Appendix AUnits and Dimensions

It is essential to distinguish between units and dimensions. Broadly speaking,physical parameters have the separate properties of size and dimension. Aunit is a, more or less arbitrarily defined, amount or quantity in terms ofwhich a parameter is defined. A dimension represents the definition of aninherent property, independent of the system of units in which it is expressed.For example, the dimension mass expresses the amount of material of whicha body is constructed, the distance between the wing tips (wingspan) of anaeroplane has the dimension length. The mass of a body can be expressed inkilograms or in pounds, the wingspan can be expressed in metres or in feet.

Many systems of units exist, each of which with their own advantagesand drawbacks. Throughout this book the internationally accepted dynamicalsystem SI is used, except in a few places as specially noted. The Imperial setof units still plays an important roll in aviation, in particular in the UnitedStates.

Fundamental dimensions and units

Dimensions can be written in symbolic form by placing them between squarebrackets. There are four fundamental units in terms of which the dimensionsof all other physical quantities may be expressed. Purely mechanical rela-tionships are derived in terms of mass [M], length [L], and time [T]; thermo-dynamical relationships contain the temperature [θ] as well.

A fundamental equation governing dynamical systems is derived fromNewton’s second law of motion. This states that an external force F actingon a body is proportional to the product of its mass m and the acceleration a

produced by the force: F = kF ma. The constant of proportionality kF is de-

511

512 A Units and Dimensions

Table A.1 Dimensions and SI units used for dynamical systems.

Quantity Dimension Unit name Symbol Explanation

length [L] metre m fundamental unitmass [M] kilogram kg fundamental unittime [T] second s fundamental unitarea [L2] – m2 length×lengthvolume [L3] – m3 area×lengthvelocity [LT−1] – m s−1 length/timeacceleration [LT−2] – m s−2 velocity/timemoment of inertia [ML2] – kg m2 mass×areadensity [ML−3] – kg m−3 mass/volumemass flow rate [MT−1] – kg s−1 mass/timeforce [MLT−2] Newton N, kg m s−2 mass×accelerationmoment [ML2T−2] – N m force×lengthpressure, stress [ML−1T−2] Pascal Pa, N m−2 force/areamomentum [MLT−1] – N s, kg m s−1 mass×velocitymomentum flow [MLT−2] – N, kg m s−2 mass×velocity/timework or energy [ML2T−2] Joule J, N m force×lengthpower [ML2T−3] Watt W, N m s−1 work or energy/timeangle 1 radian rad length/lengthangular velocity [T−1] – rad s−1 angle/timeangular acceleration [T−2] – rad s−2 angular velocity/timefrequency [T−1] Hertz Hz 1/time

termined by the definition of the units of force, mass and acceleration usedin the equation. In general, if the system of units is changed, so also is theconstant kF . It is useful, of course, to select the units so that the equationbecomes F = ma. In a consistent system of units, the force, mass, and timeare defined so that kF = 1. For this to be true, the unit of force has to be thatforce which, when acting upon a unit mass, produces a unit acceleration.

International System of Units

In most parts of the world the Système International d’Unités, commonlyabbreviated to SI units, is accepted for most branches of science and engi-neering. The SI system uses the following fundamental units:

Flight Physics 513

• Mass: the kilogram (symbol kg) is equivalent to the international stan-dard held in Sèvres near Paris.

• Length: the metre (symbol m), preserved in the past as a prototype, ispresently defined as the distance (m) travelled by light in a vacuum in299,792,458−1 seconds.

• Time: the second (symbol s) is the fundamental unit of time, defined interms of the natural periodicity of the radiation of a cesium-133 atom.

• Temperature: the unit Kelvin (symbol K) is identical in size with thedegree Celsius (symbol ◦C), but it denotes the absolute (or thermody-namical) temperature, measured from the absolute zero. The degree Cel-sius is one hundredth part of the temperature rise involved when purewater is heated from the triple point (273.15 K) to boiling temperatureat standard pressure. The temperature in degrees Celsius is thereforeT (C) = T (K) − 273.15.

Having defined the four fundamental dimensions and their units, all otherphysical quantities can be established, as in Table A.1. Velocity, for exam-ple, is defined as the distance travelled in unit time. It has the dimension[LT−1] and is measured in metres per second (m s−1, or m/s). The followingadditional remarks are made in relation to Table A.1:

• The SI system defines the Newton (symbol N) as the fundamental unitfor force, imparting an acceleration of 1 m s−2 to one kilogram of mass.From Newton’s equation, its dimension is derived as [MLT−2]. By con-trast to some other systems of units, the definition of a newton is com-pletely unrelated to the acceleration due to gravity. Clearly, the SI systemforms a consistent system.

• The fundamental unit of (gas) pressure or (material) stress is denotedpascal (symbol Pa). The bar is defined as 105 Pa, the millibar1 (mb)amounts to 102 Pa. A frequently used alternative unit of gas pressure isthe physical atmosphere (symbol atm), which is equal to the pressure un-der a 760 mm high column of mercury: 1.01325 × 105 Pa. The standardatmosphere is set at an air pressure of 1 atm at sea level. The techni-cal atmosphere (symbol at) is equal to the pressure under a 10 m highcolumn of water, g × 104 Pa. This requires a definition of the accelera-tion due to gravity, which is taken as the value at 45◦ northern latitude:g = 9.80665 m s−2.

• The (dimensionless) radian is defined as the angle subtended at the centreof a circle by an arc equal in length to the radius. One radian is equal to180/π = 57.296◦.

1 The preferred symbol is the hectopascal, hPa.

514 A Units and Dimensions

Fractions and multiples

Sometimes, the fundamental units defined above are inconveniently large orsmall for a particular case. In such cases, the quantity can be expressed interms of some fraction or multiple of the fundamental unit. A prefix attachedto a unit makes a new unit. The following prefixes may be used to indicatedecimal fractions or multiples of SI units.

