Module 2

Aircraft Performance

Module 2

2Aerodynamics, air data measurements

Where are we?

1 : Introduction to aircraft performance, atmosphere

2 : Aerodynamics, air data measurements

3 : Weights / CG, engine performance, level flight

4 : Turning flight, flight envelope

5 : Climb and descent performance

6 : Cruise and endurance

7 : Payload-range, cost index

8 : Take-off performance

9 : Take-off performance

10 : Enroute and landing performance

11 : Wet and contaminated runways

12 : Impact of performance requirements on aircraft design


Aerodynamics

Incompressible Bernoulli equation

Speed of sound and Mach number

Compressible Bernoulli equation

Flow relations near the speed of sound

Airfoil properties

Viscosity effects

Lift and drag

High-lift devices


Incompressible Bernoulli equation Describes variation of pressure and velocity in a streamtube

• F = ma

• Net force : pA - (p + dp)A = -dp A

• Mass of the element is A ds

• F = ma can be transformed into -dp A = A ds dV/dt

• Can be rewritten as -dp A = A ds/dt dV or dp = - V dV

• Assuming is constant, integration of the equation gives

p + V2/2 = constant

• (valid for incompressible flow only)


Incompressible Bernoulli equation (Cont’d) P is the static pressure

V2/2 is the dynamic pressure (q)

Sum of static and dynamic pressures is the total pressure

• ps + V2/2 = pt

• ps + q = pt

• p1 + q1 = p2 + q2

Direct application of the Bernoulli equation is the pitot-static tube


Incompressible Bernoulli equation (Cont’d)

Tube is aligned with the flow

Freestream static pressure and velocity are p0 and V0

At point 1, V1 = 0 (stagnation point)

At point 2, V2 = V0

Applying Bernoulli equation between points 0 and 1

• p1 = po + Vo2/2

Applying Bernoulli equation between points 0 and 2

• p2 = po

p = p1 - p2 = Vo2/2


Speed of sound and Mach number a = speed of sound a = (p/)0.5 = (RT)0.5 (T= absolute temperature)

= ratio of specific heats (constant pressure and constant volume) = Cp/Cv = 1.4 for air and R remain constant in the atmosphere

a = constant x T0.5

a = ao0.5

• ao is the speed of sound under SL ISA conditions (T= 15oC)

• ao = 661.48 knots = 1116.45 ft/sec = 340.28 m/s = 1225.0 km/h

• note : 1 knot = 1 nautical mile (nm) / h, 1 nm corresponds to an arc of one minute (1/60th of a degree) over the earth surface

Mach number (M) is the ratio of local air velocity to local speed of sound

M = V / a


Compressible Bernoulli equation

Air moving at speeds below 200 knots may be treated as an incompressible fluid

At higher speeds, it is necessary to consider the variation of density as the airflow is compressed an another form of the Bernoulli equation must be used• dp = - V dV (presented earlier)

• p/ = constant = C (derived from Boyle’s Law for adiabatic flow)

• The two equations above can be combined to give

C 1/ p-1/ dp = - V dV

• Integration of the equation gives

C 1/ p-1/-1/(-1/ -1) + V2/2 = constant

• Can be rewritten as

(/( -1))p(C/p)1/ + V2/2 = constant


Compressible Bernoulli equation (Cont’d)

• Substituting (C/p)1/ = 1/, the Bernoulli equation for compressible fluids becomes

(/( -1))p/ + V2/2 = constant

• The flow equation may be written for any two points in the fluid

(/( -1))p1/1 + V12/2 = (/( -1))p2/2 + V2

2/2

• Since = 1.4 for air, equations can be rewritten as

3.5 p/ + V2/2 = constant

3.5 p1/1 + V12/2 = 3.5 p2/2 + V2

2/2


Flow relations near the speed of sound Behaviour of fluid flow near the speed of sound is of

primary importance Classification of high speed flight

• Subsonic M < 1

• Sonic M = 1

• Supersonic M > 1

• Transonic 0.80 < M < 1.3 (Approximately)

