THE NATURE OF AIR 95
versions with the same aerofoil sections Fig 3.10 shows the differences in lift and drag coefficients that occur with Reynolds number with the same aerofoil section Thus, a narrow chord can cause a ‘peaky’ lift curve, with a sudden hard stall; but increase the chord and the stall characteristics are softened Furthermore, at a given angle of attack a broader chord can generate a higher lift/drag, bringing with it better cruise efficiency Litt high R low R Lift and drag Coefficie nts laminar drag ‘bucwee! / = Angle of atback
Fig 3.10 General effect of Reynolds number upon lift and drag characteristics of the same aerofoil surface
The disadvantages of narrow chords can be alleviated somewhat by the introduction of controlled turbulence, and vortices Fig 3.11a and c show vortex generators at a flap knuckle, to delay separation, anda vibrating rubber band in front of the leading edge of the wing of a model sailplane, to generate turbulence, so improving the flow
Trang 3|
THE NATURE OF AIR 97
Bats and many insects have thin membrane wings that are not efficient devices in themselves But such wings contain networks of fine blood vessels, or are covered with tiny hairs, which appear to cause favourable roughening at the Reynolds numbers involved, enabling the creatures to fly better and more efficiently than would otherwise be the case The bee, which has membrane wings, has to vibrate them before the first flight of the day to warm them up, increasing blood flow The bumble bee, in spite of a high and seemingly inefficient wing loading, nevertheless carries nearly half of its weight as payload through such natural boundary layer control
Table 3-1 compares the lift coefficients of a number of wings (based upon ref 3.3) with those of aeroplanes Birds flapping their wings generate lift more efficiently than many aeroplanes employing advanced high-lift devices
TABLE 3-1
Maximum Lift Coefficients of Animal and Aeroplane Wings (based upon ref 3.3) Maximum Cz Isolated locust forewing 1.3 Gliding birds 1.5 to 1.6 Gliding bats 1.5 Flapping birds up to 2.8 Basic aeroplane 1.3 to 1.6
The Frontispiece shows clearly a number of aerodynamic features discussed in this chapter The gannet was photographed while flying slowly A reversal of flow by the trailing edge vortex has lifted the downy feathers near the trailing edge of its wings to delay further separation The process is automatic The body of the bird is well cambered with tail and feet slowing the air underneath, increasing loca] pressure and lift, while also trimming out the nose-up pitching caused by the wings being swept forwards, which brings the lift ahead of the centre of gravity (a change of geometry which delays tip-stall by causing the boundary layer to drift towards the root, which is thus caused to stall first) A small slat, called the a/u/a (bastard wing) lying like a thumb along the leading edge of the wing at the structural wrist, has opened to deflect high pressure air from beneath the leading edge into the flow over the upper surface, again delaying separation
Just as down ruffles the airflow over the wing of the bird, causing favourable turbulence, so too are vortex generators attached to wings and other surfaces to draw in free stream air and wash away stagnating air carried along with the aircraft A vortex generator is a small plate aerofoil surface set at an angle of attack to the local flow, with its tip far enough away from the skin to protrude through the boundary layer An intense tip vortex is established with suction in its core, in an area of adversely rising pressure Each suction source opposes the stagnation of the flow, marked by rising static pressure, and acts like a broom which lifts air clinging to the skin to deposit it away in the free stream
Trang 498 THE DESIGN OF THE AEROPLANE
—— s›t>¬:““t= m=m=.e=.=.=nsiiriiililElElaEaEEaSESSsSlaSSAa0A0SaSaaaIBH
ae TAR een
AEOLIAN re sa ce
Plate 3-1 Streamlined flow breaking down into eddies and vortices: (Top) at zero angle of attack (note the stagnation streamline at the leading edge): (Bottom) at the stall (note the'way in which the stagnation streamline has migrated to a point below and behind the leading edge) (National Maritime Institute, UK)