Fraction Prefix Symbol Multiple Prefix Symbol

10−1 deci d 10 deca da10−2 centi c 102 hecto h10−3 milli m 103 kilo k10−6 micro µ 106 mega M

Imperial units

Until about 1968, the Imperial (or British Engineering) set of units was inuse in some parts of the world, the United Kingdom in particular. It uses thefundamental units foot (symbol ft) for length and pound (symbol lbm) formass, the unit for time is the second. The corresponding unit for force, thepoundal, produces an acceleration of 1 ft s−2 to 1 lbm. The Imperial Systemis therefore a consistent one. Since the poundal is considered as an unpracti-cally small force, it is often replaced by the pound force (symbol lbf), whichis defined as the weight of one pound mass. The pound force is therefore g

times as large as the poundal. However, used with 1 pound mass and 1 ft s−2,it does not constitute a consistent set of units. Therefore, the slug has beendefined as a mass equal to g times the pound mass, dictating that a standardvalue is used for the acceleration due to gravity (32.174 ft s−2). The Imperialsystem uses the Kelvin or the degree Celsius (“centigrade”) as the standardunit of temperature.

Although the SI system constitutes the generally accepted internationalstandard, many Imperial units are still in use, especially in the practice ofaircraft operation and in the US engineering world. For example, use is stillmade of the temperature scales Fahrenheit (F) and Rankine (R). The Rankineis an absolute temperature coupled to the Fahrenheit scale and is not to beconfused with the former Réaumur temperature unit. The conversion fromdegrees Fahrenheit to Kelvin is as follows: T (K) = 273.15 + 5/9{T (F) −32}. The system of units based on the foot, pound, second and rankine is

Flight Physics 515

Table A.2 Table for converting British FPSR units into SI units.

Quantity Symbol Multiply by to obtain SI units

length inch (in) 2.54 × 10−2 mfoot (ft = 12 in) 3.048 × 10−1 mmile 1.6093 kmnautical mile (nm) 1.8532 km

volume cubic ft 2.8317 × 10−2 m3

UK gallon 4.5461 × 10−3 m3

US gallon 3.7854 × 10−3 m3

velocity ft/s 3.048 × 10−1 m/smile/h 1.609 km/hUK knot = nm/h 1.853 km/h

mass slug 1.4594 × 10 kgpound mass (lbm) 4.5359 × 10−1 kgUK ton 1.0165 × 103 kgUS short ton 9.0718 × 102 kg

force pound force (lbf) 4.4482 Npoundal 1.3826 × 10−1 N

pressure lbf/in2 (psi) 6.8948 × 103 Palbf/ft2 (psf) 4.7880 × 10 Pa

temperature Rankine (R) 5/9 Kwork ft lbf 1.355 Nmenergy BTU 1.055 × 103 Jspecific energy BTU/slug 7.2290 × 10 Nm/kgpower slug ft2/s3 1.356 Nm/s

horsepower∗ (hp) 7.457 × 102 Wviscosity coefficient slug/ft/s 4.788 × 10 kg/m/skinematic viscosity ft2/s 9.290 × 10−2 m2/s

∗The unit of power in the (former) Technical System of Units is also known as the(metric) horsepower. It was derived from the kilogram as a fundamental unit forforce (kgf). Its value of 735.5 W is marginally smaller than the horsepower of theImperial system.

sometimes called the FPSR system. If their units are used in engineeringcomputations, it is recommendable to convert them into SI units with thehelp of Table A.2.

Appendix BPrinciples of Aerostatics

Ballooning originates from the early 18th century, and it is the oldest – andfor more than a century the only – form of aviation; see Sections 1.2 and 2.2.Despite recent competition from (more expensive) satellites, scientific andmeteorological balloons have preserved their place, while the popularity ofrecreational ballooning continues to grow. Because the physical principlesof ballooning form a clarifying illustration of the equation of state, someattention will be paid in this appendix to aerostatics.

Gross and net lift

From the equilibrium of a volume element of air in a static atmosphere, wederived in Section 2.6 that the pressure on the upper side of the element islower than on the lower side. This pressure difference is compensated by theweight of the air contained by the element and it is still present if the elementis replaced by an arbitrary body with the same geometry. The atmospherewill therefore exert a force on the body equal to the weight of the removedair. Using the aerostatic equation, we have thus given an explanation of thefamous law of Archimedes (287–212 BC). Applying this law to a balloonwith a volume Q, it says that the gross lift LG exerted on the balloon is equalto its volume multiplied by the specific weight of atmospheric air

LG = watQ = ρat g Q, (B.1)

with w and ρ denoting the specific weight and the density, respectively, ofthe atmosphere (index at). The weight of the internal lifting gas (index gas),forming the contents of the balloon, has to be subtracted from the gross liftto obtain the net lift LN ,

517

518 B Principles of Aerostatics

LN = LB − Wgas = g Q(ρat − ρgas). (B.2)

The net lift is positive if ρgas < ρat. To comply with this condition the fol-lowing methods can be distinguished:

1. The balloon is filled with hot air. Because the air pressure in the balloonexceeds the ambient pressure only marginally, the difference betweenthe densities of the atmosphere and the hot air follows directly from theequation of state.

2. The balloon is filled with a gas which is “lighter than air”, in other words,the lifting gas – such as helium (He) – has a smaller molecular mass thanair. Hydrogen (H2) is the lightest gas but, in view of its high flammability,it is no longer used in manned balloons.

Hot-air balloons

A hot-air balloon has an inlet opening at the bottom so that the internal airpressure is equal to the ambient air pressure: pgas = pat. Lift results from thedifference in density between the hot internal air and the atmospheric air. Theinlet air is heated by means of a (LPG) gas burner flame below the opening.The gas is burnt intermittingly to control the average internal temperature.

The temperature difference between the hot air and the atmosphere �T =Tgas − Tat, is used to rewrite Equation (B.2) as follows:

LN = ρat g Q

(1 − ρgas

ρat

)= ρat g Q

(1 − Tat

Tgas

)= ρat g Q

�T

Tat + �T.