• Hypersonic 5 < M < 10

• Hypervelocity M > 10

Relationships between total and static temperature, density and pressure can be derived from Bernoulli compressible equation for compressible isentropic flow


Flow relations near the speed of sound (Cont’d) (/( -1))p1/1 + V1

2/2 = (/( -1))p2/2 + V22/2

Point 1 = reservoir (subscript T for total) : VT = 0 Point 2 = some point in the channel (no subscript)

(/( -1))pT/T = (/( -1))p/ + V2/2

Knowing that a2 = p/ and aT2 = pT/T :

aT2 / ( -1) = a2 /( -1) + V2/2

Dividing each side of the equation by a2

(1/( -1)) aT2 / a2 = 1/( -1) + ½ V2/ a 2

Knowing that aT2 / a2 = TT / T , V/a = M and rearranging :

TT / T = 1 + (( -1)/2) M 2 = 1 + 0.2 M 2

From pT/p = (T / ) (TT / T ) and pT/p = (T / ) :

T / = (1 + (( -1)/2) M 2) 1/( -1) = (1 + 0.2 M 2) 2.5

pT/p = (1 + (( -1)/2) M 2) /( -1) = (1 + 0.2 M 2) 3.5


Airfoil properties Physical properties of the wing

• Wingspan (b) is the tip to tip dimension of the wing

• Chord (c) is the distance from the wing leading edge to the trailing edge

• Wing area (S) is the projection of the outline of the plane of the chord

• Aspect ratio (AR) of a wing is defined asAR = b/c for a wing with constant chord (rectangular

wing)AR = b2/S

• Taper ratio () is the ratio of the tip chord (ct) to the root chord (cr)

= ct / cr


Airfoil properties (Cont’d) Physical properties of the wing (Cont’d)

• Mean aerodynamic chord (MAC) is the chord of a section of an imaginary airfoil on the wing which would have force vectors throughout the flight range identical to those of the actual wing

- Can be determined graphically or by integration

S

dbcMAC

2


Airfoil properties (Cont’d) Physical properties of the wing (Cont’d)

• MAC is used as a reference for locating the relative positions of the wing center of lift and the airplane center of gravity (CG)

• Center of lift is normally located at the quarter chord (c/4) of the MAC

• Sweepback () is the angle between a line perpendicular to the plane of symmetry of the airplane and the quarter chord of each airfoil section


Airfoil properties (Cont’d) Aerodynamic properties of the wing

• Pressure distribution around an airfoil in an airflow is a function of the airfoil shape (camber) and the angle of attack ()

• The angle of attack is the relative angle between the freestream velocity (Vo) and the chord (or the fuselage)

• The integration of the pressure distribution around the airfoil can be resolved in two component forces acting at the center of pressure

Lift (L) is perpendicular to the freestream velocity

Drag (D) is parallel to the freestream velocity

V

L

D


Airfoil properties (Cont’d)

L = CLqS

CL is the lift coefficient, CL = L / (qS) , dimensionless

S is the wing area (ft2)

q is the dynamic pressure (lb/ft2)

q = 0.5V2 (q in lb/ft2 , in slugs/ft3 and V in ft/sec )

or

q = 1481.3 M2 (q in lb/ft2)

Under level flight conditions, L = W and CL can then expressed as

CL = W / (qS)


Airfoil properties (Cont’d)

Lift is normally defined in terms of the load factor normal to the flight path Nz

L = NzW or Nz = L / W

Under level flight conditions, Nz = 1.0 (I.e. 1 g)

A more general equation defining CL can be introduced

CL = NzW / (qS)

D = CDqS

CD is the drag coefficient, dimensionless

CD = D / (qS)


Viscosity effects Viscosity is the result of shear forces acting on the fluid, or the tendency of one layer of fluid to

drag along the layer next to it The boundary layer is a finite thickness of fluid next to a surface which is retarded relative to the

free stream velocity Flow in the boundary layer can be laminar or turbulent depending upon

• the smoothness of flow approaching the body, the shape of the body, the surface roughness, the pressure gradient in the direction of flow and the Reynolds number (RN) of the flow (dimensionless)

