji/p . . . . (69)
II] AIR FLOW AND AERODYNAMIC FORCE 59
called the kinematic coefficient of viscosity, whose dimensions are (cf. Article 23) M/LT -r M/L* = L*/T. The Aerodynamic force, since it results from the surface integration of pressure and skin friction, will also depend upon the size of the body, which is specified by any agreed representative length /, because the geometrical shape is constant.
It is concluded, then, that :
A depends upon p, U, I, v . . . (70)
and on nothing else. This conclusion, which is essential to the investigation, can be arrived at in other ways ; e.g. by appeal to simple experiments. Thus, if a bluff shape, such as a normal plate, is moved by hand through air, drag can be felt to depend upon size and velocity. If it is then moved through water, a great increase occurs mainly as a result of increased density. Moving the plate finally through thick oil instead of water shows that drag also depends upon viscosity, for density need scarcely have changed. The importance of the more careful consideration that we have given to the question lies in the assurance that no important factor has been omitted.
It is desired to obtain a general formula for A , connecting it with p, U, I, v. This may contain a number of terms, any one of which can be written in the form :
?pUqlrv5 (71)
Now A, being a force, has the dimensions of mass x acceleration, i.e. ML/T2. The principle of homogeneity of dimensions asserts *-!;;:[ all terms in the formula for A must have the same dimensions. Writing (71) in dimensional form :
ML (M\*(L\< /L-V
r» ViV \r/ \T/
For the dimensions of the term to be ML/T*, it is required that on account of the M's, p = 1, on account of the L's, — 3p + q + r + 2s = l, on account of the T's, — q — s = — 2, giving
/> = !
q =y = 2 — s.
Hence the formula for A is :
A - 2p[/2~'/2~*vs
60 AERODYNAMICS [CH.
or A =P W. /, . • • (72)
where f(Ul/v) means some particular function of the one variable OT/v.
This important relationship is the simplest case of Rayleigh's formula. The investigation equally leads to
\ V
.... (72a)
an alternative form of particular use where changte of fluid is involved. It will be noted that Aerodynamic force cannot vary with the area of the body or the square of the velocity exactly unless it is indepen- dent of viscosity, which is absurd.
48. Reynolds Number — Simple Similar Motions
The quantity £7//v is called the Reynolds number after Osborne Reynolds, who first discovered its significance, and is written R. Writing (72) as
.... PD
we have, on the left-hand side, a coefficient of Aerodynamic force .vnose value for any shape of body and value of R can be found if required by actual measurement.
Still keeping shape constant, let us investigate what similarity exists in the flow of different fluids at different velocities past bodies of different sizes, subject to the restriction that R remains constant.
Considering any particular position in the field of flow past the particular shape, the method of Article 47 readily gives for any velocity component there, for instance u,
Hence, from consideration of velocity components at right angles at geometrically similarly situated points, called corresponding points, one in each of a series of fields of flow past bodies of the same shape (and attitude) at the same Reynolds number, the resultant velocity there is the same in direction. Since this is true of all sets of corres- ponding points, the streamlines present the same picture, though to different geometric scales. The magnitude of the velocity at corres-
Il] AIR FLOW AND AERODYNAMIC FORCE 61
ponding points oc U ; and the pressure oc pt/f, as may be shown directly.
It follows that at corresponding points on the contours of the bodies the pressure oc pt/2 and the skin friction oc y.U/1 oc pvt///, and that part of A resulting from pressure variation oc pt/2/2, while that part due to skin friction oc pvl// oc p£/2/2, since v oc Ul, because R is constant. Hence A oc pC72/2, or the left-hand side of (73) is constant.
Example : if also the fluid is constant, show that A is constant.
The foregoing assumes the motions to be steady. Now let them have frequencies ~ (dimensions : 1/T). With frequency assumed to depend only on p, U, I, v, the method of Article 47 gives :
~ = -
While R remains constant, ~ oc U/l in periodic motions. If also the fluid be given so that v, and therefore Ul, remain constant, ~ oc U* oc l//a. The streamlines pass through the same sequence of transient configurations but at different rates ; if cinema films were taken of the motions, any picture in one film would be found in the others, but it would recur at a different, though related, fre- quency. Similarity of streamlines, etc., as described above, then occurs at the same phase. The result : A oc p[72/2 is now true of the Aerodynamic force at any phase and also of the mean value, with which we are usually concerned.
The motions considered in this article provide an example of what are termed dynamically similar motions. Constancy of the left-hand side of (73) is also found by experiment for R constant when the bodies produce flow that varies rapidly in an irregular manner.
49. Aerodynamic Scale
When the Reynolds number changes, there is no reason to expect the coefficient of Aerodynamic force to remain constant, and it is found to vary, sometimes very little through a limited range of R, sometimes sharply, depending upon the shape of the body (or its attitude) and the mean value of R, Now, if by a series of experi- ments or calculations we obtain a number of values of A for a given shape, work out the coefficients and plot these against R, it is clear from Article 48 that all coefficients will lie on a single curve. This curve is the graphical representation of f(R) through the range explored.
Fig. 25 gives as an example the variation of (drag -f- p£72/2) with R for long circular cylinders set across the stream. In order to fix
62 AERODYNAMICS [CH.
the numerical scales, it has been chosen quite arbitrarily to use the diameter of the cylinder in specif yingR and the square of the diameter for /», but the drag then relates to a length of the cylinder equal to its diameter. The full line results from a great number of observations. These are not shown, but they fit the curve closely, though a cluster of points round a particular Reynolds number may include great
|
l'\~f O-8 O6 ^fp O-Zt |
\ |
/ / / / / / |
^£O O2 O15 ~l U Ol O05 |
|||
|
\ |
x " / "^/ i\ |
~~^^ |
X- -X m ' |
\~t u |
||
|
/ \ |
^x |
\ |
||||
|
*T O2 r\ |
1 |
|||||
1O \OZ 103
_
R.
1O5 1O6
FIG. 25. — DRAG OF LONG CIRCULAR CYLINDERS SET ACROSS STREAM AND FREQUENCY OF FLOW IN WAKE (/ «== DIAMETER).
variation in, for instance, diameter. The rapid rise of drag at R = 10e flattens again at 1-3 x 10« with a value of about 0-3 for the co- efficient. The broken line gives the variation of frequency, the flow eddying for R > 100.
Similar success has been obtained experimentally in many other cases, and we conclude that the theory of Article 47 can be accepted with confidence. When observations at constant Reynolds number disagree with one another, the cause is to be sought in the particular circumstances of the experiments ; if geometrical similarity is truly realised and velocities are demonstrably too small for appreciable
II] AIR FLOW AND AERODYNAMIC FORCE 63
compressions and expansions, the cause may be traced to consider- able variation of unsteadiness in the oncoming streams.
Finally, it becomes evident that, with moderate velocities, the Reynolds number provides a proper scale for Aerodynamic motions. Circumstances in which this scale is not suitable are described in the following articles.
The principle of dimensional homogeneity is often employed to express in a rational formula the results of a series of experiments on a given shape. The process usually depends upon discovery of a constant index for one of the variables, although this restriction is not necessary. It should carefully be noted that such formulae apply only through the range for which they have been shown to hold ; large errors often result from extrapolation. Thus, such formulae amount to no more than a convenient mental note of the results from which they are derived ; they constitute merely an approximation to part of the f(R) curve for the shape concerned.