Related by function if not form are dorsal fillets, fuselage, tailplane and other strakes Their quite highly swept leading edges, which must be relatively sharp-edged, generate favourable vortices which increase tail and fuselage damping, or which re- attach separating flows, at large angles of pitch and yaw (as when sideslipping and spinning)
Slats, slots and flaps
Trang 5
THE NATURE OF AIR 99
slot is one of the best known The lift curve is extended to larger angles of attack, while the increment gain in lift depends upon slat chord/ wing chord (one of 30 per cent more or less doubles Cumax)
All flaps are camber-changing devices of one form or another, although some also increase wing area Camber change is harder to visualise with a split flap, but it serves the purpose by pushing air downwards, increasing the pressure underneath, intensifying the circulatory component of flow around the wing leading edge The nose flap behaves very much like a slat, by introducing a strong camber near the leading edge The leading edge suction peak is reduced at large angles of attack, boundary layer is thinned and stall angle of attack is increased
Trailing edge flaps are the most obvious camber-increasing devices The increased slope of the top surface, towards the trailing edge (seen most clearly with the geometry of the plain flap in table 3-2) increases the potential swept volume to be filled by the surrounding air Large lift coefficients are achieved, but separation is hastened The flap increases the angle of attack of the basic section (an approximate truth which can be seen by drawing a straight line between the leading edge of the wing and the trailing edge of the flap) Many and varied flap forms are to be found, some employing slats and slots, so as to delay separation
Normal flying control surfaces are plain flaps (see chapter 12)
Plate 3-2 Wind-tunnel model of author’s Warren-winged S37-3, which needed roughened leading edges to cause representative attached flow at test Reynolds numbers It was impossible to induce a conventional stall and g-break with this wing, based on the work of Norman Hall-Warren (Shell Aviation News, UK)
Aerofoil section families
There have been many aerofoil sections designed in many countries Some of the more important have been grouped in rational families and series (usually having common
thickness distributions, but with variations of their mean lines and relative
Trang 6100 THE DESIGN OF THE AEROPLANE TABLE 3-2 Flap and slat characteristics tye x? L/D Description Profile Ch max “at at Cmac Reference Cumax Comox NACA
Basic aecsfoil Ctark Y C———_ 1.29] 15 7.5 |-o.o85|TN4S4
o.2c Plain fiab NACA deflected 45° ` 1.45 12 4o " TR427 ©.3c Stetted flab aa " NACA deflected 45° IN L8 | 12 4.0 TR427 O.3¢ Split fla NACA defiecked + =———>- 2.16] 14 4.32 |-o:25o|TN422
o.Be, hinged at 0.8e NACA
Split (Zab) Flap > 2.26 I3 4.43 |-o.3oo| TN422
deflecteq 45°
O.3c, hinged at 0.9¢ NACA
Split (Zap) Flap 2.32 Ì !2:5 | 4.45 |-o.285|TN422 deflected 45° = O.3¢ Fowler flap Cee NACA : Aeflecked 40° N 2.82 13 4.55 |-0.6b60 |rasa4 / Oc Fowler flap co |a oO : | ~© foséo|tas: @|JTR5A deflectea 4o® N OF I4 4 4
O3e Nese flap Based
deflected Bo%to 40° CC 2.09 28 4.0 - upon
ị Best for Shark cased Ret a0
|
Naca
| Fixed Stak forming Z—— TT I.77 24 S.25 — T427
aA SloE
Handley Page ¬ NACA
Automatic Silat (oe 1.34 1 28 | 41 - TN454
Fixed Slat and NAcA
O.Be Pla flap « 2.1% 19 3.7 — T8427
deflected 45°
Fixed Slot aud NACA
O.3¢ Slotted flap ⁄— ——Tw 2.26 I8 377 — TR427
deflected 45° N
NACA
Handley Page slat Coo
ana °.c Fowler flap £ 3.36 1G 3.7 |-0.740 | TN459 deflected 40° O.1e Kruger flap EsEixcee Cretracts backwards —_—_—_ 1.82 — — — forming LE profile) based on frighe test
Ole Kroger flab 3 Cesulks in
e.®< Fowlee flab £—————- 41 — _— _
defiecked FO? NV 8«c 3.1I
NACA References : aspect Tatio & ReynotdS number 609 000
Trang 7
THE NATURE OF AIR 101 a Basic Seckion Stalled
b Section with leading edge droop (orc flap) ak Same qugle of attack AS A, effeck of leading edge flops C>Z¬ orlaEs CL / trailing sdqae flab dowh
