Aerodynamics for Students
www.aerodynamics4students.com (c) 2002-2014 Auld,
Aerospace Mechanical and Mechatronic Engineering
The University of Sydney. NSW. Australia
ABN: 15 211 513 464. CRICOS number: 00026A
Aircraft Performance
Sections :
•
•
•
•
•
•
•
•
•
•
•
•
Properties of the Atmosphere
Aircraft Weight and Geometry
Airspeed Measurement
Lift and Lift Coefficient
Drag and Drag Coefficient
Engine Thrust and Power
Flight Envelope
Take-Off and Landing
Climb and Descent
Range and Endurance
Manoeuvres
Performance Envelopes
International Standard Atmosphere (ISA).
The International Standard Atmosphere (ISA) has been agreed to by many representative countries and entities
in the aviation field. It is a representative model of the atmosphere and not an average. Most of the participants
in aviation are in the northern hemisphere so this standard is based on the range of local weather conditions that
occur in their locations. The standard results will need to be adjusted to suit weather conditions elsewhere.
Sea Level Conditions
Property
Metric unit
Imperial Unit
Pressure, P
101.3 kPa
2116.7 lbf/ft2
Density,ρ
1.225 Kg/m3
0.002378 slug/ft3
Temperature,T
15oC , 288.2 K
59 oF , 518oR
Speed of Sound, a
340.3 m/s
1116.4 ft/s
Viscosity,μ
1.789 x 10-5 Kg/m/s
3.737 x 10-7 slug/ft/s
Kinematic Viscosity, ν
1.460 x 10-5m2/s
1.5723 x 10-4ft2/s
Thermal Conductivity, k
0.02596 W/m/K
0.015 BTU/hr/ft/oR
Gas Constant, R
287.1 J/Kg/K
1715.7 ft lbf/slug/oR
Specific Heat, Cp
1005 J/Kg/K
6005 ft lbf/slug/oR
Specific Heat, Cv
717.98 J/Kg/K
4289 ft lbf/slug/oR
Ratio of Specific Heats, γ
1.40
Gravitational Acceleration, g
9.80665 m/s2
32.174 ft/s2
ISA Variation with Altitude
Pressure, temperature, density, viscosity and speed of sound variation for the international standard atmosphere
(ISA) can be calculated for a range of altitudes from sea level upward. This is done using an exact solution to
the hydrostatic equation for a column of air. The air is assumed to be a perfect gas. In the lower region, the
troposphere, the atmosphere has a lapse rate (L) of 6.5K/Km . At an altitude of 36089 ft the stratosphere starts
and the temperature remains constant at 217K. The hydrostatic equation, perfect gas law and the lapse rate
equations are,
,
and
where the variables used are,
P -- Pressure (Pa)
T -- Temperature (K)
g -- Gravitational acceleration (9.8
m/s2);
TO -- Standard sea level temperature (288 K);
R -- Gas constant for air (287 m2/s2/K); h -- Altitude above sea level (m),
L -- Lapse rate (0.0065 K/m) and
ρ -- Air density, ( Kg/m3);
Figure: Atmospheric Layers and Temperature Variation with Altitude.
Solving the hydrostatic equation with a constant lapse rate gives the resulting pressure variation in the
troposphere.
where sea level pressure, Po , is set at the standard 101.3 kPa.
Solving the hydrostatic equation with a constant temperature gives the resulting pressure variation in the
stratosphere.
where conditions with subscript (s) are values of altitude (hs), pressure (Ps) or temperature (Ts) at the
tropopause, the start of the stratosphere; the line dividing the two distinct atmospheric regions.
Once pressure has been calculated at a particular altitude, density is then calculated using the perfect gas law.
Viscosity and kinematic viscosity are found by applying the Sutherland law
And finally speed of sound is found based on the temperature,
Variation due to Local Ground Conditions.
On many occasions the ground temperature and pressure will not exactly be equal to ISA standard conditions.
In these cases it is possible to adjust the atmosphere model to suit local conditions. As the depth of the
atmosphere is very small (125Km-150Km) local variations in temperature and pressure will substantially effect
the full depth of the atmosphere.
Where local sea level temperature is above 15oC an ISA+ model is used. In this model the complete
atmosphere is incremented by the temperature difference between the current sea level temperature
and the standard value of 15oC. For example, on a 20oC day, an ISA + 5 model is used.
Temperature at all levels of the atmosphere model are incremented by 5 oC. With this adjustment, the
previous formula for pressure, density, viscosity and speed of sound variation can still be used.
No adjustments are made for ground surface altitude, all calculations are done based on model
starting at sea level. If the local ground level locally is well above sea level then the atmospheric
model will still be based on sea level and will start at the altitude of the ground compared to sea level.