The outstanding practical significance of general formulae such as (72) is to establish the basis on which single experiments on scale models of aircraft or their component parts should, if possible, be carried out. Provided the model is tested at the same Aerodynamic scale, experimental measurements are accurately related to corre- sponding quantities at the full scale ; otherwise corrections ire necessary. The proviso can by no means always be satisfied even when the gauge of Aerodynamic scale is simply the Reynolds number. The more complicated formulae completing this chapter will show that the Reynolds number alone is often insufficient ; the position then becomes more difficult and experiment requires planning with judicious care.
49 A. Rayleigh's Formula — High Speeds
If the compressibility of the air cannot be neglected, its modulus of bulk elasticity E must be admitted and the typical term in the new formula for Aerodynamic force becomes
ftVWE* ... (i)
The dimensions of E are M/LT*, and the method of Article 47 gives (M) I = p + t
(L) l = —3p + q+r + 2s — t (T) -2 = - q - s - 2t, i.e.,
p = 1-*
q = 2 — s — 2t
r = 2 — 5,
64 AERODYNAMICS [Cfi.
giving the result —
By Article 20, E — pa2, where a is the velocity of sound, i.e. pt/2/£ = (U/a)*. The ratio t//0 is called the Mach number and de- noted by M, and the formula becomes finally
A = 9UW.f(R, M) ...... (73A)
There being five unknowns and only three dimensional equations to relate them, the new function has two arguments ; A depends upon both and, theoretically, this dependence cannot be separated. From calculation of the stagnation pressure, an inference has been made in a preliminary way that compressibility can be ignored for speeds little greater than 250 m.p.h. at low altitude, i.e. for values of M less than J, and considerably greater values produce in some cases only negligible effects on A . Formerly, the airscrew provided almost the only occasion calling for a formula of the type (73A), speeds towards the tips of their blades being so high as to make M approach unity. In modern Aeronautics, however, the importance of the formula is much wider. Considering, for example, a strato- spheric aeroplane flying at the moderate indicated air speed of 200 m.p.h. at an altitude of 40,000 ft., where the relative density of the air is J, U = (22/15) . (200/Vi) = 58? ft- P^r sec., whilst a is reduced by the low temperature to the value 975 ft. per sec., giving M > 0*6 for every part of the aeroplane. Now in full-scale flight at little more than this Mach number the effect of varying M may be much more important than that of varying R. Thus, while (72) can still be relied upon in a great variety of practical circum- stances, the occasions on which it is superseded by (73A) are multiplying.
For two motions 1 and 2 to be dynamically similar, both R and M must be the same, leading to
For a dynamically similar experiment on a model of an aircraft it will be plain from the next chapter that the power required to produce the artificial wind is economised chiefly by reducing the speed. But this would involve employing very cold air. In
II] AIRFLOW AND AERODYNAMIC FORCE 66
these circumstances and in view of the labour involved, the task of constructing a data sheet such as Fig. 25, which would now embrace a series of curves for a body of given shape, is abandoned, and experiments on the effect of high Mach numbers are usually carried out with no more than the precaution of avoiding very small Reynolds numbers.
498. Some Other Conditions for Similitude occurring in Aerodynamics
(1) When a seaplane float or flying-boat hull moves partly im- mersed in water, waves formed cause variation of pressure over horizontal planes due to the weight of the heaped liquid. Thus gravity comes into the problem of similarity. Approximate treat- ment ignores air drag of parts projecting above the surface and also surface tension. Then p, C7, /, v refer to the water only and, with g added, we write any term in the formula for drag as
Dimensional theory at once gives :
(M) \=p
(L) i==-3p + q + r
(T) — 2 = — q — s — 2t whence p = 1
q = 2 — s — 2* r = 2 — s + t
and the term becomes
leading to the following formula for drag :
. (73B)
The drag is made up of two parts : (a) a part akin to Aerodynamic force but modified by (b) wave-making resistance, which again is modified by (a). U*/gl is called the Froude number, F.
For dynamical similarity both arguments of the function must be kept constant. For change of size the second argument gives Ucc<\/l since g is practically constant, and then by the first v oc ^/l*, i.e. the fluid must be changed when Doc pvfoc p/8. A change from water is not
A.D. — 3
66 AERODYNAMICS [CH.
convenient, however, and it has been found sufficient, as originally suggested by Froude, to assume the two kinds of resistance to be independent of one another, i.e. to write (73B) as
D = Pl7«/« /,(*) + /, 1 . . . * (73C)
This is convenient in regard to the wave-making resistance, because a model of scale e can be towed in a ship tank at the low correspond- ing speed : U^/e, where U is the full-scale speed.
One ' ship ' tank (U.S.A.) is 1980 ft. long, 24 ft. wide, and 12 ft. deep, with a maximum towing speed of 60 m.p.h. Another tank (R.A.E.) has rather more than one-third these dimensions, with a maximum speed of 27 m.p.h. In the latter a ^th scale model of a large hull is feasible, when its maximum model speed would corres- pond to 81 m.p.h. full scale.
The wave-making resistance is assessed by subtracting from the total drag measured an estimated Reynolds resistance. The wave- making resistance is simply related to that under full-scale condi- tions, to which the Reynolds resistance is added after correction for rhange of scale.
(2) Froude's law of corresponding speeds reappears, unconnected with wave-making, in wind-tunnel tests on unsteady motions of air- craft. The subject is discussed under Stability and Control, but a simple example will shortly be provided by the ' spinning tunnel/
49C. The airscrew is a twisted aerofoil, each section of the blades moving along a helical path defined by the radius, the revolutions per second n, and the forward speed U. To secure geometrical similarity in experiments on airscrews of different sizes, each made from the same drawing, it is therefore necessary that U/nl be constant. Thus a third argument must be added to (73A). The diameter D is chosen for convenience to specify /, and the non- dimensional parameter U\nD is given the symbol /. It is also con- venient to replace U as far as possible by n. Now n1!)4 has the same dimensions as E71/1, and the formula becomes —
A==9n*D'.f(RtM,J). . . (73D)
Derivation from first principles on the assumption that A depends on p, £7, /, v, E and n presents no difficulty. But it will now have become apparent that formulae even more complicated than (73D) can be constructed from dimensional considerations almost by inspection.
II] AIRFLOW AND AERODYNAMIC FORCE 67
The following extension of Table III relates to the standard atmosphere and gives approximate values of various quantities which are constantly required in calculations of Aerodynamic scale.
TABLE III A
|
Altitude (ft.) -» 1000 |
v* |
1 v* |
V "o |
I/* (ft'/sec.)"1 + 1000 |
I/a (ft/sec.)-1 X 1000 |
|
0 |
1-00 |
1-00 |
1-00 |
6-4 |
0-89 |
|
10 |
0-86 |
M6 |
1-29 |
6-0 |
0-93 |
|
20 |
0-73 |
1-37 |
1-68 |
3-8 |
0-96 |
|
30 |
0-61 |
1-63 |
2-24 |
2-9 |
1-00 |
|
40 |
0-495 |
2-02 |
3-33 |
2-0 |
1-03 |
|
50 |
0-39 |
2-57 |
6-37 |
1-2 |
1-03 |
Chapter III WIND-TUNNEL EXPERIMENT
50. Nature of Wind-tunnel Work
The calculation of Aerodynamic force presents difficulties even in simple cases. Great progress has been made with this problem, as will be described in subsequent chapters, and designers of aircraft now rely on direct calculation in several connections. Theoretical formulae are improved, however, by experimentally determined cor- rections that take neglected factors into account, while other formulae are based as much on experiment as on theory. Yet many effects of change of shape or Reynolds number are of so complicated a nature as entirely to elude theoretical treatment and to require direct rfieasurement. Measurements can be made during full-scale flight by weighing, pressure plotting, comparison of performance, etc. This method is employed occasionally, but is economically reserved where possible to the final stages of investigations carried out primarily on models made strictly to scale. Thus model experiment, which formerly provided the whole basis of Aerodynamics, apart from the theoretical work of Lanchester in England and Prandtl in Germany, still occupies an important place.