ant C Section With leading edge slak and trailing
⁄ edge Flap
° x
A Lift curves
Fig 3.12 The effect of leading edge slat, flap and trailing edge flap upon lift and angle of attack of basic wing section
Administration)) in the United States NACA sections are perhaps the most universally known and well used
Trang 8
¡02 THE DESIGN OF THE AEROPLANE
Air displaced faster than sound (supersonic)
At low airspeeds, around 100 knots at sea level (51.4 m/s), the dynamic pressure is only about 34 Ibf/ft? (1630 N/m?) compared with the sea level static pressure of 2116 lbf/ft2 (1.0132 X 10° N/ m2), ISA But at speeds faster than M 0.3 or so, displacement of the air is accompanied by quite dramatic changes Molecules are squeezed together Their pressure and density increase Calculations of dynamic pressure by means of eqs (2-9) and (3-4) cannot be carried out by simply inserting whatever value is found from tables for air density, p The change in density with compressibility causes us to add a percentage increment:
percentage increase in dynamic pressure, gq = 25 per cent M? + 2.5 per cent M4
= M2/4+ M*/40 (3-16)
which is shown in fig 3.13 Thus, at 66 knots the incompressible dynamic pressure is about the same as the compressible value, 10 Ibf/ft? (480 N/m2) But at 400 knots, the incompressible value would be 542 Ibf/ ft? (26 016 N/ m2), and the actual compressible value nearly 10 per cent more, at 596 Ibf/ ft? (28 608 N/ m2) A change of this magnitude causes marked changes in lift, drag and pitching moment
Very few light aeroplanes have been troubled by compressibility, because of inadequate engines But modern materials which enable designers to achieve advanced, high quality, low drag profiles; automotive technology which provides cheap flap, gear and other systems; and the appearance of several small jet engines, may lead té a potent new breed of aircraft within reach of the amateur builder and private owner
It is far easier to use lift, drag and pitching moment coefficients to describe the changes that occur with a wide range of operating speeds, altitudes and Reynolds numbers By doing so units largely disappear and arithmetic becomes easier We may add pressure coefficient, Cp, to this vocabulary, which enables us to describe the way in which pressures behave, in non-dimensional terms, with variations in airspeed and angle of attack
Pressure coefficient is defined as:
C» = pressure difference/dynamic pressure = (p — p ,,)/q oo
= Ap/q, (for static pressures) (3-17)
= (g~ đ,)/4 œ — (G/¢,,)—1 (in terms of dynamic pressures) (3-17a) and, knowing too that qg varies as V2, eq (2-9), then at speeds too low for compressibility:
» = (V/V,,)? — 1 (in terms of airspeeds) (3-17b) Now, look back to fig 3.6b (or fig 3.12) which shows suction and compression lobes around a wing The same picture can be drawn more usefully by representing the chord as a straight line, and then plotting suction upwards and compression downwards, in terms of C,, as shown in fig 3.14 Eq (3-17b) demonstrates that peak suction marks maximum local airspeed; peak compression, minimum local airspeed If the local speed of flow reaches the speed of sound, the molecules of air are unable to adjust in time to changes in pressure brought about by the surface contours Their behaviour is affected critically We speak, therefore, of subcritical and supercritical flows and their associated aerodynamics
Trang 10104 THE DESIGN OF THE AEROPLANE Bor ⁄ 3 ~ $b 25Ƒ ¬= oA Mw Su 2okE u 4 6 Vệ 5 Oo ISEF Ì——— s W ne lok Sv sv v3 L = về 5 vv QO =_ oO L a 1 + 9 0:2 0.4 ob og ho Mach number M Fig 3.13 Dynamic pressure is increased by compressibility of air, the percentage being approximately: M2 M4 1 + 20 (eq (3-16))
(derived from ref 3.5)
0.6 Although the aeroplane is flying subsonically the airflow has then reached sonic speed somewhere on the airframe (where C, is most negative) As the aircraft approaches the speed of sound progressively large regions of local flow achieve supersonic speed Other regions in which airflow is gathered up and carried along with the aircraft to a certain extent (making C, positive) are subcritical, and these may still be found when the aeroplane has reached M 1.2 or even 1.4, depending upon the magnitude of the positive C, peak