Variation of Density due to Humidity.
The density of the air for a given level of humidity can be found by applying a correction factor to the above
calculated perfect gas density.
,
The correction factor, Kh, can be found by using wet and dry bulb temperature measurements to predict relative
humidity as shown in the table below. The relative humidity can be used to obtain the density correction factor
from the following graph. Note that the density correction factor produced assumes approximately sea level
pressure in the application of these formula.
Table: Relative Humidity (%) based on measured Temperature Difference
Relative Humidity (%)
Temperature Difference,oC TDRY - TWET
Dry Bulb
oC
1
2
3
4
5
6
7
8
9
10
11
12
81
64
46
29
13
2
84
68
52
37
22
7
4
85
71
57
43
29
16
6
86
73
60
48
35
24
11
8
87
75
63
51
40
29
19
8
10
88
77
66
55
44
34
24
15
6
12
89
78
68
58
48
39
29
21
12
14
90
79
70
60
51
42
34
26
18
10
16
90
81
71
63
54
46
38
30
23
15
8
18
91
82
73
65
57
49
41
34
27
20
14
7
20
91
83
74
66
59
51
44
37
31
24
18
12
22
92
83
76
68
61
54
47
40
34
28
22
17
24
92
84
77
69
62
56
49
43
37
31
26
20
26
92
85
78
71
64
58
51
46
40
34
29
24
28
93
85
78
72
65
59
53
48
42
37
32
27
30
93
86
79
73
67
61
55
50
44
39
35
30
32
93
86
80
74
68
62
57
51
46
41
37
32
34
93
87
81
75
69
63
58
53
48
43
39
35
36
94
87
81
75
70
64
59
54
50
45
41
37
Reference: "Handbook of Chemistry and Physics" Ed R.C.Weast, The Chemical Rubber Co.,
Ohio, 1964
Figure: Humidity Correction Factor.
Variation with Extremely High Altitude.
The above formula and the tabulated data below give accurate predictions for an ideal atmosphere in the lower
portions near the earth's surface, the troposphere and the lower stratosphere. This covers the first 20km which is
where almost all aviation takes place. For variations in density beyond this level to the edge of space, where
continuous gas behaviour no longer applies, a rough approximation can be used by assuming there is a
logarithmic decay from sea level to the edge of space.
Figure : High Altitude Density Approximation
While this estimate is rather crude for small changes over the atmosphere near the earth's surface, it is a simple
way to estimate density effects for drag on launch vehicles that will rapidly go from surface to orbital altitudes
of 200 to 300km or more.
Weight
The weight (W) of the aircraft and its aerodynamic properties are the primary factors determining its flight
performance.
The weight of the aircraft can be broken down into fundamental components:
• the empty weight of the vehicle;
• the weight of the pilot, passengers and payload;
• the weight of the fuel.
There will be limiting weight values due to the aircraft design and flight regulations:
•
•
•
•
maximum weight of payload;
maximum fuel load or fuel tank capacity;
maximum take-off weight (MTOW);
maximum landing weight.
It is not simply a matter of adding the components together to obtain a final answer for the aircraft weight. For
example it may be necessary to remove fuel weight so that additional payload may be carried while still
maintaining the requirement of a maximum take-off weight. For stability and hence flight safety considerations
an accurate "weight and balance" calculation should be performed prior to the flight of the aircraft.
In flight the aircraft weight will change as fuel is burnt by the propulsion system or possibly dumped in an
emergency situation. All the weight specifications will need to be identified from data given by the aircraft
manufacturer before force equilibrium calculations can be applied.
Aircraft Geometry.
A typical aircraft planform layout is shown in the following Figure.
S -- Wing Planform Area,
s -- Wing Semi-span,
b -- Wing Span, b = 2s .
The wing planform area (S) is shaded as shown. The wing taper ratio can be calculated as the ratio of tip chord
to root chord,
The mean aerodynamic chord can be found by integrating the individual section chords across the span.
Where
For wings with simple linear taper, the mean aerodynamic chord will roughly equal the mean geometric chord,
The aspect ratio of the wing will be,
and for wings with rectangular planform this will simplify to,
Aspect ratio will play a major part in the efficient performance of an aircraft.
Airspeed Measurement
The speed of an aircraft relative to the surrounding air is a critical part of operation and performance analysis.
No direct method is available for measurement of velocity relative to the air so instead the velocity is inferred
from measurements of dynamic pressure.