In early days of the science, models were sometimes studied out-of- doors when flying freely (cf. Lanchester 's experiments), suspended from a balance in a natural wind (Lilienthal), during fall from a considerable height (Eiffel), or towed. Calm days are few, however, and unsteadiness of winds was soon found to cause large errors, so that experiments came to be carried out in laboratories. In the Whirling Arm method (Langley and others), models attached to a balance were swung uniformly round a great horizontal circle ; a disadvantage, additional to mechanical difficulties arising from cen- trifugal force, lay in the swirl imparted to the air by the revolving apparatus and the flight of models in their own wakes after the first revolution. Experiments are now nearly always made in an artificial wind generated by or within a wind tunnel. This method was introduced during the second half of the nineteenth century and wind tunnels were built in various countries during the first decade
CH. Ill] WIND-TUNNEL EXPERIMENT 69
of the present century. A matter of great historic interest is that the Wright Brothers carried out numerous experiments in a diminu* tive wind tunnel, less than 2 square feet in sectional area, in prepara* tion for their brilliant success in the first mechanically propelled aeroplane, which flew in 1903. The tunnel method of experiment has since been developed to a magnificent degree.
The artificial wind should be steady and uniform, for otherwise superposing a velocity cannot change the circumstances of experi- ment exactly to those of flight through still air. Tunnels can be designed to achieve a fair approximation to this requirement. Through the part of the stream actually used for experiment, the maximum variation of time-average velocity need not exceed ± 1 per cent, and the variation of instantaneous velocity at any one point, though more difficult to suppress, can be reduced to ± 2 per cent. This standard of steadiness may be relaxed for experiments in which it is not of prime importance. The wide range of modern experiment has led on economic grounds to the evolution of several specialised forms for the wind tunnel, as will be described, although in a small Aeronautical laboratory a single tunnel must serve a variety of widely different uses.
In elementary Aerodynamics it is advisable to carry out many experiments* which mathematical treatment renders unnecessary in a more advanced course, but there still remains unlimited scope for wind-tunnel work on scientific matters in which analysis is of little avail or particularly complicated. Questions of this nature will appear as the subject proceeds, and it will only be remarked here that their investigation invites originality of method and ingenuity in the design of special apparatus.
Another and equally important domain of model experiment lies in direct application to specific designs of aircraft. The Aero- dynamic balances and other measuring apparatus surrounding the working section of a tunnel have usually been installed with this purpose primarily in view.
A more or less complete model of an aircraft can be suspended in a wind-tunnel stream of known speed and its reaction measured. It can be pitched, yawed, rolled about its longitudinal axis, or oscillated in imitation of a variety of circumstances arising in free flight, and its response accurately determined. A special technique described in a later chapter enables due allowance to be made for the limited lateral extent of the stream. Yet with every precaution
* A programme of experimental studies requiring only simple apparatus is given in a companion volume to this book.
70 AERODYNAMICS [CH.
the interpretation of the observations in terms of full-scale flight is attended with uncertainty. Two outstanding reasons are as follows. Experiments on complete models, even in national tunnels, can only cross the threshold of large full-scale Reynolds numbers, and fall far short in more modest tunnels. Secondly, the initial turbulence remaining in an artificial wind is sufficient to produce a marked difference in some connections from flight at the same Reynolds number.
The first difficulty can be circumvented in the case of small component parts of an aircraft by employing enlarged models ; for test in an artificial wind of normal density they would be larger than full-scale. The drag of the complete aircraft is then built up from piecemeal tests on its parts. A new problem introduced is to determine how each part will affect a neighbouring part or one to which it is joined. Such mutual effect is called interference and becomes familiar in wind-tunnel work, for in principle it enters into all experiments in which a model is supported in the stream by exposed attachments. The same device may be applied to wings and tail-planes by testing short spanwise-lengths of large chord under two-dimensional conditions. The consequent problem in this case is the change from two- to three-dimensional conditions and is left to calculation. For reliable data on wings at greater inci- dences or on long bodies, there is no alternative to large or costly wind tunnels except flying tests.
The above expedients leave the second main difficulty still to be faced, viz. the effect of initial turbulence. This question is many- sided and its consideration must be deferred, but there is evidently need to ascertain by suitable tests the degree of turbulence character- ising the particular tunnel employed.
Finally, fast aircraft are considerably affected, especially at high altitudes, by the compressibility of the air. It was found in the preceding chapter that for dynamical similarity under these conditions both the Reynolds and Mach numbers require to be maintained. Tunnels capable of realising even moderate Reynolds numbers at high speeds are particularly expensive to construct and operate, and experiments are usually carried out in small streams, Reynolds numbers being ignored, and the effects of compressibility determined as corrections of a general nature.
It will be seen that, whilst the principles and phenomena of Aerodynamics can be illustrated qualitatively with ease in a modest wind tunnel, the constant need for quantitative information makes more serious demands and creates a study within itself.
in]
WIND-TUNNEL EXPERIMENT
71
51. Atmospheric Wind Tunnels — Open-Return Type
The cross-section of the experimental part of a wind-tunnel stream may be square, round, elliptic, oval, octagonal, or of other shape. The size of a tunnel is specified by the dimensions of this cross-section. Apart from small high-speed tunnels actuated by a pressure reservoir, the flow past the model is induced by a tractor airscrew located downstream. The airscrew is made as large as possible, if only to minimise noise, and its shaft is coupled direct to the driving motor, the speed of which is controlled preferably by the Ward-Leonard, Kramer, or similar electrical system. If C "is the cross-sectional area and V the velocity of the experimental part of the stream, the ' power factor ' P is usually defined as
__ 550 X input b.h.p.
-
But in some publications the reciprocal of this ratio is intended.
The term atmospheric applied to a wind tunnel means that the density of its air stream is approximately the same as that of the surrounding atmosphere. Some tunnels employ compressed or rarified air, but they are few, and so the term is commonly omitted in referring to the atmospheric class.
For some years many of the wind tunnels built were of the type shown in Fig. 26, described as ' straight-through ' or ' open-return/
cas-
.-:£ \>
SCALE OP FEET
FIG. 26. — 4-FT. OPEN-RETURN WIND TUNNEL.
H, inlet honeycomb ; P, plane table ; S, guard grid ; D, regenerative cone ; W, honeycomb wall.