We say, therefore, that the aeroplane is flying transonically as long as there is a jumble of mixed subcritical and supercritical flows somewhere on its skin
When air is forced to supersonic speeds it objects, and tries to readjust to subsonic motion as quickly as possible If able, it does so with a sharp deceleration, characterised by a shock wave which, under normal flight conditions, shows as a violent rise in static pressure taking place in a distance of a few thousandths of an inch This is accompanied by an equally sharp rise in temperature Shock waves can be heard on the skin of an aircraft by microphones inside the wing Many readers will have heard the sonic bang of a supersonic aeroplane, when standing at a point on the ground where coalescing shock waves pass in the form of a strong pressure front
Trang 11THE NATURE OF AIR 105 gration ota , Straamline seeonati Ce | a Suction and Compression Suction - plotted in berms of Cp at various Stations along
upper Surfoce Ehe chord, Iw Ehis way
the pressure Aistributrons are Uniquue Eo one augle
of aktack , And Indeprtndent of Erue airspeed Ambient
Static pressure ana density Stag nat tow Pernts occur where Cp =O lower Surface Compare Wirth Figs 4 bb and Comabression 2.12 + Wing chord Pressure coefficient Cp b Litt, or crossforce, 1% brobortranal to the area between Ehe upper and lower surface pressure profiles ina, The
resulkant diagram is more of less Erianguiac, whieh Suggests that Lhe centre of pressure Ciks cenlroid)
lies about Ya chord aft
of the leading edge Lift oe crossforce , of ACp = CP wee CPupper © Le Wik chord Te
Fig 3.14 Conversion of pressure diagrams into lift distribution across chord (needed for structural design and stressing) These diagrams apply to an essentially slow-flying aeroplane with subcritical flow everywhere
To see how it all works, imagine a streamlined solid of revolution like that shown in fig 3.15, moving through the air like a javelin At very low speeds the sharp point merely nudges molecules gently, so that they are able to transmit their warning pulses at the local speed of sound in all directions Pressure waves radiate outwards in spherical ripples
Trang 12106 THE DESIGN OF THE AEROPLANE (==— Q) Body hacdly MoVIhg Myo —— (2) Speed abovt MO.5 , Shock ; wave | Zone
of () Speed MIO: body
Silence has Caught up with
iks OWN pressuce Waves Zone of
\ ACkION
bo < P
Fig 3.15a Generation of a normal shockwave at high speed after von Karman (ref 2.1) But as the solid body moves faster and faster, it gradually catches up with the pulses, crowding them together ahead of itself When the body travels at the speed of sound (M 1.0) warning of its advance cannot be transmitted ahead, because it is travelling with its waves, which coalesce to form a normal shock wave at right angles to the line of flight Ahead of the normal shock is an undisturbed region that von Karman calls the ‘zone of silence’ (ref 2.1) Behind it is his ‘zone of action’ In front of the shock are the ambient conditions of the free stream Behind the shock there has been a rise in pressure and temperature (kinetic heating), caused adiabatically by displacement, such that:
temperature rise AT = (true airspeed, mph/ 100)?°C (3-18) to within three parts in 1000 Alternatively, the temperature rise of the boundary layer is given by:
AT = 75 M2°F (3-18a)
Trang 13
THE NATURE OF AIR 107
@ Body moving about M 3.0 Angle 8
diameter of related Ea Mach
refecence fKumbec by M=cosec @
Cylinder, A (See Eq.(3- 19)) When
bhe cone augie is Small Nose Tad era a’ - < ree beay, om (2) Distribution of Cross-Sectional area Cress Sectional aren
Na? Of Femaining AIC in 4 Ce[aceuce eylrote forming a Convergent ~ divergen—e aucl ° Nose Tail Suction (3) Pressure Aistribution Ch ° ' ' values inereasing Comapressi Ị In Eq (2-14) + | values decreasing x duct duct | Contracting t | expanding
PAV = Lea vì = QAVv > pM - pAV = Constant, Eq ( 2-14)
Fig 3.15b | Generation of a conical shock wave by a body at supersonic speed The reference cylinder enables us to imagine the disturbance isolated from the remaining air Note what is happening within eq (2-14)