(B.3)By varying the gas burner heat added the value of �T is adjusted, making theballoon to ascend or descend. Equation (B.3) shows that the net lift largelydepends on the atmospheric air temperature. For example, we assume a bal-loon to be launched at an outside air temperature of 17◦C, and the inside airto be heated by �T = 80◦C. For an atmospheric density ρat = 1.25 kg/m3 itis found that LN = 2.65Q. At sea level the balloon will lift 2.65 N per cubicmetre of hot gas. However, if this balloon were to be launched on a hot daywith an ambient temperature of 37◦C and the same ambient pressure, we thenfind �T = 60◦C for the same hot air temperature, and ρat = 1.17 kg/m3 forthe ambient density, Equation (B.3) now indicates that the net lift per cubicmetre is merely 1.86 N or 30% less than for the earlier case. If the balloon’sempty weight is assumed to be the same in both cases, then the availableuseful load is reduced by the same 30%. Such a significant temperature de-

Flight Physics 519

pendence must be thoroughly taken into consideration when preparing for ahot-air ballooning flight.

Gas balloon

As for a hot-air balloon, the pressure of the lifting gas in a gas balloon isapproximately equal to the ambient pressure.1 By contrast, the lifting gastemperature is not much different from the outside air temperature, thoughit may be heated up appreciably by the sun, or cooled down when the bal-loon drifts below clouds. The equation of state dictates that the lifting gasdensity ρgas and the atmospheric air density ρat have a ratio similar to themolecular masses,

ρgas

ρat= Rat

Rgas= M̂gas

M̂at

. (B.4)

The net lift can be expressed according to Equation (B.2) either proportionalto the gas volume

LN = ρat g Q

(1 − ρgas

ρat

)= ρat g Q

(

1 − M̂gas

M̂at

)

(B.5)

or to the gas weight,

LN = ρgas g Q

(ρat

ρgas− 1

)= Wgas

(M̂at

M̂gas

− 1

)

. (B.6)

According to Equation (B.5), the net lift at sea level for a balloon filled withhelium gas (M̂ = 4) amounts to about 10 N per cubic metre. For an arbitrarygas volume, the lift is proportional to the ambient density and therefore de-creases at higher altitudes. Conversely, the net lift for a constant gas weightaccording to Equation (B.6) is also constant. By using the previous relation-ships, the altitude control of a gas balloon will be explained hereafter.

Open gas balloon

The gas in an open balloon is in contact with the surrounding atmospherevia a nozzle at its bottom, which is permanently open during flight. In level

1 Some gas balloons can accommodate a significant overpressure which allows them to attainan altitude up to 40 km without tearing. Their skin is manufactured from an extremely lightmaterial reinforced with high-strength fibres.

520 B Principles of Aerostatics

flight the net lift and the balloon’s weight Wb, including the useful load, are inequilibrium: LN = Wb. When a balloon ascends, ρat decreases and, becausethe (fully inflated) volume remains the same, gas escapes from the balloon sothat ρgas also decreases. According to Equations (B.5) and (B.6), the net liftdecreases so that LN < Wb. This counteracts the ascending motion and helpsthe balloon to maintain a steady rate of ascent. Conversely, in a descendingmotion, the gas weight is kept constant and the balloon is allowed to take onatmospheric air which does not contribute to the lift. For a constant amountof gas, the lift stays constant and – apart from the air drag on the balloon –the descending motion is not counteracted. An open gas balloon is thereforeindifferent to the rate of descent, which can only be reduced by off-loadingballast (sand). A fast descent – for example, while landing – can be executedby opening a gas valve at the top of the balloon.

Closed gas balloon

During its launch, a closed balloon will only be partially filled with gas, sothat the net lift is marginally greater than the weight: LN > Wb. Initially theballoon will ascend with constant acceleration, though the increasing speedwill magnify the air drag and cause the acceleration to reduce. After a while,the balloon will ascend at a steady rate. Due to the decreasing air pressure,the balloon will begin to expand until it becomes fully inflated. To preventthe balloon from tearing open, the gas valve is opened and the ascendingflight is continued as an open balloon, until the altitude limit is reached, asexplained below.

Ceiling of a gas balloon

Open balloons are used in ballooning sport at relatively low altitudes. Bycontrast, the purpose of closed balloons is to reach high altitudes, often pen-etrating the stratosphere. The ceiling of a closed balloon is reached whenthe net lift equals the balloon’s all-up weight. Expressed as the minimumatmospheric density achievable, this is determined by Equation (B.5)

ρat = Wb

g Qmax(1 − M̂gas/M̂at). (B.7)

For example, let us assume that we have a balloon with a volumeQmax = 576,000 m3 and a mass of Wb/g = 2,000 kg. Using M̂gas = 4 and

Flight Physics 521

M̂at = 28.96, we derive the density altitude at the ceiling from Equa-tion (B.7): ρat = 0.0040 kg/m3. According to the data for the standard at-mosphere (Section 2.6), the corresponding altitude is approximately 40 km.

Index

accelerate-stop distance, 310Ackeret, J., 476active controls, 79actuator disc, 197, 207, 237, 238, 240,

408Ader, C., 14adiabatic flow, 438adiabatic process, 438advance angle, 236, 242advance ratio, 242, 245, 418, 421adverse yaw, 19, 392aero-elasticity, 332, 475aerobatics, 329aerodynamic balance, 379aerodynamic centre, 351–353, 356, 360,

381, 435, 463, 478, 483aerodynamic efficiency, 176, 268, 276,

300, 317, 343, 369, 435, 497aerodynamics, 8, 15, 19, 57, 90, 103,

127, 160, 267, 438aerofoil, 13, 52, 79, 126, 130aerofoil section, 56, 91, 100, 117, 126,

128, 131, 133, 142, 145, 155,158, 236, 434, 459

aeroplane, 52aerostatic equation, 70, 96, 517aerostatics, 49, 517afterburning, 39, 89, 194, 221, 488, 493aileron, 23, 305, 346

differential, 392Frise, 392

high-speed, 475spoiler, 475

aileron reversal, 392air, 62, 65

intake, 409air brake, 59, 287air breathing engine, 61, 182, 196air density, 49, 89, 112, 185, 212, 233,