Though the design has been superseded, numerous examples are still in use. Air is drawn from the laboratory into a short straight tunnel through a faired intake and wide honeycomb, the location of the latter being adjusted to spread the flow evenly over the working section. Subsequently the stream has most of its kinetic energy reconverted into pressure energy in a divergent duct D, from
72 AERODYNAMICS [CH. Ill
which the airscrew exhausts the air into a ' distributor,1 a large chamber enclosed by perforated walls W. The distributor returns the air, with disturbances due to the airscrew much reduced, over a wide area to the laboratory, which conveys it evenly arid slowly back to the intake and thus forms an integral part of the circuit. A consequent disadvantage is that the laboratory requires to be reasonably clear of obstructions, symmetrically laid out, and also large ; approximate dimensions for a tunnel of size x are : over-all length, including diffuser, 14* ; height and width, 4£#. A second disadvantage is lack of economy in running, the power factor P having the high-value unity. In small sizes, however, the type is simple to construct and convenient in use.
A boundary layer of sluggish air lines the tunnel walls, but away from this Bernoulli's equation holds closely, showing a wide central stream almost devoid of vorticity. This stream slightly narrows along the tunnel owing to increasing thickness of the boundary layer. Thus the streamlines are slightly convergent ; velocity increases and pressure decreases along the parallel length. To compensate for tnio characteristic variation, tunnels are sometimes made slightly divergent.
The static pressure is obviously less within the tunnel than out- side. At first sight it may appear feasible to calculate the velocity at the working section from a measurement of the difference in static pressure between there and some sheltered comer of the laboratory. But losses in total energy occurring at the intake, principally through the honeycomb, prevent this. The pressure in a pitot tube within the working stream is less than the static pressure in the room. A small hole is drilled through the side of the tunnel several feet up- stream from the working section, and the pressure drop in a pipe connected with it is calibrated against the appropriate mean reading of a pitot-static tube traversed across the working section (excluding, of course, the boundary layer). By this means velocities can after- wards be gauged without the obstruction of a pitot-static tube in the stream.
52. Closed-Return Tunnels
In the more modern tunnels of Fig. 27, the return flow is con- veyed within divergent diffuser ducts to the mouth of a convergent nozzle, which accelerates the air rapidly into the working section. A ring of radial straighteners is fitted behind the airscrew to remove spin and the circulating stream is guided round corners by cascades
(c)
FIG. 27. — RETURN-CIRCUIT WIND TUNNELS. (a), enclosed section ; (b), lull-scale open jet ; (c), compact open jet ; (d), corner vane.
A.D.— 8*
73
74 AERODYNAMICS [CH.
of aerofoils or guide- vanes (see (d) in the figure for a suitable section), which maintain a fairly even distribution of velocity over the gradually expanding cross-section. The experimental part of the stream is preferably enclosed, as at (a), but sometimes takes the form of an open jet, as at (b) and (c). An open jet is distorted by a model and is resorted to only when accessibility is at a premium. These tunnels are often known as of ' closed-return ' or ' race- course ' type. They effect a great economy in laboratory space, only a small room being required round the working section, and also in running costs, P having approximately the value J. Wood is not a suitable material for construction, though often used, because during a long run the air warms and produces cracks which are destructive to efficient working since the ducts support a small pressure.
A characteristic of prime importance is the contraction ratio of the tunnel, defined as the ratio of the maximum cross-sectional area attained by the stream to the cross-sectional area of the experimental par*, A large contraction ratio effectively reduces turbulence but increases the over-all length of a tunnel of given size, since divergent ducts must expand slowly to prevent the return flow separating from the walls. A rather long tunnel has the advantage of prevent- ing disturbances from a high-drag model being propagated completely round the circuit. Modern designs usually specify a contraction ratio greater than 6 ; values for the tunnels (a), (b), (c) in the figure are 6£, 5, and 3|, respectively, (a) may be regarded as suitable for general purposes, (b) illustrates the full-scale tunnel at Langley Field, U.S.A., which has an oval jet 60 ft. by 30 ft. in section, an over-all length of some 430 ft., and a speed of 175 ft. per sec. with a power input of 8,000 h.p. (c) indicates the maximum possible compactness for this type of tunnel ; developed at the R.A.E., it has been used for sizes up to 24-ft. diameter.
» » 1 1 ' » i * i »
FIG, 28. — SPINNING TUN- NEL.
M, flying model; O, observation window ; N, net for catching model ; H, honeycomb.
53. Spinning Tunnel
A few vertical tunnels have been built, as shown schematically in Fig. 28, for spinning tests. An aeroplane may fly in a vertical spiral with a velocity of descent VT, say. A
Ill] WIND-TUNNEL EXPERIMENT 75
question arising is whether operation of the aerodynamic control surfaces will steer the craft into a normal flight path. To investigate this, a light model of balsa wood, similar in disposition of mass as well as in form, is set into corresponding spiral flight, a camera mechanism operating the controls after a delay. Ignoring the effects of vis- cosity, the Froude number V*/lg must be the same for craft and model. If the latter is made to T^th scale, its velocity of descent = JFF. This is a small speed, and it is feasible to employ a wide vertical tunnel with an upwardly directed stream, so that the model does not lose height and the action can be observed conveniently. The difficulty with these tunnels is to prevent the model from (a) flying into the wall, (b) spinning upwards or downwards. Accord- ing to tests carried out on model tunnels, (a) can be overcome by a suitable distribution of velocity along the radius, and (b) by making the tunnel slightly divergent, which gives stability in respect of vertical displacement, since the rising model then loses flying speed, and vice versa.
54. Coefficients of Lift, Drag, and Moment *
In the general case of a body suspended in a wind tunnel Aero- dynamic force is not a pure drag, but is inclined, often steeply, to the direction of flow. This inclination is not constant for a given shape and attitude of the body, but is a function of the Reynolds number.
When the flow has a single plane of symmetry for all angles of incidence of the body, the Aerodynamic force can be resolved into two components in that plane, parallel and perpendicular to the relative wind — the drag and lift, respectively. By Article 47 we find for any particular shape and incidence a lift coefficient :
*CL=*L= -/.(*) . . . (74)
and a drag coefficient :
D - p
* There are two systems of coefficients in Aerodynamics. , In the now prevailing system, associated with the symbol C, forces and moments are divided by the product of the stagnation pressure for incompressible flow, viz. JpF* (cf. Art. 32), and /* or /* ; in an earlier system, distinguished by the symbol k, the quantity pV* takes the place of the stagnation pressure. Thus a ^-coefficient = J x the corres- ponding C-coefficient, as indicated in (74), (76), and (77). Neither system has an advantage over the other, but to secure a universal notation (^-coefficients have superseded ^-coefficients in this country since 1937. They are generally adopted in this book, but some matters are still expressed in the older system.
76 AERODYNAMICS [CH.
Most bodies tested are parts of aircraft, and L is then positive if it supports weight when the aircraft is right way up. For any chosen Reynolds number, we have
Aerodynamic force (4) = foV*PVCL* + CD«
and, if its inclination to the direction of lift is y (Fig. 29),
tan Y = CD/CL . . (76)
AX/&D = CJCD is called the lift-drag ratio and = LjD.
Without a plane of symmetry as above, A will have a third component, called the crosswind force.
Again assuming this plane of symmetry, the line of action of A can be found from its magnitude and direction and the moment about some axis in the body perpendicular to the plane, usually through the quarter-chord point. This moment is called the pitching moment Af. The method of Article 47 gives for any parti- cular shape and attitude a coefficient :
M - . . . (77)
FIG. 29.
M is positive when it tends to increase angle of incidence, i.e. to turn the body clockwise in the figure.