At supersonic speeds the body travels faster than the pressure pulses, and a conical shock wave is formed (containing the zone of action) If the half-vertex angle of the cone subtends an angle @ deg it may be shown that when @ is small:
a/V = sin@=1/M (from eq (2-51))
M = cosec 0 (3-19)
Now, if we draw a reference cylinder to just contain the shock cone and the body at any instant, we can see clearly that the air within it is squeezed by the passage of the body volume The resulting pressure distribution is caused by:
O Distribution of body volume from nose to tail (i.e distribution of the cross-sectional areas) which is intimately related to the changing s/ope of the body surface along its length
Trang 14
i
108 THE DESIGN OF THE AEROPLANE
As the slope of the surface, velocity and pressure distribution are all linked, for a given set of boundary layer conditions, the aerodynamicist has a powerful tool for calculating idealised shapes needed to produce best lift/drag at different design Mach numbers This is the method of area-ruling an aircraft from nose to tail, so that the curve of total cross-sectional areas (i.e including wings, tail surfaces, canopy, engine ducts and fairings) approximates to that of a solid of revolution generating minimum drag at the same speed
There is an important difference between sub- and supercritical flows that is worth remembering The key lies in eq (2-14), which must hold for continuity because flow is neither lost nor gained, regardless of what is done to it during displacement The changes in air density brought about by compressibility affect the exchange rate between cross-sectional area of a given streamtube and the flow velocity along it In fig 3.15d the cylinder is a gross streamtube, from which e and f are derived Thus, subsonically:
t t incompressible (constant density): 4 V=A V
- and we see that a contraction in cross-sectional area squeezes the flow to higher speed,
and vice versa But, when the air is compressible, as when supersonic, its density
changes and the following situation occurs:
t t
compressible (density variable): p A y =pA vy |
so that a decrease in cross section chokes and slows the flow; and an opening out of the
cross section expands and accelerates it
The effect of such behaviour is shown in fig 3.16, which represents pressure distributions over the upper surface of a wing at subsonic and supersonic speeds In the transonic regime the wing would suffer an indeterminate mixture of the two
The general speed range applicable to design data is: low speed M&S 0.2
subsonic M = 0.2 to 0.9 transonic M = 0.7 to 1.5 supersonic M2 1.0 Area-ruling
In recent years much work has been done to avoid the worst effects of mixed flows In the 1950s Dr Richard T Whitcomb of the (then) NACA Langley Research Centre in the USA developed his area-rule concept, by means of which he fattened and thinned fuselage cross sections from nose to tail, and added bulged fairings where required, to smooth the contours of the enclosed volume The method is now part of design history, because of the dramatic change in performance bestowed on the Convair YF-102 Before area-ruling, the aeroplane generated too much drag to fly supersonically After area-ruling, the aeroplane went supersonic in a climb
Similar work proceeded towards the same ends in Britain, at the Royal Aircraft Establishment, Farnborough There Professor Dietrich Kiichemann (1911-1976), with others, approached the problem from a slightly different angle, but with much the same result
Trang 15THE NATURE OF AIR 109 Suction Compression Subsonic + (siowec Ehan about Mob) a ms bhickness _ ——- Suctio w Compress iow Supersonic Vth drag-broduci hg regions
[ ChrosE producing ceaìoeS
Fig 3.16 Typical pressure distributions over an aerofoil at subsonic and supersonic speed
special aerofoil sections evolved by RAE, and Hawker Siddeley Aviation (now part of British Aerospace), which have rear-loading and flat (roof-top) pressure distributions Such sections have flatter curves on their upper surfaces, extending further aft, and deep curving bellies which sweep upwards and become concave towards the trailing