RN = Vl/ • l is length from leading edge (ft)

• is the dynamic viscosity (lb-sec/ft2)

• = 0.3125 x 10-7 T1.5 / (T + 120) where T is in oK

Friction drag of laminar flow is smaller than friction drag of turbulent flow


Viscosity effects (cont’d)

RN at the transition is approximately 530,000


Lift and drag Airplane lift and drag vary as a function of

•Linear portion of lift curve is where the aircraft normally operates• For performance analysis, it is more practical to look at the variation of CD in terms of CL


Lift and drag (Cont’d) In the range of CL corresponding to normal low speed operation, CD can be closely approximated

by :

CD = A + B CL 2 = CDP + CDI

CDP = parasite drag

CDi = induced drag = CL 2 / ( AR e)

CDi= K CL 2, K is the induced drag factor e is the Oswald efficiency factor

• CL / CD = L / D = Lift to drag ratio• L / D max = max. L/D or min. drag


Lift and drag (Cont’d) Drag increment due to compressibility - ΔCD comp

• Drag increases as a result of separation induced by shock waves during flight at high speed

• Compressibility effects can also happen at relatively slow aircraft speeds (at high CL)

• Drag rise occurs at the onset of shock wave separation - related to critical Mach number (lowest flight speed at which sonic speed is attained on the wing)

ΔCDcomp

CL

CD M

CD = CDP + KCL2 + ΔCD comp

Fixed CL

CD REF

(CD – CD REF)


Lift and drag (Cont’d) Sweep can be used to alleviate compressibility effects

Drag critical Mach number (MCR) is usually defined as the Mach number that causes a drag coefficient increase of 0.0020 (20 drag counts)

Other factors that can alleviate compressibility effects :• Thin supercritical wing section (aft loaded)

• Area ruling

• Vortex generators


Lift and drag (Cont’d) Other factors affecting airplane drag

• During operation with one engine inoperative, additional drag results from - Windmilling engine – CDWM - Data provided by engine manufacturers – typically a

function of M and

- Airplane control deflections and sideslip required to control an asymmetric thrust condition - CDCNTL - function of yawing moment due to thrust

• Drag increment due to deflection of spoilers – CDSP

- In flight or on the ground (ground lift dumpers)- Function of CL and M

• Drag increment due to landing gear - CDLG

- Function of CL

• Deployment of leading edge and trailing edge high-lift devices- Discussed in the next section


High-lift Devices Two types of devices are commonly used to increase CLMAX, the maximum lift

coefficient, and to reduce the stall speed VS

• Trailing edge flaps

• Leading edge slats

Trailing edge flaps provide an increase in camber• Increase in CL at a given angle of attack – increase is essentially proportional to flap

deflection

• Drag increase

• Reduction of the angle of attack at the stall

Leading edge slats provide smoother air on the upper surface of the wing • Slats take high pressure air from under the wing leading edge through a slot to the

upper surface

• Results in greater CLMAX and greater angle of attack at the stall

• Drag increase

Trailing edge flaps and leading edge slats may be used in combination in order to maximize CLMAX


High-lift Devices (Cont’d)

Effect of flaps and slats

Effect of slat on air flow


High-lift Devices (Cont’d)

Lift and drag increments from various types of trailing edge flaps relative to clean wing

ΔCDP

ΔCL


Air data measurements

Introduction

Airspeed

Mach number

Altitude

Temperature

Relationship between flight parameters

Angle of attack

Typical Pitot-static system

Position errors


Introduction

Air data measurements relate atmospheric parameters to the motion of the aircraft

• Airspeed, Mach number, altitude, temperature and angle of attack are important parameters for performance analysis

The objective of this section is to describe :

• The physical principles normally used for these parameters

• The methods of measurement

• The calibration procedures

• The applicable regulations


Airspeed – True airspeed The airspeed V that has been introduced previously is called the true airspeed

• Sometimes also defined as TAS

The true airspeed is the speed of the aircraft relative to the undisturbed air mass. V is the sum of

• The aircraft ground speed Vg (i.e. speed relative to the earth)

• The wind speed vector

Example : An airplane flies in level flight at a ground speed Vg of 500 knots in a tailwind of 50 knots -> V = 450 knots

V has only limited applications operationally.