Other moment coefficients will be introduced later when the motion of aircraft is considered in greater detail. It should carefully be noted that Q, CD," CM are different functions of R ; we shall often omit a distinguishing suffix to / without implying equality.
It has been stated that any agreed length may be adopted for I to specify the size of a body of given shape and attitude. More generally, any agreed area may be used for /*, or volume for I9. Practice varies in the choice made. CL, CD are always calculated for single wings on the area S projected on a plane containing the span and central chord (line drawn from nose to tail of median section). The length of the chord c is introduced as the additional length required for CM (although not for other moment coefficients, when the semi-span is used). Thus for wings :
CL « L/$PF«S, CD « D/*pF«S, CM =
Hi] WIND-TUNNEL EXPERIMENT 77
The parasitic, or ' extra-to-aerofoil/ drag of a complete aeroplane, i.e. the drag of all parts other than the wings, may sometimes for convenience be referred to 5. But usually for fuselages (aeroplane bodies), struts, and the like, and sometimes for airship envelopes, /* is specified by the maximum sectional area across the stream. Another area frequently used for airship envelopes is (volume)2/8, enabling the drags of different shapes to be compared on the basis of equal static lift. It is seldom suitable to employ the same / to specify both R and the coefficients ; for R, the length from nose to tail is usually chosen.
55. Suspension of Models
It is evident that the foregoing and other coefficients can be determined through a range of R by direct measurement, given suit- able balances. These are grouped round the working section of the tunnel, and the model is suspended from them. Their design and arrangement are partly determined by the following consideration.
Suppose the true drag D of a model in a tunnel is required. Let the suspension attachments (called, for short, the holder) have a drag d when tested alone. Let the drag of holder and model be D'. Except under special conditions we cannot write : D = Dr — d ; the combination represents a new shape not simply related to either part. The mutual effect of d on D, or vice versa, is termed the mutual interference. An example is as follows. If a 6-in. diameter model of an airship envelope be suspended by fine wires, and a spindle, the size of a pencil, made to approach its side end-on, the drag of the airship may increase as much as 20 per cent, before contact occurs.
The approximation used in general depends upon the interference being local. A second holder is attached to a different part of the model and a test made with both holders in place. Removing the original holder and testing again with only the second holder fitted gives a difference which is applied as holder correction to a third test in which the original holder alone is present. The approximation gives good results, provided neither holder creates much disturbance, to ensure which fine wires or thin streamline struts are used.
Fig. 30 shows as a simple illustration an arrangement suitable for a heavy long body having small drag. Near the nose the body is suspended by a wire from the tunnel roof, while a ' sting ' screwed into the tail is pivoted in the end of a streamline balance arm, for the most part protected from the wind by a guard tube. If the guard
78
AERODYNAMICS
[CH.
tube is of sufficient size to deflect the stream appreciably, a dummy is fixed above in an inverted position. Sensitivity, in spite of the heavy weight of the body, is achieved by calculating the fore-and- aft location of the wire to make, following small horizontal displace- ment, the horizontal component of its tension only just overcome that of the compression in the balance arm. To find the effective
FIG. 30. — TESTING A HEAVY MODEL OF Low DRAG. G, guard tube ; P, scale pan ; S, sting ; T, turnbuckle ; W, cross-hair.
drag of the wire, another test is made with a second wire hung from the nose as shown at (a) and attached to the floor of the tunnel. Next, the sting is separated slightly from the balance arm, support being by the wires (6) from the roof, and the effective drag of the balance arm measured with the body almost in place. Finally, the model can be suspended altogether differently, from a lift-drag balance as at (c), the wires and original balance arm being removed, and the small effective drag of the sting estimated by testing with it
Ill] WIND-TUNNEL EXPERIMENT 79
in place and away. At the same time special experiments can be made to investigate the interference, neglected above, between the sting and the original balance arm. It will be appreciated that the reason why the arrangement (c) is avoided except for corrections is that the spindle, although of streamline section, would split the delicate flow near the body, and artificially increase its drag. The model fuselage shown may have a small lift. To prevent consequent error in drag measurement, the wire and balance arm must be accurately vertical for a horizontal wind. This is verified by hanging a weight on the body without the wind, when no drag should be registered.
56. The Lift-drag Balance
V-
When several force and couple components act on a model it is desirable for accuracy to measure as many as possible without dis- turbing the setting of the model. Omnibus balances designed for this purpose tend to be complicated, and reference must be made to original descriptions. An indispensable part of the equipment of a tunnel, however, is an Aerodynamic balance that will measure lift and drag simultaneouslyjand preferably at least one moment at the same time.
Aerofoil Kyn^ Diaphragm v
Tunnel Wall — %''W
FIG. 31. — SIMPLE LIFT-DRAG BALANCE.
A simple form of lift-drag balance is illustrated in Fig. 31. The main beam passes through a bearing B centrally fixed to a hard copper diaphragm, 5 in. diameter and 0-003 in. thick, clamped to a flange of a casting which abuts on a side wall of the tunnel through soft packing to absorb vibration. The diaphragm gives elastically, permitting the beam to deflect in any direction almost freely between the fine limits imposed by the annular stop O which is opened by the
80 AERODYNAMICS [CH.
lever T while observations are being taken. The diaphragm suspen- sion prevents leak into the enclosed-type tunnel assumed ; it may be replaced, if desired, by a gymbals with an open-jet tunnel, but sensi- tivity is then more difficult to maintain with large forces. The sensitivity of the balance described is 0-0003 Ib. The bearing permits of turning the beam about its axis quickly and accurately by means of the worm gear W, an angular adjustment that is often useful, e.g. when testing an aerofoil of the form which can be sus- pended by screwing a spindle into a wing-tip as shown in the figure. Lift is measured by adjusting a lift rider on the main beam and by weights on the scale pan L. The free end of the main beam carries a knife-wheel E, engaging a hardened and ground plate at one end of a horizontal bell-crank lever, of sufficient leverage to ensure that the ainaH end movement of the main beam is negligible. This lever is mounted on vertical knife edges, and transmits drag to a subsidiary balance, with a drag rider and scale pan D.
Horizontal lift-drag balances are simple to construct and also particularly convenient for testing square-ended aerofoils, negligible interference occurring between the aerofoil and a spindle screwed into its tip. They are inconvenient for aerofoils having thin tips and are not readily adaptable to measure pitching moments. Their usefulness is enlarged in combination with a simple steelyard mounted on the roof of the tunnel, as described in the next article. But experience with this double-balance method of testing suggested the more adaptable modern types of balance described in principle later.
57. Double Balance Method of Testing an Aerofoil
The distinguishing feature of a good aerofoil, or model wing, at fairly large Reynolds numbers is that its Aerodynamic force A is, at small angles of incidence, nearly perpendicular to the stream ; LjD may then be 25 and y of (76) 2-3°. The point P (Fig. 32), at which A intersects the chord, of length c, is called the centre of pressure and NP/c the centre of pressure coefficient &CP. The method described enables L, D and P and consequently M to be determined with only a simple roof balance and a lift-drag balance. The aerofoil is suspended from the former by wires attached to sunk eye-screws at W and from the latter through a sting pivoted at E. A drum carried by the roof balance enables the length of the wires to be adjusted and hence the incidence a. The model is suspended upside down to avoid the use of a heavy counterpoise, although a small one is desirable with a
WIND-TUNNEL EXPERIMENT
81
HI]
light model for safety, to keep the wires taut, and to permit measure- ment of small upward forces — negative lifts.