259, 268, 284, 309, 313, 317,370, 460, 499, 508, 519

air intake, 61, 80, 92, 185, 193, 198,215, 217, 231, 440, 475, 488–490,494

air traffic control, 77air-cooled engine, 185airborne distance, 308, 313airborne phase, 308aircraft

canard, 19, 343, 346, 367, 370, 376fixed-wing, 8, 52high-wing, 28hypersonic, 88jet, 38low-wing, 30propeller, 33, 40subsonic, 88supersonic, 88tail-first, 343tailless, 133, 343, 360, 371, 486tilt-rotor, 406, 422tilt-wing, 54

523

524 Index

transsonic, 88aircraft configuration, 257, 323, 330,

332, 343airship, 5, 6, 11, 36, 49–51, 183airspeed, 52, 258

calibrated, 461equivalent, 258, 460indicated, 258, 461maximum, 53, 266, 273, 275–277,

281, 420, 501minimum, 151, 266, 276, 280, 303,

312, 394true, 258, 278, 460

airspeed indicator, 258, 461Alembert, J. le Rond d’, 102all-flying tail, 346, 347, 382altimeter, 76, 259altitude, 68, 69

critical, 212, 235density, 76, 508energy, 286geopotential, 73pressure, 76, 259scale, 72

altitude sickness, 81angle of attack, 8, 33, 58, 135, 136, 237,

260, 336absolute, 136critical, 151, 152, 171, 267, 304, 320,

395angle of sweep, 466anhedral, 157apparent mass, 167approach, 80, 287, 302, 312, 391Archimedes’ law, 5, 517area

reference, 113wetted, 56, 59

area rule, 474, 484artificial feel, 330aspect ratio, 19, 156, 164, 165, 170,

176, 270, 365, 385, 468, 479, 483atmosphere

exponential, 72isothermal, 72

physical, 513standard, 69, 73, 75, 258, 259, 461,

513technical, 513

autogiro, 34, 416autorotation, 397, 416, 420autothrottle, 287aviation

dynamic, 48, 52static, 5, 48

axis system, 333, 334aerodynamic, 262, 335body, 260, 335earth, 335flight path, 262section, 133, 145wind, 145, 166wing, 166

backside of the drag curve, 287balanced field length (BFL), 311balloon, 49

gas, 49, 519hot-air, 5, 49, 518hydrogen, 6

bank angle, 314, 315, 336, 386Baumhauer, A.G. Von, 35Bennett, J.A.J., 418Bernoulli’s equation, 97, 101, 109, 139,

239, 259, 436, 438, 447, 461Bernoulli, Daniel, 97Betz, A., 167biplane, 18, 22, 25Blériot, L., 23blade pitch, 236, 243, 247, 288, 312,

414, 423blade twist, 236, 237blade-element theory, 241Blasius, H., 120blind flying, 79boundary layer, 24, 91, 108, 111, 120,

140, 151, 160, 257, 306, 352,435, 458, 503

laminar, 108, 111, 118, 120, 471, 504turbulent, 108, 118–120, 472, 504

Flight Physics 525

Bréguet’s equation, 296, 298Bréguet, L., 296brake power, 273Brayton cycle, 215Bryan, G.H., 24, 368buffet boundary, 500buffet margin, 500buffeting, 432, 470Busemann, A., 466by-pass engine, 193, 222by-pass ratio, 42, 193, 223, 224, 226,

250

calibrated airspeed, 258camber, 21, 132, 133, 152, 303

negative, 133positive, 133

camber linemean, 132, 146, 352S-shaped, 133, 352, 360

cambered aerofoil, 12, 13, 129, 147,352

CAS, 258, 461Cayley, George, 7, 8, 48, 58, 129ceiling, 50, 188, 284, 520

absolute, 278, 284aerodynamic, 500service, 285, 499

centre of gravity, 8, 15, 55, 328, 333,382, 425

centre of pressure, 57, 350, 351, 379,463, 470, 478

centripetal acceleration, 262Certificate of Airworthiness, 255Chanute, O., 16Charles, J.A.C., 6, 64chord, 106, 121, 128, 156, 236

mean aerodynamic, 156, 356, 357mean geometric, 156root, 156, 357, 483tip, 156, 357

chord length, 132, 147, 148, 165chord line, 132, 145, 158, 357Cierva, J. de la, 34, 416

circulating flow, 24, 92, 104, 138, 141,160, 169, 408

circulation, 137, 139, 141, 146, 160,165, 303, 367, 462, 464

clear air turbulence, 79climb

dynamic, 286rate of, 83, 262, 281, 283, 310, 318,

419, 420, 486time to, 281, 284, 286

climb angle, 262, 279, 286, 308climb gradient, 280, 310, 311climb ratio, 273, 275, 282, 291, 295Coanda effect, 408cockpit, 12, 347cockpit controls, 330, 346coffin corner, 500collective pitch, 422, 423combustion chamber, 219compound helicopter, 54compressibility, 88, 89, 208, 332, 421,

431, 432, 434compressible flow, 460, 462compression ratio, 186, 207, 210, 230compressor, 217

axial-flow, 191, 218centrifugal, 191, 217

compressor spool, 219, 220, 228computational fluid dynamics, 57, 109,

131consistent system of units, 512continuity equation, 93, 94, 99, 115,

436, 449control, 328, 329

directional, 423lateral, 15, 18, 23, 339, 422longitudinal, 23, 339, 371, 422pitch, 15

control area, 99, 167, 198, 238control column, 12, 347, 349, 395, 397control force, 347, 368, 378, 431control reversal, 432control stick, 23, 347, 366, 390, 424control surface, 80, 310, 330, 343, 345,