Other airspeeds must be defined : Calibrated airspeed Equivalent airspeed

Indicated airspeed


Airspeed – Calibrated airspeed

The airspeed indicator uses the pressures obtained from a Pitot-static probe normally located on the forward fuselage

Pitot-static probes are normally used in pairs (one on each side of the fuselage) and are protected from ice accumulation with an electrical heating system

• Static pressures obtained from pilot and co-pilot sides are cross-coupled so as to minimize effect of sideslip on airspeed and altitude indications

Typical Pitot-static probe

Staticports

Total pressureport


Airspeed – Calibrated airspeed (Cont’d) The compressible Bernoulli equation is the basis for calibrating the airspeed indicator

pT - p = q c = p [ (1 + (( -1)/2) (V/a) 2) /( -1)- 1]

The indicator is only driven by the pressure difference (pT – p) or impact pressure (q c) obtained from a pitot-static installation

• Static pressure (p) and speed of sound (a), which is a function of temperature, are not known

• True air speed V can not be related directly to impact pressure

Solution is to define the calibrated airspeed Vc that is based on standard sea level values for p and a :

q c = po [ (1 + (( -1)/2) (Vc/ao) 2) /( -1)- 1]

q c = po [ (1 + 0.2 (Vc/ao) 2) 3.5- 1]

Vc = ao { 5 [ (q c/po + 1) 0.2857 – 1]} 0.5


Airspeed – Calibrated airspeed (Cont’d)

Vc is equal to V under SL/ISA conditions

Flight at constant calibrated airspeed is equivalent to flight at constant q c

q c is close to q during take-off and landing operations at low altitudes (difference is < 2 % typically)

For a given weight, flight at constant Vc at low altitudes ensures that CL and angle-of-attack are nearly constant even if altitude or air density changes• A simple means to maintain a satisfactory margin to the stall

• It would not be the case during flight at constant true airspeed


Airspeed – Calibrated airspeed (Cont’d)

(qc-q)/qc versus Vc

02468

10121416

0 50 100 150 200 250 300 350

Vc

(qc-

q)/

qc

(%

) 40,000 ft

30,000 ft

20,000 ft

10,000 ft

SL


Airspeed – Equivalent airspeed

The equivalent airspeed, Ve, is equal to Vc corrected for adiabatic compressible flow for the particular altitude

Ve, is based on SL ISA density o

Ve = { [2/( -1)] (p/o) [(qc/p + 1) ( -1)/ –1]}0.5

• Ve from above equation is in ft/sec with pressures in lb/ft2 and o in slugs/ft3

Equivalent airspeed is a function of qc and p

Flight at constant Ve is equivalent to flight at constant q

Ve can be defined as the answer to : How fast do I have to travel in SL ISA air to have the same q that I currently have?

0.5 o Ve 2 = 0.5 V2

Ve = V 0.5

True airspeed V is a function of Ve (I.e. qc and p) and


Airspeed – Equivalent airspeed (Cont’d) Ve is not used operationally to fly the aircraft but it is sometimes used for low speed

performance calculations as it results in simpler calculations

• q = Ve 2 / 295.37 (Ve in knots, q in lb/ft2)

• CL = 295.37 L / (Ve 2 S) (Ve in knots)

Ve = Vc - Vc Vc is the compressibility correction

Vc is always positive because qc is greater than q when compressibility effects are present

Vc is equal to 0 at SL (i.e. Ve = Vc at SL)

Vc is less than 1 knot for take-off and landing operations (i.e. altitude less than 10,000 ft and Vc

less than 200 knots Vc ranges between 10-20 knots for typical cruise conditions

Relationship between Vc , Vc and pressure altitude is presented graphically on the next page


Airspeed – Equivalent airspeed (Cont’d)