Part L' of the lift L is taken at W, the remainder U at E. The wires are set truly vertical at some small incidence a, when they will be also vertical at a small negative a, but at no other incidence. Let
FIG. 32.
6 be their small inclination to the vertical, the stream being assumed truly horizontal, and T that part of their tension due to A. They support a part T sin 0 of the drag D, only the remaining part d being supported at E. The lift-drag balance connected to E provides the only means of measuring D. Thus 0 must be corrected for accur- ately and the method adopted is as follows. At any setting of the aerofoil the zeros of the lift-drag balance are observed, before starting the wind, with and without a known weight hooked on the model. An apparent drag is thus found for a known value of T at the parti- cular value of 0 corresponding to a, but which need not be known. A proportionate correction appropriate to the value of T measured when the wind is on can then be applied to drag observed at E. This correction requires to be determined for all values of a.
Measurements of drag must further be corrected for (a) part of the drag of the wires, for which purpose the measurements may be repeated with additional wires attached in a similar manner, or a calculation may be made based on Fig. 25, the geometry of the rig and the thickness of the tunnel boundary layer ; (6) the effective
82 AERODYNAMICS [CH.
drag of the lift-drag balance 'arm, determined as in Article 55, a being varied through the complete range studied ; (c) the effective drag of the sting, obtained by measuring drag with and without the sting in place at all incidences with the model suspended in some other manner, e.g. by a spindle fastened to a wing-tip.
Referring to Fig. 32, L = L' + L* , and taking moments about E we have
U . I cos 0 = A . a = Tl cos (p - 6) giving
L9 =7(1 + 0 tan (3)
since 6 is small. Also D = TQ + d
Tl a = -j(cos (3 + 6 sin p)
A
where A = V(if + &)
and y = tan ~l (D/L).
Finally NP = c — {a sec (a — y) — s}-
58. Aerodynamic Balances
The foregoing method is simplified by fixing W and adjusting a by displacing E ; the front wires may then form two longitudinal vees, and a vertical sting wire at E replace the lift-drag balance arm. The whole of the drag, as well as the major part of the lift, is taken by the vee-wires, and the sting wire supports only the remainder of the lift.
This in brief is the principle of the Farren balance, shown schematically at (a) in Fig. 32A. Part of the lift and the entire drag are communicated by two parallel pairs of vee-wires, inter- secting at W, to the frame F located above the tunnel and pivoted vertically above W. The drag is transmitted by an increase of tension in the front wires of the vees and a decrease of tension in the back wires, and thus a counterpoise must be suspended from a light model of high drag in order to keep the back wires taut. Such a counterpoise is advisable in any case as a safeguard, and then care need not be taken to locate W well in front of the centre- of-pressure. The frame is weighed in the balances L and D for the lift and drag communicated to it. The sting wire, shown fastened at E to the fuselage of a complete model in the figure, remains truly vertical with change of incidence by virtue of being raised or lowered by a stirrup R which is parallel to EW and pivoted vertically above W. The familiar problem is to measure the remaining part
Ill] WIND-TUNNEL EXPERIMENT
of the lift supported by the sting wire without interfering with the drag balance. This is achieved by pivoting the bell- crank lever, which sup- ports the stirrup, level with the pivot of the frame F. Thus these two pivot lines are coincident, although in the figure they are shown slightly displaced from one an- other for clearness. If the pivot of the bell- crank lever is carried on the lift beam, the whole of the lift is transmitted to that beam, and the balance marked M is used only to determine the pitching moment of the Aerodynamic force about W.
The balance shown at (ft) in the figure makes use of a different system, enabling all pivots to be located outside the tunnel. The model is suspended from the platform F by any convenient means and, provided the two lift beams shown are of equal length, the true lift and drag are measured whatever the position of the model relative to F. However, the pitching moment is determined about the line joining the intersections of the
M
(a)
(b)
w/////
FIG.— -32A. AERODYNAMIC BALANCES
84 AERODYNAMICS [CH.
centre-lines produced of the two inclined pairs of sloping struts which support F, i.e. about W in the figure. This is readily verified by considering the effect of a load acting in any direction through W ; it would evidently cause tensions and compressions in the sloping struts but no force in the moment linkage. Hence the suspension from the platform will in practice usually be so arranged that the pitching moment is measured about a significant point in the model. The linkage connecting the drag and moment balances should ensure that these give the drag and moment separately, i.e. without interfering with one another.
The third balance (c) is an inverted form of (b) with other modifications. The moment balance is mounted on the lift plat- form G instead of being attached to a fixed point, a step which eliminates the necessity for a linkage to prevent interference between the moment and drag measurements. All weights used on the moment balance are stored on the lift platform so that their adjustment will not affect the lift reading. The drag frame H is supported in a parallel linkage so that fore and aft movements can occur without vertical displacement, and in consequence excessive static stability is avoided without the use of counterpoises.
The foregoing illustrates only a few of the many devices put to use in the design of a modern Aerodynamic balance. For clearness, the three balances have been described in 3-component form, but all are readily adaptable to cope with additional components. The following constructional features may also be noted. Elastic pivots are preferred to knife-edges or conical points and commonly take the form of two crossed strips of clock-spring. The amount of damping required is extremely variable, and therefore the electro- magnetic method is preferred to a plunger working in oil. When a balance is inaccessible or there is need to save time in operation, weighing and recording can be carried out mechanically.
59. Given tunnel determinations of lift, drag, etc., freed from parasitic effects, various corrections are necessary before they can be applied to free air conditions at the same Reynolds number. These are in respect of : (1) choking of the stream by a body of rela- tively considerable dimensions, (2) deviation of the undisturbed stream from the perpendicular to the direction in which lift is measured, (3) variation of static pressure in the undisturbed stream, (4) effects of the limited lateral extent of the stream, applying principally to wings, and developed in Chapter VIII. A further cause of difference is introduced in Article 65.
(1) It is possible to express the argument of Article 35 in approxi-
WIND-TUNNEL EXPERIMENT
85
III]
mate numerical form for a given shape, when it is seen to follow that the choke correction is small. For a body whose diameter is J that of the tunnel the correction is usually < 1 per cent.
(2) Let the stream be inclined downward from the horizontal at a small angle p, and, taking the familiar case of an aerofoil upside down, let its aerodynamic force be A and its true lift and drag L and D, respectively. The apparent lift and drag measured, however, are La and Da (Fig. 33), We have, assuming p small —
L = A cos y D = A sin y
La = A cos (Y - P) Da = A sin (Y - p)
= A (sin Y — P cos y)
= D - pi.
Thus the error in La is negligible, but this may be far from true of Daf for we have
D
(78)
FIG. 33.
Upward inclination of the stream leads to an error in drag of the same magnitude but opposite in sign.
Example : If p == £°, and L/D ==>20, the error in D is ± 17 J per cent.
This error can be removed by testing the model right way up and upside down, and taking the mean. The process is laborious, how- ever, and a correction factor for general use is worked out by an initial test of this kind. Where their design permits, balances are carefully set on installation so as to eliminate the error as far as possible.
(3) Convergence or divergence of the stream leads to an error due to the pressure gradient that exists in the direction of flow prior to introducing the model. In the former, the more usual case, pressure decreases downstream (x increasing). Owing to the short length of the model dp/dx may be assumed constant, and to this approxima- tion is easily determined experimentally. The maximum con- vergence in a parallel-walled tunnel is only about J°.