348, 376, 379, 432, 475, 488

526 Index

control system, 330, 346, 347, 370, 424,476

control variable, 264, 290control wheel, 267, 308, 320, 347, 476cooling

air, 30, 212liquid, 26, 212

core engine, 193, 216, 223corner velocity, 321cross-flow, 159crosswind, 60, 80, 257cruise number, 297cruise speed, 40cruise technique, 293cruise-climb, 298Curtiss, G., 23cyclic pitch, 36, 423

damping, 341dead air region, 110dead man’s region, 416decision height, 80decision speed, 310deep stall, 396density, 52, 64, 88, 90, 436, 517

relative, 74, 258total, 440

descentangle of, 262, 287, 289, 312, 321,

416rate of, 77, 83, 262, 287, 289, 291,

312, 520design cruising speed, 321design diving speed, 321diesel engine, 186dihedral, 9, 12, 27, 157, 331, 363, 386,

387longitudinal, 363, 367

dihedral effect, 387, 390dimension, 511dimensional analysis, 112direct-lift control, 376disc loading, 240, 409dive, 55, 58, 189, 247, 287, 319, 321,

329, 371, 390, 400, 431, 476

Dornier, C., 28down-burst, 79, 257downwash, 52, 157, 159–161, 163, 307,

365, 366rotor, 409

drag, 8, 58, 89, 102, 145form, 59, 111, 121, 173friction, 59, 64, 100, 108, 110,

118–120, 143, 151, 172, 177,300, 458

induced, 59, 119, 121, 157, 162, 163,165, 166, 173, 175, 275, 297,389, 501

intake momentum, 221, 234interference, 173, 473parasite, 173, 417pressure, 59, 111, 117, 119, 151, 172,

435, 459, 478profile, 121, 122, 150, 164–166, 173,

417, 463thickness, 460trim, 59, 174, 256, 377, 435, 487wave, 435, 450, 459zero-lift, 175, 177

drag area, 113drag coefficient, 113, 116, 117, 121,

145, 166, 174, 268, 410, 470, 496drag divergence, 470drag due to lift, 459, 478, 482drag polar, 128, 152, 174, 257, 266,

282, 291, 496, 501ducted fan, 37, 250Dutch roll, 389, 390, 428

EAS, 258, 460efficiency

Froude, 201intake, 490jet, 239propeller, 20, 186, 203, 207, 229,

239, 244, 245, 247, 274, 295propulsive, 201–203, 207, 224, 240,

489thermal, 206, 209, 224, 235total, 205, 225, 249, 298, 489, 497

Flight Physics 527

elevator, 9, 12, 18, 21, 59, 256, 342, 346empennage, 9, 329, 342–344, 389, 401endurance, 21, 291–294

specific, 293energy equation, 97, 436, 438, 447, 452engine control, 231engine failure, 215, 245, 280, 307, 310,

339, 393, 394, 416engine nacelle, 61engine noise, 44, 227engine operation, 59, 66, 232, 332, 346engine rating, 209, 232, 297, 321enthalpy, 438entropy, 439equation of motion, 262, 307, 369equation of state, 64, 70, 77, 115, 259,

436, 462, 517Euler angle, 335Euler’s equation, 96, 109, 436, 476Euler, Leonhard, 96, 97expansion flow, 457

fan, 194, 197, 216, 223feathering hinge, 414feathering position, 245, 310fenestron, 408figure of merit, 410fin, 22, 23, 342flap

double-slotted, 304plain, 345single-slotted, 303slotted, 33, 303split, 21, 33, 303

flap angle, 175, 303, 309flapping hinge, 34, 36, 414flat plate, 107, 119flight

cruising, 169, 232, 260, 264, 292,293, 298, 369, 489, 496, 497

gliding, 288hovering, 53, 408quasi-steady, 60, 264, 284steady, 60, 264

steady level, 48, 59, 60, 267, 278,497

straight and level, 52, 277, 299, 316,371, 419

symmetric, 59, 60, 260, 286, 315,336, 497

turning, 19, 55, 314, 329, 390flight control computer, 349flight control system, 348flight corridor, 508flight dynamics, 257, 328, 333flight envelope, 278, 321flight level, 77flight manual, 255flight mechanics, x, 15, 55, 257, 260flight planning, 255flight-path angle, 260, 262, 263, 336flow

compressible, 97hypersonic, 440ideal, 90, 107, 384, 436incompressible, 89, 94, 97, 101, 259,

436, 446, 447, 460, 462inviscid, 90, 96, 102, 107, 109, 138,

139isentropic, 439, 444, 446laminar, 104, 107steady, 90, 92, 95, 98, 142, 198subcritical, 464subsonic, 447supercritical, 464supersonic, 447turbulent, 105unsteady, 90

flow separation, 110, 111, 117, 118,121, 143, 151, 152, 166, 171,217, 267, 304, 458, 469, 486, 499

flutter, 321fly-by-wire, 330, 349flying qualities, 153, 328, 333Fokker, A.H.G., 27, 30, 131force

aerodynamic, 56–59field, 55ground, 56

528 Index

inertial, 55pressure, 57shear, 57surface, 55water, 56

foreplane, 343, 367, 376, 486Fowler flap, 304Franklin, Benjamin, 4friction, 57, 103frictionless flow, 90, 141Froude, W., 197, 201fuel

reserve, 301trip, 301

fuel consumption, 205specific, 208, 225, 230, 294, 296,

420, 495

Garnerin, Jacques, 6gas, 62gas constant, 64gas generator, 193, 216, 225gas turbine engine, 61, 190Gibbs-Smith, C.H., 2, 14Giffard, H., 11Glauert, H., 146, 167, 418, 462glide ratio, 289glider, 9

hang, 15, 372ground effect, 307, 311, 332, 412ground speed, 258gust, 79, 322gust load, 319, 322

Handley Page, F., 33handling qualities, 328Hargrave, L., 17, 25, 184heating

aerodynamic, 503, 509heating value, 205, 299, 497Heinkel, E.H., 192helicopter, 7, 34, 35, 52, 195, 257, 406

compound, 422Henson, W.S., 10high-lift device, 127, 303, 329, 332, 382

hinged nose, 304Hooke, Robert, 4horizontal tail, 361hydroplane, 26hypersonic speed, 88