Dynamic pressure is defined as
Various forms of dynamic pressure can be obtained depending on the instruments used to measure the pressure
and the flight speed range of the vehicle. A simple airspeed indicator is shown in the section on Aircraft
Instruments. All instruments rely on knowledge of Standard Atmosphere properties.
The primary atmospheric properties can be expressed in the form of ratios compared to Sea Level conditions.
– Pressure Ratio.
– Temperature Ratio.
– Density Ratio.
The approximate dynamic pressure can be measured by a Pitot-Static system,
This pressure difference is known as the indicated impact pressure. For subsonic aircraft operating in a
normal angle of attack range, total pressure (p0) can be measured accurately. Static pressure (ps) is a less
accurate measure of the surrounding atmosphere pressure (p) and calibration factors may need to be applied to
correct for measurement errors due to the position of the device.
where qi is defined as the impact pressure.
Applying the compressible flow Bernouli equation gives the following expression for velocity,
where γ = 1.4 is the ration of specific heat for air.
For incompressible flow this simplifies to
As air density cannot be directly measured and static pressure measurements are subject to error, airspeed
indicators use ISA conditions instead. The airspeed displayed on cockpit indicators is thus,
or for incompressible flow,
For transonic and supersonic vehicles additional compressibility corrections may need to be applied. If shock
waves form in front of the pitot tube then total pressure (p0) will become inaccurate due to shock losses.
There is thus a translation sequence between the observed speed (VO) on the aircraft's instruments and the
actual airspeed of the vehicle (V). For subsonic vehicles,
VO , Observed airspeed on instrument … + Δ VIC (instrument correction)
= VI , Indicated airspeed. …
+ Δ Vpos(static port position error)
= VC , Calibrated Airspeed. …
+ Δ VC (compressibility correction)
= VE , Equivalent Airspeed. …
x 1/ sqrt(ρ) (Altitude correction)
= VT , True Airspeed = V
For incompressible flows a rough estimate can be made that
At high speed where compressibility correction becomes a large component then a Mach meter rather than an
airspeed indicator may be used.
Indicated Mach Number would thus be,
or
.
For supersonic flight when a shock forms infornt of the pitot tube an additional compressibility correction is
required.
When
then
Based on these airspeed relationships, dynamic pressure can be formulated in several ways,
Lift and Lift Coefficient
The aircraft generates lift by moving quickly through the air. The wings of the vehicle have aerofoil shaped
cross-sections. For a given flow speed with the aerofoil set at an angle of attack to the oncoming airstream, a
pressure difference between upper and lower wing surfaces will be created. There will be a high pressure region
underneath and a very low pressure region on top. The difference in these pressure forces creates lift on the
wing. The lift produced will be proportional to the size of the aircaft; the square of its velocity; the density of
the surrounding air and the angle of attack of the wing to on-coming flow.
To simplify the problem, lift is typically measured as a non-dimensional coefficient.
In the normal range of operations the variation of lift coefficent with angle of attack of the vehicle will be
approximately linear,
where
Lift coefficient increases up to a maximum value at which point the wing flow stalls and lift reduces.
The values of the lift curve gradient and maximum lift coefficient are effected by the shape of the wing, its
twist distribution, the type of aerofoil section used, the flap configuration and most importantly by the amount
of downwash flow induced on the wing by the trailing wing tip vortices.
A simple approximation for straight, moderate to high aspect ratio wings is to assume an elliptical spanwise
load distribution which gives the following result,
where it is assumed that the rate of lift coefficient change for a two-dimensional section is 2 π / radian
Calculation of zero angle lift coefficient or zero lift angle can be done by crudely assuming that the zero lift
angle for the aircraft equals the zero lift angle of the aerofoil section adjusted for the wing incidence setting.
Section zero lift angle can be calculated from analysis of the section geometry using a method such as thinaerofoil theory or panel method analysis.
A rough approximation is that zero lift angle for the section lies between -3o and -1.5o. Calculation of
maximum lift coefficient can be again take as approximately equal to the two dimensional section value minus
5% due to the negative lift needed at the tailplane to maintain moment equilibrium. A typical CL versus α graph
is shown in the following Figure. Results for the two-dimensional section and an aspect ratio 7 rectangular wing
using this section are shown.
For swept wings, wings with complex taper or wings with flaps, a more accurate calculation needs to be
undertaken using either lifting line theory or the vortex lattice method.
Minimum Flying Speed
From the the typical lift coefficient graph, it can be seen that there exists a maximum lift coefficient (CL(max))
for the aircraft. This sets the absolute lower speed limit for flight. If the aircraft attempts level flight below this
minimum speed then, for L=W, with other parameters ( such as density and area ) fixed, then the aircraft would
require an attitude that produced a lift coefficient larger than the maximum possible.