Complete analysis of the problem presents difficulty, but an inferior limit to the correction is readily calculated by a method that will now be familiar. Considering an element cylinder of the body, of cross-section AS and length /, parallel to the direction of flow and
86 . AERODYNAMICS [CH.
coming to ends on the surface of the body, the downstream force on it, if we apply a method analogous to that of Article 8, is readily found to be — (dp/dx) (AS . /). The whole volume V can be made up of such cylinders, giving for the downstream force on the model — (dp/dx)V, which is essentially positive for convergence. This force has nothing to do with drag, vanishing when the stream is parallel or the model moves through free air, and measurements must be decreased on its account. The correction is important for low resistance shapes such as airship envelopes and good aeroplane bodies and wings at high-speed attitudes.
Further analysis shows that the volume should be greater than that of the body, an increase of 5-10 per cent, being required for long bodies of revolution, 10-15 per cent, for wings, and 30 per cent, for compact strut shapes, approximately. The correction does not vanish in the case of bluff shapes of small volume, but it is then numerically unimportant. (See also Article 230B.)
59A. Pitot Traverse Method
The drag of a two-dimensional aerofoil can be estimated from an exploration of the loss of pitot head through a transverse section of its wake. This loss will be denoted by a non-dimensional co- efficient h, as follows. Let U, pQ be the undisturbed velocity and pressure, respectively, and q, p the corresponding quantities at any point in the wake. Then the loss of pitot head at the point is
P* + *pt/« - (P + *P?a) = h . iplT*.
It is much more marked close behind the aerofoil than farther down- stream, where the wake has diffused outward.
Consider first a section of the wake sufficiently far behind the aerofoil for the pressure to be equal to pQ and the velocity to have become parallel again to the relative motion, a state distinguished by writing q = u. Through an element 8y of this section, of unit length parallel to the span of the aerofoil, the mass passing in unit time is p«8y, and the rate of loss of momentum parallel to the relative motion is pwSy . (U — u). Hence the drag Z)0 of unit length of the aerofoil is given by
Dg = p/«(C7 — u)dy or
«
Ill] WIND-TUNNEL EXPERIMENT
But u/U = (1 - A)*. Hence
87
Far behind the aerofoil h will be small and the term in the square brackets can be expanded as follows —
1_(1- tA-i*i + ...) = t*. approximately, so that in this case
Defining a drag coefficient C^Q as equal to DQlfoU*c, where c is the aerofoil chord, the result can be written
CDO =
(iii)
This coefficient is known as the profile drag coefficient of the aerofoil. It includes the entire drag under two-dimensional conditions but only part of the drag under three-dimensional conditions, except at the incidence for zero lift ; at other incidences a wing has in addition an inditced drag coefficient, arising from the continuous generation of lift Aerodynamically and appearing as a modification of the
pressure distribution for two-dimensional flow. The pitot traverse method finds uses in the wind tunnel, where two-dimensional flow can be simulated, but its chief application is to flight experiments. Exploration on the above lines of the wake of a wing can give only its profile drag, but its induced drag can be estimated separately by calculation, as will be found in Chapter VIII. A difficulty arising in flight is that the pitot traverse must be made close behind the wing, so that the pressure differs from pQ and a correction to (iii) becomes necessary. The experimental section near the wing will be distinguished by suffix 1, see Fig. 33A.
This correction is rather uncertain. Jones* has suggested ignor- ing the turbulence in the wake and relating the pressure and velocity
* Jones (Sir Melvill), A.R.C.R. & M. No. 1688, 1936.
88 AERODYNAMICS [CH.
at section 1 to those at the distant ^section, where p «= pQ, by Bernoulli's equation, applied along a suppositions mean stream- tube. Then :
Pi + iPft1 - Po +
Writing
* -
*~
this gives
Again,
- (A
so that
| = (1 - h, - ktf ... (V)
Let w denote distance perpendicular to the direction of mean motion at section 1. Then for incompressible flow q£n = «Jy and (i) becomes
s - (^(i-u &- }~UV U
Substituting from (iv) and (v) and again introducing the drag coefficient,
1 - (1 - AJ*J rfn . (79)
This result is known as Jones1 formula. Tested in a full-scale wind tunnel, it was found* to be accurate within experimental errors along the middle three-quarters of the span of a rectangular wing. Nearer the wing-tips the induced flow associated with the produc- tion of lift under three-dimensional conditions makes the method inapplicable. Restrictions of another kind have been discussed by Taylor, f
A different treatment of the problem has been given by Betz.f His formula includes provision for dealing with the induced flow caused by three-dimensional production of lift.
* Goett, N.A.C.A. Report No. 660, 1939. t Taylor (Sir Geoffrey), A.R.C.R. & M. No. 1808, 1937.
j Betz, Z.F.M., vol. 16, 1925 ; see Arts. 79-81 by Prandtl in Tietjens, ' Applied Hydro- and Aero-Mechanics/
Ill] WIND-TUNNEL EXPERIMENT 89
The pitot traverse method of drag measurement offers such manifold advantages that the subject is an old one and has received attention on many occasions. The exact theory is complicated, however, and formulae obtained by simple means require to be established by experiment, The method can be relied upon to give a close estimate of drag under fairly favourable conditions ; viz. briefly when the pressure in the wake differs little from pQ and the velocity trough is rather shallow. These conditions imply, especially if CDO has a considerable value, that the traverse should be made well downstream, but this is obviously inconvenient in flight, whilst in tunnel experiment it may sometimes vitiate the two-dimensional assumption. Again, the section behind which a traverse is made may not be truly representative of the average section of a wing or aerofoil. Such difficulties partly explain discrepancies that are found to exist.
The exploring pitot tube should be fine in order to avoid a system- atic experimental error. The^ effect of compressibility on the method has also been examined.* The estimates are not affected by any pressure gradient that may exist in the empty tunnel.
60. An example of wing characteristics obtainable by the method of Article 57 at small scale, e.g. in a 6-ft. open-jet tunnel at 100 ft. per sec., is given in Fig. 34, corrections noted in Article 59 having been made so that the results apply to free air conditions at a small Reynolds number. The aerofoil is of the section shown, known as Clark YH, with a ratio of span to chord, called aspect ratio, of 6.
Features fairly typical of aerofoils in general may be noted.
Zero lift occurs at a negative incidence, usually small. The value — 3° shown in the present case is arbitrary, in the sense that it depends upon how a is measured. The present aerofoil has a partly flat lower surface, which is used to define inclination to the wind. Another aerofoil might have a slight concavity in the lower surface, when the common tangent would be employed. Most aerofoils are bi-convex, however, when the line joining the centres of curvature of the extreme nose and tail defines a.
Lift attains a maximum at a moderate angle, 15° in the present instance, after which it falls. This incidence is known as the critical or stalling angle and, combined with the maximum value attained by CL, is of importance in connection with slow flying. The open- jet tunnel determines this feature more reliably f than the enclosed-
* Young, A.R.C.R. & M. No. 1881, 1938.
f Bradfield, dark, and Fairtfcorne, A.R.C.R. A M., 1363, 1930.
075
0° 5° 10* 15*
INCIDENCE, <x
20°
FIG, 34.— CHARACTERISTICS OF CLARK YH WING (ASPECT RATIO 6) AT SMALL SCALE (6-rr. OPEN-JET TUNNEL AT 100 FT, PER SEC.).