Delta Vc versus Vc

0

5

10

15

20

0 50 100 150 200 250 300 350

Vc

Del

ta V

c

40,000 ft

30,000 ft

20,000 ft

10,000 ft

SL


Airspeed – Indicated airspeed

The actual airspeed displayed to the pilot is the indicated airspeed, VI, or IAS

VI, is essentially equal to Vc but contains inherent system errors :

• Instrument error Vi (error in instrument calibration)

• Position error Vp (error due to the fact that p and pT are not equal to free stream values, will be detailed later)

VI = Vc - Vp - Vi

Vc = VI + Vp + Vi

Note : Vi is assumed to be zero in our analyses


Mach number

The compressible Bernoulli equation is also the basis for calibrating the Mach number displayed to the pilot (Mach meter)

pT - p = q c = p [ (1 + (( -1)/2) (V/a) 2) /( -1)- 1]

pT - p = q c = p [ (1 + (( -1)/2) M 2) /( -1)- 1]

M = { (2/( -1)) [ ( 1 + q c/p )( -1 )/ - 1 ] } 0.5

M = f (q c , p)

Many aerodynamic effects are function of M

M is independent of static temperature at a given pressure altitude and calibrated airspeed


Altitude

As discussed previously, the altimeter measures static pressure and converts it into an altitude based on equations for the standard atmosphere

The altimeter is connected to the static pressure source and picks up any existing position error hp (error in instrument calibration, pressure leak, …)

• Position error hp will be detailed later

hp = hpI + hp


Temperature

The free air temperature indicator is very important since the indicated temperature has two specific uses associated with performance

• Determination of true air speed V

• Determination of engine pressure ratio (EPR) or engine fan speed (N1) for the required thrust settings

Free air temperature gages are usually operated by an electrical resistance thermometer probe located on the forward portion of the fuselage


Temperature (Cont’d)

Because of the adiabatic temperature rise due to compressibility, the thermometer probe picks up a temperature reading higher than the static temperature

Tt / T = 1 + (( -1)/2) M2 = 1 + 0.2 M2

The equation relating total and static temperatures must be modified to include a temperature probe recovery factor, K, to account for the fact that the probe may not be able to recover the full temperature rise

Tt =T ( 1 + 0.2 K M2)



The probe recovery factor must be determined by flight testing and its value is normally very close to 1.0

• Aircraft must be flown at constant altitude in a stable air mass with constant temperature

• Aircraft is stabilized at various Mach numbers over the operational speed range

• For each stabilized test point, indicated total temperature (TTI) and Mach number (M) are recorded

• K can be determined by plotting 1/Tt as a function of M2 / Tt

• Slope of fitted test points is equal to –0.2 K or – K/5

• Example is presented on the next page



Determination of temperature probe recovery factor K


Relationship between flight parameters

To summarize, all of the data parameters that can be derived Pitot-static and temperature probes are defined in terms of qc, p and T

hp = = f (p)

Vc = ao { 5 [(qc/po + 1) 0.2857 – 1]} 0.5 = f (qc)

Ve = { 7 (p/o) [(qc/p + 1) 0.2857 –1]}0.5 = f (qc,p)

V = a { 5 [ (q c/p + 1) 0.2857 – 1]} 0.5 = f (qc,p,a or T)

M = { 5 [ (qc/p + 1)0.2857 - 1] } 0.5 = f (qc,p)

Be careful with units!


Angle of attack Aircraft angle of attack (AOA) is normally measured with vane-type AOA

transmitters that can be located on the fuselage, on the wing or on a boom

AOA vanes provide AOA information to stall warning / protection systems, flight controls and flight displays

AOA vanes are calibrated on prototype aircraft in order to determine the relation between local AOA at the vane location and aircraft AOA, normally defined relative to the fuselage longitudinal axis


Typical Pitot-Static System


Position errors Total and static pressures sensed by the Pitot-static

system may not be equal to free stream values for various reasons

Position errors are inaccuracies in static and/or total pressures that result in inaccurate airspeed and altitude indications unless corrections are applied


Position errors (Cont’d) Position errors for total and static pressures are defined in terms of pressure coefficients