Using angles of attack that exceed the maximum lift coefficient causes the wing flow to separate and the
aircraft to stall. So, the minimum speed where the aircraft is at maximum attitude and maximum lift coefficient
is called the stall speed.
By applying the equilbrium equation at this speed, the stall conditions can be calculated.
,
so stall speed will be
Drag and Drag Coefficient
In moving through the air an aircraft experiences a resistive drag force. This force is made up of several distinct
components
Friction drag + Pressure drag + Compression drag + Lift induced drag
In theory the drag can be predicted by using a simple parabolic drag assumption
If the small offset due to camber is neglected then this can be written simply as,
This lumps friction and pressure components into a constant base drag term and then treats the variation due to
friction, pressure and most importantly lift induced drag, as a quadratic function of lift.
The effect of compressibility can be predicted by the use of a correction factor for speeds ranging from M=0.4
up to transonic.
Once a critical Mach number, Mcrit is reached the flow becomes more complex and drag coefficent increases.
Typically Mcrit will be in the range 0.8 to 0.9. For these flows in the transonic region (M = 0.8 to 1.5 ), mixed
subsonic and supersonic areas exist in the flow field and analysis will only be possible using high order
numerical codes or by referencing experimental data. After the transonic region (M > 1.5), fully supersonic
flow is established and CD can again be predicted using compressible flow gas dynamics equations (see Gas
Dynamics Section) for details.
The lift dependant component of drag coefficient can be approximated as K.CL2 where
Here e is the wing planform efficiency factor. Values for these drag constants for various categories of aircraft
are shown in the following table.
Table: Nominal Drag values for several class of aircraft.
Aircraft Type
Cdo
e
Single Engine
Light Aircraft
-No Struts
0.023
0.8
Single Engine
Light Aircraft With Struts
0.026
0.8
Multi Engine
Widebody
Aircraft
0.019
0.84
Twin Engine
Widebody
Aircraft
0.017
0.85
Twin Engine
Commuter
Aircraft
0.021
0.85
Military Aircraft 0.028
with external
stores
0.70
Vintage Bi0.038
planewith struts
and bracing wire
0.70
The consequence of this physics is that most aircraft have a roughly quadratic drag curve for level flight. The
graph below shows that there is a minimum drag speed for level flight. Flying slower results in increased drag
due to lift induced effects, flying faster increases drag due to friction effects. Attempting to fly below the
minimum drag point is problematic and is a “back the front situation”... the slower you go the more drag
results.
Drag versus Airspeed (Typical Light A/C).
Supersonic Drag
For supersonic aircraft the simple parabolic drag approximation is not accurate. Compressible flow effects
including shock and expansion waves cause complex variations in drag for both speed and altitude. The
following figure shows a typical drag variation for a supersonic aircraft plotted against Mach number.
To get performance results in this case experimental or flight test data or complex CFD analysis would be
required.
Minimum Drag Speed
For level flight Lift = Weight, so the required lift at the various airspeeds shoud be constant. The speed at
which minimum drag occurs is the same as the point at which max L/D or minimum D/L occurs.
The minimum D/L location can be found by the standard method of differentiation to look for a point of
inflection, that minimum D/L or minumum drag occurs when
this is when
or when
hence
By substituting this back into the steady flight equiblibrium equation of L=W then an airspeed for minimum
drag can be found,
so that
Production of Thrust
Thrust to overcome drag is produced by engines generally using one of the following configurations,
•
Reciprocating Piston Engine driving a propeller.
•
Gas Turbine Engine driving a propeller (turbo-prop).
•
Podded High By-Pass Gas Turbine (subsonic) (turbo-fan).
•
Internal Low By-Pass Gas Turbine (supersonic) .
Propeller Thrust
For cases (1) and (2) engine horsepower performance data will be provided from the engine manufacturer. To
find thrust, a reasonable estimate of propeller efficiency is required.
Propeller efficiency can be measured against advance ratio (J) , the ratio of forward to rotational speed of the
propeller,
where V is forward speed of aircraft, n is the propeller rotation speed in revs/sec and D is the diameter of the
propeller.
A typical fixed pitch propeller performance graph will be as shown in the following Figure.
Figure : Propeller Efficiency versus Advance Ratio.
For detailed methods of calculating propeller performance see the chapter on Blade Element Theory.
Typically maximum efficiency will be around 80% and for a fixed pitch propeller this will be achieved at an
advance ratio corresponding to cruise speed and engine RPM settings. As the advance ratio at take-off is lower
this may result in a take-off propeller efficiency of only 50%.