90
WIND-TUNNEL EXPERIMENT
91
CH. in]
section tunnel, which tends to flatter compared with free air. The flow is often delicate in this region, some lift curves branching, and different coefficients being obtained according as to whether a is increasing or decreasing.
Minimum drag occurs when the lift is small but maximum LfD at a considerably larger incidence. Drag and the angle y begin to increase rapidly at the critical angle.
At — 1-3° the centre of pressure is midway along the chord. It moves forward as incidence is increased up to the critical angle, and then back. This travel results from striking changes which occur in the shape of the pressure diagram, illustrated for a rather similar aerofoil in Fig. 35. Between — 2-7° and — 4-5°, approximately,
D5r
of
-O5 -1-0 -15
-20
OUO°
10C
FIG. 35. — LIFT PRESSURE DIAGRAMS FOR THE MEDIAN SECTION OF AN AEROFOIL (BROKEN LINE APPLIES TO LOWER SURFACE).
the C.P. is off the aerofoil. CCP = ± oo at a = — 3°, meaning that when the resultant force is a pure drag, lift being zero, there is a couple on the aerofoil ; the two loops seen in the pressure diagram for 0° become so modified at — 3° as to enclose equal areas. Thus the C.P. curve has two branches asymptotic to the broken line in Fig. 34 ; part of the negative lift branch is shown near the left-hand zero of the scales.
The travel of the C.P. for a < 16° indicates a form of instability. To see this, imagine the aerofoil to be pivoted in the tunnel about a line parallel to the span and distant 0-3 chord from the leading edge, and to be so weighted that it is in neutral equilibrium for all angles without the wind (an experiment on these lines is easy to arrange). If now the model be held lightly at oc = 5°, the incidence at which the C.P. is cut by the pivot line, and the wind started, a couple tending to increase or decrease a will instantly be felt, small disturbance of a displacing the C.P. in such a direction as to increase the disturbance. The aerofoil will ride in stable equilibrium, however, at a = 20°. It would also be stable
AERODYNAMICS
[CH.
if pivoted in front of 0*25 chord, but this case is only of interest in connection with the auxiliary control surfaces of aircraft. The C.P. curve is physically indefinite, in so far as it would have a different shape if we defined the C.P. as the intersection by A of some line parallel to the chord but displaced from it. Thus in the above experiment different results would be obtained if the pivot line were displaced from the chord plane. Fig. 36 contains the essential information of Fig. 34 plotted in
more compact and practical form, CL being more generally useful than a as the indepen- dent variable. The moment coefficient given defines the pitching moment about a line one-quarter chord behind the leading edge of the aero- foil, often preferred for greater precision to that of Article 54. Its middle point is called the Aerodynamic centre.
61. Application of Complete Model Data
Where f(R) has been found in the tunnel through a sufficient range, calculations may be made for the shape of body concerned in a variety of practical circum- stances, as illustrated in the two examples following.
(a) A temporary wire, J in. diameter and 2 ft. long, is fixed parallel to the span of a wing just outside its boundary layer, above * a position where the pressure
drop amounts to 15*36 Ib. per sq. ft. when the aeroplane is flying at
100 m.p.h. at low altitude. Find the drag of the wire under these
conditions. First determine the relative velocity of the wire. Writing p, V
for the local pressure and velocity (ft. per second) where it is ex-
-0-4
-002
-0-04
-006
FIG. 36. — CHARACTERISTICS OF CLARK YH WING (ASPECT RATIO 6) AT THE SMALL SCALE OF FIG. 34.
CM «= pitching moment coefficient about a line J chord behind the leading edge.
in] WIND-TUNNEL EXPERIMENT 93
posed, and distinguishing normal values by suffix 0, by Bernoulli's theorem ^^ _ ^ ^ ^ _ ^ ^ ig<36 ft ^ ft^
giving V =s 185-5 ft. per sec.
Now / = diameter of wire = TV ft., v = 1-56 x 10~ * sq. ft. per sec. Hence :
giving, from Fig. 25, CD = M6
0-58 X 0-00238(185-5)' X 2
Drag ==
96
= 0-99 Ib.
(b) The lift coefficient of a wing of the section shown (known as R.A.F. 48) and span/chord ratio (aspect ratio) = 6, set at 15° incidence, is found in the wind tunnel to vary as in Fig. 37, through
FIG, 37, — APPROXIMATE SCALE EFFECT ON LIFT COEFFICIENT OF R.A.F. 48 WING AT 15° INCIDENCE.
the range of R given. A parasol monoplane fitted with a wing of this shape, chord = 6 ft., is required to approach a landing field situated at 6000 ft. altitude at the same incidence and 60 m.p.h. A.S.I, (indicated air speed). What lift will the wing exert when standard atmospheric conditions prevail ?
94 AERODYNAMICS [CH.
From Table III, Article 14, temperature is 6'1° C. and relative density 0*862.
p \ 273 / 0-00238 X 0-862
= 1*77 x 10 ~4 sq. ft. per sec. (cf. Article 25). 60 m.p.h. = 88 ft. per sec. and the true velocity
88
V = , = 94-8 ft. per sec.
Vo-862
_ 94-8 X 6 X 10*
R = - — - = 3-21 X 10*.
Hence, from the figure CL = 1-248, and, since wing area = (6 X 6)6 = 216 sq. ft.,
L = 0-624 X 0-00238(88)* X 216 = 2484 Ib. or 11-5 Ib. per sq. ft.
The following may also be verified. A scale model of 1 ft. chord would have, at 60 m.p.h. at sea-level, a Reynolds number = 0-65 X 106. Its CL would be 1-164, and it would lift 10-63 Ib. per sq. ft. or a total of 63-8 Ib. The same fairly low Reynolds number would apply to the full-scale wing held in a natural wind of 10 m.p.h., when the total lift would be the same and its mean intensity 0-296 Ib. per sq. ft.
62. Arrangement of Single Drag Experiment
Such complete data as in the last article are rare. Frequently the drag of some aircraft part is desired accurately only under particular conditions, e.g. at top speed at a certain altitude. From these specifications and the size of the part the full-scale Reynolds number can be calculated, and sometimes a single decisive test arranged in the wind tunnel under dynamically similar conditions.
Examples : the drags are required of the following aircraft parts exposed at A.S.I. = 150 m.p.h. at 10,000 ft. altitude : (a) a stream- line static balance weight of 2 in. diameter, (b) a long strut whose streamline section is 6 in. in length. Arrange suitable experiments in a 4-ft. wind tunnel at sea-level working at 50 ft. per sec.
The true relative velocity of the craft is, from Article 33,
150 88
X — « 256 ft. per sec.
VO-738 60
Assuming standard atmospheric conditions, the temperature = — 4-8° C. and, as in Article 61 (6), v is found to be 2-01 x 10 ~ *. For the tunnel 15° C. may be assumed, so that v = 1-56 x 10 -«.
WIND-TUNNEL EXPERIMENT
95
ni]
Models geometrically similar to the parts will be tested in the tunnel. Distinguishing experiment by suffix T and full scale by suffix F, for dynamical similarity we have JRT = J?F.
(a) Let d be the diameter of the model of the balance weight. Then
d X 50 (1/6) X 256
1-56
2-01
or
d = 0-662 ft. = 8 in.
The drag coefficient measured on a model