IAS, 258, 461ice protection, 80in-line engine, 183, 185, 188incidence

angle of, 12, 157, 158, 363, 365, 367indicator diagram, 211instrument landing system, 80, 287, 312interference, 173internal combustion, 9, 183, 215International System of Units, 512ISA, 74–76

jet engine, 37, 193, 216, 235, 489jet stream, 78Jones, R.T., 171Joukowski, N.E., 24, 139Junkers, H., 28

Kármán, Th. von, 29, 110Kelvin, 513kilogram, 513Krueger flap, 304Kutta condition, 142, 147, 480Kutta, W.M., 24, 139Kutta–Joukowski relation, 139, 142,

147, 160

lagging hinge, 414Lana-Terzi, Francesco, 5, 50Lanchester, F.W., 24, 141, 160, 368,

370landing, 12, 77, 80, 127, 175, 186, 302,

312, 332, 416landing distance, 54, 312, 486landing flare, 256, 312landing run, 313Langley, S.P., 17, 183lateral motion, 338, 369lateral stability, 339

dynamic, 388, 427

Flight Physics 529

static, 427Laurent, F., 6leading edge, 33, 88, 98, 128, 153, 155,

166sonic, 480subsonic, 480, 481supersonic, 479, 480

leading-edge suction, 166, 175, 482Leduc, R., 38Lenoir, J.E., 11Leonardo da Vinci, 4, 7, 93lift, 8, 50–52, 58, 137, 145

gross , 517net, 517

lift coefficient, 52, 140, 146, 161, 172,266, 354, 410, 467, 470, 509

lift curve, 150, 152, 165, 171, 172, 257,268, 305, 359

lift divergence, 470lift dumper, 312, 347lift gradient, 147, 150, 164, 323, 359,

463, 468, 477, 481lift-off speed, 308lifting line, 24, 161lifting surface, 52, 58, 128, 159, 330,

363Lilienthal, Otto, 15, 129, 372Lindbergh, C.A., 31, 187liquid, 62liquid-cooled engine, 185load

limit, 321manoeuvre, 319ultimate, 321

load factor, 189, 315, 319, 371, 376,500

loading condition, 319, 329, 332, 400longitudinal axis, 260longitudinal motion, 338, 369longitudinal stability, 21, 339, 349, 487

dynamic, 369, 426static, 358, 359, 426

Lord Rayleigh, 87

MAC, 156, 356, 357

Mach angle, 445, 455Mach cone, 445, 479Mach meter, 462Mach number, 88, 209, 249, 257, 442,

476critical, 297, 464, 465, 468

Mach trimmer, 476Mach wave, 445Mach, Ernst, 88, 442Magnus effect, 140Magnus, H.G., 140Manly, C.M., 17, 183manoeuvrability, 28, 50, 51, 286, 314,

318, 371manoeuvre margin, 378manoeuvre point, 321, 378manual control, 79, 330, 347, 368, 379mass, 55mass flow, 93Maxim, H., 14metre, 513minimum control speed, 394minimum drag speed, 268, 280, 287,

297minimum power speed, 273, 295mission analysis, 255molecular mass, 64, 519moment curve, 354, 359, 373, 396momentum equation, 100, 122, 167,

198, 221, 238, 409, 418, 436momentum flow, 99, 198, 202, 241monoplane, 10, 25Montgolfier, Jacques, 5Montgolfier, Joseph, 5Moss, S.A., 190motion

aperiodic, 340directional, 338periodic, 340

MTOW, 268, 302Munk, M.M., 146

NACA cowling, 185NACA/NASA, 29Navier–Stokes equations, 109

530 Index

negative camber, 352neutral point, 364, 366, 377

stick-fixed, 364, 374neutral stability, 340Newton, 513Newton, Isaac, 4, 65, 103, 112, 182Newtonian fluid, 65no-slip condition, 103, 108, 139, 503normally aspirated engine, 212Northrop, J.K., 33nose circle, 132, 134nose point, 132, 141, 154, 355, 456nose radius, 134, 169, 304, 472NOTAR, 408nozzle, 197, 216, 449

convergent-divergent, 492de Laval, 446exhaust, 61, 193, 220, 229, 492, 494

Ohain, H.J. Pabst von, 38, 190one-dimensional flow, 93, 99, 441, 451operational conditions, 233orbital velocity, 508ornithopter, 4, 8, 15oscillation, 328, 340, 367, 369, 391,

427long-period, 369, 428short-period, 370, 382, 428

Oswald factor, 175, 177, 270Otto cycle, 210Otto engine, 13, 26, 183Otto, N.A., 13, 183overpressure, 101, 122oxygen deficiency, 81oxygen limit, 82

Pénaud, Alphonse, 12, 273parachute, 4, 26paradox of d’Alembert, 102pascal, 513payload, 255, 300, 332payload-range diagram, 301perfect gas, 64performance

path, 264, 293, 420

point, 264, 420performance analysis, 253performance diagram, 273, 275, 281,

419, 502period, 341, 427petrol engine, 13, 17phase of flight, 254, 329Phillips, Horatio, 13phugoid, 370Pilcher, P.S., 16piston engine, 33, 37, 61, 182, 189, 196,

209, 273, 294, 406pitch angle, 243, 260, 279, 336, 396pitch axis, 59, 335pitch control, 347pitch rate, 337, 426pitch-up, 396pitching moment, 61, 133, 338pitching moment coefficient, 351, 356,

362, 463pitot intake, 490pitot tube, 259, 460pitot-static tube, 259plain flap, 303Platz, R., 27Plesman, A., 29Poisson’s equation, 440positive camber, 352potential flow theory, 102, 109power

available, 201, 229, 239, 273, 275,282, 419

equivalent, 208, 229, 273, 296induced, 239, 409, 418jet, 239required, 272, 273, 408, 411, 417,

419shaft, 214

power coefficient, 243power loading, 302powered controls, 330, 382Prandtl, L., 24, 29, 91, 109, 142, 160,

462Prandtl–Glauert correction, 462pressure, 63, 90, 436

Flight Physics 531

dynamic, 97, 113, 231, 258, 409,460, 475

impact, 460relative, 74stagnation, 98, 231, 509static, 63, 259total, 97, 98, 100, 109, 207, 217, 228,