Trang 16110 THE DESIGN OF THE AEROPLANE
Plate 3-3 Natural control of lift and drag through control of the boundary layer is a feature of many creatures, including bats,
owls, beetles, flies, moths In the sea cetaceans (€.g., porpoises (often called dolphins)) can reduce drag by subtle changes of cross- sectional area to keep flows attached (Left) Blood vessels in the wings of a worker bee stabilise the membranes and turbulate the flow (Right) This Small Elephant Hawk Moth is covered with fine down and has dusty wing scales, which help to turbulate and attach the flow, aiding silent flight (Natural History Photographic Agency, UK)
Trang 17THE NATURE OF AIR III
Plate 3-3 (Right)
However, rear-loading moves the centre of lift of the wing about 10 per cent of the chord further aft, which causes the wing to be mounted further forwards on the fuselage Furthermore larger, stronger and heavier elevators are needed to cope with the pitching moments Loading of the rearward portion of such wings necessitates stronger structures toward the trailing edge, but as thicker wing sections of less area can be designed, this does not necessarily lead to heavier wings There is often less room for fuel inside such smaller wings
We have reached the end of a chapter that started by talking about the nature of air, and has ended by bringing us face to face with the much wider issue of arranging and shaping surfaces to make use of air in a disciplined and economical way
This is the threshold of aeroplane design
Trang 19THE NATURE OF AIR
Plate 3-4(Top) Schlieren photograph of double wedge aerofoil section at Mach number 1.8, showing dark leading and trailing edge shock waves, with light coloured expansion waves at the crests in between (Jmperial College, Department of Aeronautics,
UK)
Plate 3-4 (Bottom) Handley Page Victor with ‘Kiichemann carrots’ at wing trailing edges, to provide rudimentary favourable area-distribution, so reducing wave drag and unpleasant transonic side-effects (John Fricker, UK)
Trang 20114 THE DESIGN OF THE AEROPLANE
Plate 3-5 Rutan Model 72, Grizzly, explores a more extreme tandem configuration (like a Warren-Wing with the foreplane reversed) The tail appears to bestow control authority in pitch which the $3/-3 lacked (see plate 3-2) (Rutan Aircraft Factory, USA)
References
3.1 Sutton, O G (1949) The Science of Flight Harmondsworth, Middlesex: Penguin Books
3.2 Bede Aircraft, Inc Bede Design No 7 (Jan 1971) Sport Aviation, Wisconsin: Experimental Aircraft Association International
3.3 Pennycuick, C J (1972) Animal Flight (Studies in Biology No 33) London: Arnold
3.4 Perkins, C D and Hage, R E (1967) Airplane Performance Stability and Control New York, London and Sydney: John Wiley & Sons, Inc
3.5 Dommasch, D O Sherby, S S and Connolly, T F (1951) Airplane Aerodynamics New York, Toronto and London: Pitman
3.6 Abbott, I A and von Doenhoff, A E (1959) Theory of Wing Sections, Including a Summary of Airfoil Data New York: Dover Publications Inc
3.7 NASA TN D7428 (Dec 1973) Low Speed Aerodynamic Characteristics of a 17- Per cent-Thick Airfoil Section Designed for General Aviation National Aeronautics and Space Administration
3.8 NASA TMX-72697 (1973) Low Speed Aerodynamic Characteristics of a 13-Per cent Thick Airfoil Section Designed for General Aviation National Aeronautics and Space Administration
3.9 Sunderland, L D (1977) ‘What’s New in Low Speed Airfoils? Sport Aviation, Experimental Aircraft Association
3.10 Lee, G H (1953) ‘High Maximum Lift’, The Aeroplane
3.11 Kohlman, D L (1979) Flight Test Results for an Advanced Technology Light Airplane Journal of Aircraft 16 (No 4), American Institute of Aeronautics and
Trang 21CHAPTER 4
Arrangement of Surfaces
‘There can be no doubt that the inclined plane, with a horizontal propelling apparatus, is the true principle of aerial navigation by mechanical means and there is nothing new in it, the principle has as yet remained dormant, for want of sufficient power.’
Sir George Cayley (1843)
(From ‘Sir George Cayley’s Aeronautics 1796-1855’, Charles H Gibbs-Smith, Her Majesty’s
Stationery Office.)