Static pressure coefficient Cp

Cp = (plocal – p)/qic

Where plocal = local static pressure

p = free stream static pressure

qic = indicated impact pressure (ptlocal- plocal)

ptlocal = local total pressure

Total pressure coefficient Cpt

Cpt = (ptlocal – pt)/qic

Where pt = free stream total pressure


Position errors (Cont’d)

Total pressure can be measured accurately as long as the lips of the Pitot-static probe are very sharp and the local AOA at the probe is less than approximately 25 degrees

Measurement of free stream static is much more difficult

Static pressure varies significantly along the fuselage

Only a few locations where Cp is equal to zero and these locations change with Mach number and AOA

It is not possible to find one location on the aircraft where plocal = p under all flight conditions





Aircraft manufacturers normally install Pitot-static probes where pressure variation is minimum for most normal flight conditions (e.g. Point 2 on figure presented on last page)• Fuselage mounted flush static ports can also be used in combination with a

Pitot probe but flush static ports are more sensitive to skin waviness effects

Other considerations must also be taken into account in order to select Pitot-static probe location• Interference with stall vanes, temperature probe, ice detectors, doors, …

• Skin waviness effects : airframe to airframe variations may be more important in some areas

Calibrations are done on prototype aircraft to determine the position error (static pressure coefficient Cp) for all flight conditions



• Trailing cone is used to measure reference free stream static pressure• Noseboom is used to measure reference total pressure • Reference pressures are compared with ship pressure to determine Cp

• Calibrated pacer aircraft can also be used to determine position error



Total pressure errors are normally negligible• Errors are normally only a concern at high AOA (stall) if probes are

properly located on the airplane

Static pressure errors are affected by different parameters depending on the speed regime

• Low speed regime (M < 0.3): Cp is typically only a function of AOA or CLI, where CLI = (295.37*NZ*W) / (VKIAS^2*S)

• High speed regime (M > 0.3): Cp is typically a function of M but AOA effects may also be present

Cp can be converted in airspeed, altitude and Mach number errors (Vp

, hp and Mp)


Position errors (Cont’d) Typical problem : Determine Vc and hp knowing Cp, indicated airspeed VI and indicated pressure altitude hpI

• qic is determined from VI :

q ic = po [ (1 + 0.2 (VI/ao) 2) 3.5- 1]

• plocal is determined from hpI :

For hpI< 36,089, plocal = po(1 - 6.87535 x 10-6 hpI)5.2559

• Since Cp qic = plocal – p, p = plocal – Cp qic

= p/po

hp = (1 - 1/5.2559)/6.87535 x 10-6

• qc = qic + Cp qic

Vc = ao { 5 [ (q c/po + 1) 0.2857 – 1]} 0.5


Position errors (Cont’d) Example : Errors associated with Cp = -0.01 for hpI = 5000 ft

VI hp hp VcVp

100 kts 4993 ft - 7 ft 99.5 kts -0.5 kts

200 kts 4977 ft - 23 ft 199.0 kts -1.0 kts

300 kts 4952 ft - 48 ft 298.6 kts -1.4 kts



Static pressure error can either be compensated aerodynamically or electronically in order to minimize errors in indicated airspeed and altitude values

• Design of the pitot-static probe can be modified to compensate (in part) the position error (aerodynamic compensation)

• A Static Source Error Correction (SSEC) can be programmed in the Air Data Computer (ADC) to compensate the position error (electronic compensation)

• Electronic compensation is used to compensate position error on most modern aircraft

• SSEC is typically a function of Mach number, AOA and flap position



Residual altitude and airspeed errors, once compensation is applied, must be presented in the AFM

FAR/JAR 25 defines limits for airspeed (FAR 25.1323) and altitude (FAR 25.1325) errors

Altitude error at SL must not exceed 30 ft per 100 ktsAltitude error at SL must not exceed 30 ft at speeds < 100 kts



Airspeed error must not exceed 3 knots per 100 knotsAirspeed error must not exceed 5 knots at speeds < 166 knots

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Module 2