259, 440, 450, 453, 460, 490, 495pressure cabin, 33, 42, 82, 185, 212pressure coefficient, 101, 148, 351, 463,

476pressure gradient, 107propellant, 61, 182, 200propeller, 3, 11, 12, 26, 54, 61, 101,

197, 236, 406adjustable-pitch, 186, 247constant-speed, 33, 186, 247, 248fixed-pitch, 186, 247

propeller blade, 61, 80, 186, 197, 236,242

propeller diagram, 246propeller plane, 236propeller torque, 197, 242, 245, 247propfan, 249propulsion, 6

jet, 37, 61, 197propeller, 61, 197reaction, 61, 196, 221rocket, 39, 61

propulsion system, 59propulsive jet, 197

quarter-chord line, 157, 387, 468quarter-chord point, 356, 383

radial engine, 17, 30, 185, 187radius of action, 302ram compression, 235, 493ramjet engine, 37, 61, 196, 494range, 293, 294, 296, 297

harmonic, 301specific, 293, 295, 497

range parameter, 300, 498Rankine, W.J.M., 197Rayleigh, Lord, 112

reaction principle, 6, 181reaction torque, 34, 411reheat, 194, 205, 492, 493resonance

air, 429ground, 429

reverse flow, 110, 414reverse thrust, 221, 231, 307, 312, 492reversible process, 439Reynolds number, 106, 119

critical, 119Reynolds, O., 91, 105Robins, Benjamin, 8, 112rocket engine, 61, 200, 432roll angle, 336, 376, 392roll axis, 335roll damping, 389roll rate, 337, 388, 391rolling mode

aperiodic, 389, 428rolling moment, 34, 338, 386, 414root section, 154rotary engine, 26, 184rotor, 4, 7, 34, 406

coaxial, 407hingeless, 415main, 52, 406tail, 35, 36, 52, 406, 411tamdem, 407teetering, 425

rotor blade, 53, 80rotor control, 423rotor tilt angle, 414rotorcraft, 34, 406Rozier, P. de, 6rudder, 9, 12, 19, 342, 346

sailplane, 177, 288scale effect, 118schlieren optical system, 451scramjet engine, 495seaplane, 33Sears–Haack body, 474second, 513section code, 134

532 Index

section thickness, 132SEP, 285, 371shear stress, 64, 103, 111shell structure, 26, 31shock stall, 470, 499shock wave, 88, 434, 450, 490, 505

λ, 456, 469bow, 434, 451, 456, 469normal, 451, 490oblique, 451, 454

shock wave angle, 454side-force, 384, 386, 388, 389side-slip, 60, 336, 384Sikorsky, I.I., 26, 36slat, 33, 304slender body, 456slipstream, 54, 61, 197, 237, 274, 306,

332, 366, 393, 410slug, 514sonic boom, 44, 434, 505sonic flow, 447sonic line, 456, 468sonic speed, 88, 431, 441sound

velocity of, 37, 39, 44, 65, 411, 435sound barrier, 432specific excess power, 285, 318, 371specific heat, 65, 437, 438specific impulse, 209specific volume, 64, 88spin, 267, 339, 392, 397

flat, 400inverted, 401steep, 400

spiral mode, 389spoiler

flow, 59, 117, 312, 392stability, 328

directional, 22, 384, 390, 488dynamic, 24, 340hovering, 425inherent, 329, 361, 488lateral-directional, 386speed, 287, 358, 426spiral, 390

static, 340stability margin, 366stagnation point, 91, 98, 101, 109, 139,

141, 258, 439stall, 18, 170, 245, 267, 268, 276, 320,

332, 352, 395stalling, 15, 33, 128, 151, 329, 395,

397, 421stalling speed, 268, 280, 302, 312, 360,

395, 486state variable, 62, 436, 443stator blade, 223stealth technology, 487stick force, 330, 375stick pusher, 397stick-fixed stability, 358, 361stick-force stability, 375, 378stick-free stability, 368stick-position stability, 375, 378Stokes, G.G., 91, 112STOL, 53, 54, 127, 306stratosphere, 40, 69, 72, 297, 489, 498,

520stream filament, 93, 436stream tube, 92–94, 98, 198, 238, 436,

437, 446streamline, 91, 92, 94–96, 105, 138,

159, 434, 439, 462, 463dividing, 92, 109

streamlining, 6, 8, 9, 31, 59, 111, 236stressed skin, 28, 31, 33Stringfellow, J., 11subsidence, 340, 389subsonic speed, 44, 88, 148, 203, 345,

434suction, 101, 122suction force, 143, 166, 222, 399, 459,

481supercharger, 185, 209, 212, 213

turbo-, 190, 213supercritical section, 131, 472, 501supersonic speed, 39, 44, 88, 195, 203,

432, 434Sutherland’s equation, 74swash plate, 423

Flight Physics 533

sweep angle, 126, 128, 157, 161, 169,360, 365, 387, 479, 501

variable, 486symmetric section, 133, 146, 151, 351,

355

T-tail, 344tab

geared, 381servo, 381spring, 381trim, 368, 381

tail angle, 134, 135tail load, 172, 174, 256, 365, 376, 383tail moment arm, 361, 384tail point, 132, 141, 151, 456, 469tail volume

horizontal, 362vertical, 384

tailplane, 342, 348adjustable, 256, 381controllable, 256

take-off, 12, 52, 77, 80, 127, 175, 257,302, 305, 307, 332, 382, 383,406, 416

take-off distance, 54, 308, 486take-off rotation, 256, 308take-off run, 302, 308take-off safety speed, 308take-off weight, 268, 301, 302, 321taper ratio, 156, 170, 356, 484TAS, 258, 460temperature, 63, 90, 436

absolute, 513relative, 74thermodynamic, 63total, 438, 442, 443, 462, 503

temperature lapse, 71thermal, 257, 290, 292thermodynamics, 436thickness, 25, 132, 133, 146, 236, 459thickness ratio, 133, 150, 169, 466, 471,