‘We shall use results as long as they have not been refuted.’ Professor Dietrich Kitichemann (ref 4.1)
In the last chapter we looked at airflows moving essentially in two dimensions, in the plane of the paper Here we consider them moving in three dimensions Lift is derived from a cross-force on an aerofoil surface, and this in turn involves circulation, in the form of a major, bound vortex (fig 3.5d(2)) The bound vortex lies along the span of the aerofoil, almost from tip to tip, although it is not of constant strength along its length Near the tips the vortex bends around like a horseshoe and trails downstream in the wake of each, where it can sometimes be seen when air is moist, or heard in the hiss and whisper of collapsing vortices after a low-flying aeroplane has passed, plate 4.1 Circulation along the span is hard to visualise unless helped by a camera The closest the pilot comes to physical awareness of the presence of this strange phenomenon which both supports and retards him, is through changes in stick force needed to trim out the pitching moment from the lifting surfaces when flap is selected, or when attitude is changed Even then the force is an amalgam of many other complicated pitching effects
Strength of circulation is measured by the product of velocity and length of the curved path over which the velocity applies:
I = Tangential velocity X circumference of curve around which flow moves
which, for a purely rotational motion around a vortex core, at any radius outside the
core, becomes:
T=27 Rv ft2/s (m2/s) (4-1)
This equation tells us that when circulation is constant, the tangential velocity, v, varies
inversely as R, the radius, as we move outwards from the core
Trang 22
116 THE DESIGN OF THE AEROPLANE
Wingspan: airflows in three dimensions
If we look up the wake of a rear-engined jet airliner that has passed overhead on its approach to land, we sometimes see a pattern of tip vortices and downwash traced in smoke, like that shown in fig 4 la The spiral flows behind both tips, and the entrained downwash between them, are contained within an approximate cylinder of air which varies in diameter (depending upon how far it is behind the aeroplane), but which is more or less equal to the wingspan Each tip vortex has a Rankine form (see footnote to table 2-5) consisting of a central core in which the rotational velocity increases with radius, like a wheel Outside the core is a region in which velocity decreases as 1/R, as we have just seen
The cylinder represents the mass of air acted upon by the aeroplane in flight Early
ideas were that the mass of air was not cylindrical, but an ellipse, about 1.2 times wider
than deep But Frederick W Lanchester (1878-1946), a profoundly original English
Plate 4-1 Tip vortices shed by a wing can contain such intense suction as to condense water vapour into cloud The near-elliptic
Trang 23ARRANGEMENT OF SURFACES 117
Plate 4-2 Jim Bede in his BD-5, which is twice as long as the reclining pilot and is as quick reacting as a cat It caused great
excitement when it appeared but has suffered accidents through use of unsuitable engines by enthusiasts, as a result of several specified engines failing to materialise in time The author found it an aeroplane to treat with caution and respect (Howard Levy)
engineer, and Ludwig Prandtl (1875-1953), a German engineer at Gottingen, who brought incisive insight vision and analysis to fluid mechanics, assumed that a wing entrained a cylinder of air equal in diameter to the span In fact, the mass of air appears to have a vena contracta form, sketched in fig 4.1c, with a cylinder diameter V2 times the wingspan, at the wing; and a diameter somewhat less than the span some way behind in the wake This is borne out by the way in which tip vortices draw themselves together in the wake, until they stabilise around wingspan/x/2 to 7r /4% wingspan (about 3⁄ of the span apart) Their form is shown by vapour trails behind high flying aeroplanes, or in smoke trailed from wing-mounted canisters
Ahead of an aeroplane the air is sensibly unaffected In the plane of the wing the cylinder, with a diameter \/ 2 times the span, and length equal to the distance flown in one second, is given an impulse, a downwash velocity, w» Some way behind the wing
the downwash has a final value, w Now, by Newton’s second law:
lift = rate of change of momentum of cylindrical mass of air of density, p = mass of cylinder X downwash velocity
=n X weight of aircraft (4-2)
Trang 24
118 THE DESIGN OF THE AEROPLANE
Q The mass of air Worked ou by a Wing haS a more orless Cirtular Cress-section With a Aiameter roughly equal to the Span (see Eq (4-2) Each lobe Contains a vortex Skein from a Wikg kip Between theniaga
Cegion of dowhwash of entrained
air QublSide themisa cotresbondina obwash The energy in the motion of fhe air 18 Probockional bo the
Qnerg y needed bo sustain the Weight of the aeroblane Thus the wake of a heavy bransport flying Slowly, ora helicopter, ig” most dangerous Eo light aircraft,
b, Each trailing vertex haga Rankine form , giving a Spauwise dow n— Wash disttibution tke Eha Shsuse, x
(Only the rare el peical Hifl dige — tribution accoss the sban
gives Constant downwash ‘between the vortex cores) The average down-—
wash velocity at the wing is
about Sft/s (1.Sm/s) fora Nght
Aeroblahe , anol tl ft/s (Su/s)
+
`
fora big jet Eranspert Downwash fac downstreaaw mm Fhe Wake | LỘ overage OW Kno may have double Ehes@ values
fright
‘on of Cc The mass of Ol ShownR rR a
f2S@umnbles a cylinder in elevation WiEK A Vena Coutracta form ` \downwash "^eadle,€ `
diametes & /2 shan
odiawmeter & shaun
Fug 4.1
Fig 4.1 The working mass of air acted upon by a wing in | second (after Lanchester
and Prandtl)
(when lift is doing all of the work in sustaining the aircraft)
Trang 25
ARRANGEMENT OF SURFACES 119
Thus, taking a cylinder of diameter equal to /2 X span of the wing, b:
lift, L = (7 /4)(\/2b)°p Vw = nW (4-2a)
and downwash velocity, w = (2n/mpV)(W/b2) = (2/mpV)(L/b?) (4-3)
The angle of downwash, a: in some notations ¢ in others, especially when we are concerned with the local flow at the tail, is the angle subtended by downwash velocity / TAS w/ V, such that:
w/V = tan e deg ~e radians =(1/7q)(W/b?) (4-4) or, knowing that nW = lift =CiqS (from eq (2-3a)), we may also write:
c= CuS Jxb3 = (1lJ nạ)(LJb3) (4-4a)
And, as aspect ratio A = b2/S (from eq (2-4)):
¢ = Ci/ 7A radians = tan e deg (4-4b) When multiplied by 57.3 we obtain the angle in degrees
If we argue in exactly the same way about the downwash of a cylinder of air in the wake of the wing, of diameter equal to the wingspan, b instead of \/ 2b, we find that the downwash velocity two to three wing chords behind at the tail is roughly doubled, to 2w (which is the theoretical value reached at infinity, although viscosity reduces it to zero long before that) The reason is that at the wing we have downwash caused in the main by the bound vortex, but behind in the wake there are the additional effects of the tip vortices, each providing their own impulses, fig 4.2a, which increase the downwash angle by a further increment € to 2e
Immediately ahead of the wing there is an upwash, as shown in fig 4.2b, caused by the bound vortex This must be taken into account when calculating local angle of attack of a surface lying forward of a wing
The existence of downwash means that an aeroplane has to fly uphill aerodynamically, so as to maintain altitude The shorter the span, the steeper the aerodynamic hill for a given wing area and weight It follows that greater effort is required of the propulsion system to fly level, when a wing is short rather than long Short spans are inefficient in these fuel-conscious times (in fact, we might say that they are positively antisocial within the context of most aircraft with which we are concerned) We only use short spans for special applications, like racing, which is quite a different matter from flying for range and fuel-efficiency But one cannot extend the span of a wing indefinitely, because we run into the problem of its growing structural weight to resist bending, and the adverse effect upon the size of the fin and rudder, which must become larger and heavier in consequence of problems with directional stability and control
Downwash reduces the geometric angle of attack as shown in fig 4.2c, by the anglee, such that:
Qo —=ằẲœ_-€ (4-5)
It also has the effect of bending backwards the lift vector through an angle e, (or a),
which introduces an induced component, D, which is also referred to as a vortex drag
component by some sources The downwash angle is usually small enough for us to be able to say:
¢ in radians © tan e deg ~ sin e deg (4-6)