478, 486throat, 446

thrust, 39, 51, 52, 59, 89, 182, 197, 199,238

available, 273gross, 221, 234, 492ideal, 200, 225, 234net, 221propeller, 197required, 268specific, 201, 225

thrust angle, 59, 263, 265, 279, 315thrust coefficient, 240, 242thrust loading, 277, 302, 309tilt-rotor aircraft, 54time to double, 340, 427time to halve, 340, 370, 389time to turn, 318, 371tip speed, 410tip-path plane, 414touch-down, 312trailing edge, 33, 122, 127, 128, 142,

153, 155, 159, 330, 342, 380,414, 458, 484

supersonic, 480trailing-edge flap, 303transition point, 105transonic speed, 88trim curve, 374trimmed equilibrium, 328, 340, 358,

368, 374trimming, 372triplane, 25tropopause, 68, 72, 78, 235, 297, 502troposphere, 68, 71, 235, 443, 502tuck-under, 431turbine, 220

free power, 228, 248turbo-ramjet engine, 433, 495turbofan engine, 42, 61, 193, 196, 199,

216, 222, 227, 250, 280, 296,435, 488

turbojet engine, 38, 61, 190, 196, 216,221, 296, 432, 488

turboprop engine, 40, 61, 190, 194, 196,216, 227, 296

534 Index

turboshaft engine, 195, 196, 217, 229,406

turbulence, 78turn

coordinated, 315, 316, 390, 392horizontal, 314, 318, 392

turn radius, 315, 390turn rate, 318, 371turning

instantaneous, 315sustained, 314

two-dimensional flow, 91, 99, 102, 116,117, 137, 454

undisturbed flow, 99, 101, 108, 113,201, 239, 259, 439, 460

unit, 511upwash, 160useful load, 50, 55, 300, 301, 319, 343,

383, 518

V-n diagram, 319V/STOL, 53, 59, 265velocity, 90, 436velocity profile, 103, 104, 108, 111, 448viscosity, 64, 65, 90, 95, 102, 103

dynamic, 65, 74, 104kinematic, 65, 104

vortex, 98, 138, 143, 146bound, 143, 159, 160, 367delta, 170, 171free, 159horseshoe, 160tip, 159, 163trailing, 159, 161, 169, 243

vortex field, 243vortex theory, 146VTOL, 53, 54

wake, 90, 98, 100, 103, 105, 109, 111,118, 122, 151, 365, 395, 396,401, 469, 475, 499, 500

weight, 52, 55empty, 55, 126, 300, 301, 518take-off, 268

Wenham, F.H., 11, 12

wetted area, 111, 177Whitcomb, R.T., 131Whittle, F., 39, 190wind, 78, 257, 290wind shear, 79, 257wind tunnel, 8, 12, 56, 57, 92, 114, 129,

386, 449wing, 52, 163

arrow, 483, 484cantilevered, 26, 131delta, 11, 126, 128, 170, 343, 479,

483, 484diamond, 487flying, 343, 360, 361plank, 127, 156, 166slender, 157, 169, 171, 482straight, 126, 127, 157, 160, 163,

333, 356, 386, 466straight-tapered, 155, 165, 356, 357swept, 128, 466swept-back, 126, 157, 171, 387, 467tandem, 17, 343tapered, 127

wing area, 33, 52, 58, 121, 156, 172,304, 338

wing loading, 268, 275, 291, 292, 302,309, 313, 360, 409, 472, 509

wing planform, 126, 127, 154, 356wing section, 29wing twist, 18, 153, 158, 357, 360, 473wing vertex, 154wing warping, 18winglet, 169, 178wingspan, 126, 154, 156, 159, 167, 168,

176, 177, 270, 400Wright, Orville, 1, 18, 33, 129Wright, Wilbur, 1, 18, 129

yaw angle, 78, 336yaw axis, 335, 406yaw damper, 391yaw rate, 337, 388, 399yawing moment, 338Yeager, C., 431, 432

Flight Physics 535

Zahm, A.F., 363Zeppelin, Count von, 36zero-lift angle, 136, 150, 158

zero-lift line, 136zero-lift moment, 352, 359, 363zone of silence, 446, 458, 480

Sources of figures

The following publishers/organizations have graciously granted permissionfor the reproduction of figures:

• Airlife Publishing Ltd.: Figures 5.17a and 5.22a.• Blackwell Science Ltd.: Figure 9.1a.• Edward Arnold Publishing Co.: Figures 8.8a and 8.9.• Flight International: Figures 1.27, 2.2, 2.4b, 5.24, 9.31d, 9.31e and

9.31f.• Granada Publishing: Figures 4.29 and 9.10.• G.T. Foulis & Co.Ltd.: Figures 2.9 and 9.44.• Ian Allan Ltd.: Figures 9.31b.• John Wiley & Sons. Inc.: Figures 3.21 and 5.36.• Longman Scientific & Technical: Figures 5.19 and 9.32.• Mc Graw Hill Book Company: Figure 5.1.• Midland Publishing Ltd.: Figures 4.39, 9.1b and 9.31a.• Nationaal Lucht- en Ruimtevaart Laboratorium: Figure 3.16.• Phoebus Publishing Co.: Figures 8.1a and 8.1b.• Pitman Publishing Ltd.: Figure 9.16.• PJS Publications Inc.: Figure 8.14.• Planes of Fame Publishers: Figure 7.21.• Prentice Hall Inc.: Figures 5.2, 5.3b and 5.4.• Rolls-Royce plc.: Figures 5.18, 5.20, 5.21, 5.22b, 5.26, 5.38b and 5.39.• Science Museum, London: Figures 1.1, 1.3–1.9a, 1.10, 1.16, 1.21, 1.23

and 4.26.• Smithsonian Institution, Washington DC: Figures 1.9b, 1.12 and 1.24.• VDI-Verlag: Figures 1.2, 1.22, 2.4a, 4.27, 5.3a, 5.5–5.7, 5.33a, 7.8 and

7.10.• Verlag Werner Dausien: Figures 1.13, 1.14, 1.18b, 1.19 and 1.20.

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