By using a constant speed unit on the engine and thus varing the pitch in flight it is possible to maintain high
efficiency for a range of advance ratios. The propeller will then run at maximum efficiency (80%) from take-off
speed, through climb speed to cruise and high speed cruise. The constant speed units typically have a fixed
range of pitch change so that again below take-off speed and above high speed cruise the propeller efficiency
will rapidly decline.
More advanced turbo-prop units have a greater pitch range including the options of reverse thrust and feathered
(aligned with airflow, min drag with no rotation) positions.
Given information regarding the propeller efficiency, the engine horsepower output and the speed of the
vehicle, thrust produced by the propeller can be predicted by,
Engine Power output must be obtained from engine manufacturer's datasheets and must include the effects of
altitude and temperate to obtain an accurate value for any particular flight condition.
For Gas Turbine aircraft, thrust output will be obtained directly from the manufacturer's charts. In many cases
this will only quote static sea level thrust (Tstatic). If this is the case, then estimates of thrust at altitude or high
speed must be made. Variation of static thrust with altitude can be approximated as,
where n varies between 0.7 at sea level and 1.0 at cruise conditions in the stratosphere.
Thrust variation with speed is approximately linear between static and cruise conditions,
The gradient A is relatively small and would require detailed manufacturer's engine operating data to define
exactly.
In all cases, for all engines, when the aircraft is in equilibrium at steady level flight,
Thrust (T) = Drag (D)
Power Requirements
The power required to keep an aircraft in steady level flight is,
or
Then substituting for CD variation,
for level flight L = W, so
In terms of velocity dependent components,
This fourth order curve will have a minimum value which can be found by differentiation,
Thus for minimum power,
cancelling terms in ρ, S, V2 and ½ gives,
or
Power required versus airspeed is shown in the following Figure.
The minimum power speed of the aircraft is typically 76% of the minimum drag speed.
Power Required versus Airspeed
Flight Envelope
The operation of any aircraft is expected to meet a range of performance minimum or maximum requirements.
For different categories of aircraft the minimum requirements are determined by the aircraft's flight envelope.
Aircraft must be able to meet these performance and design specifications before they can be certificated as
being air-worthy.
A typical flight envelope is a graph of speed versus load factor. Speeds are determined by the aircraft's handling
performance and desired cruise operation. Load factors are set as the limiting design requirements for the
vehicle. The aircraft and all of its components must be designed to operate safely at any point within this
envelope.
Determination of the extremities of this envelope are based on aerodynamic performance and structural
integrity requirements. The requirements for certification are documented under the Federal Aviation
Administration Airworthiness Regulations. In this document FAR 23 (CFR Section 23) normal category
requirements are referenced. See full FAR documentation for 23-utility and aerobatic category aircraft , FAR
25 for transport categories, FAR 27 and 29 for helicopters and other rotorcraft. Information given below is the
minimum requirements for normal category light aircraft.
VC – Design cruise speed.
•
chosen by the designer and normally based on operational requirements and engine power
availability.
•
VC must not be less than
knots where W/S is wing loading (lb/ft)
VD – limiting dive speed.
•
determined by the strength of components under maximum dynamic loading.
•
VD must not be less than 1.4 ×VCmin
VS – stall speed
ft/s
VA – stall speed under peak manoeuvre load
ft/s
(+)nmax – maximum positive manoeuvre load factor
•
must be no less than
•
but does not need to be greater than 3.8
(-)nmax – maximum negative manoeuvre load factor
•
magnitude must not be less than
ng – maximum load factor due to cruise gust or dive gust.
where gust factor
and
– air density in slug/ft3
c – wing chord in ft
a – wing lift curve slope in radian-1
– gravitational constant 32.2 ft/s2
– gust velocity 50 ft/s for cruise, 25 ft/s for dive.
– Aircraft equivalent airspeed in knots.
Note : As these design parameters are based on US regulations all calculations must be done using Imperial
units not metric units. Conversions should be done afterward.
Reference : Federal Aviation Regulations.
http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgFAR.nsf/Frameset?OpenPage
Take Off Performance
Take off performance can be predicted using a simple measure of the acceleration of the aircraft along the
runway based on force equilibrium.
The forces involved will be,
•
T – Thrust of propulsion system pushing aircraft along runway.
•
D – Aerodynamic Drag of vehicle resisting the aircraft motion.
•
F – Rolling resistance friction due to the contact of wheels or skids on the ground.
During take-off run the imbalance in these forces will produce an acceleration along the runway.
where dV/dt is the acceleration along the runway and m is the mass of the vehicle.
Rotation Velocity, VR
The procedure for take-off will be that the vehicle will accelerate until it reaches a safe initial flying speed. The
pilot can then rotate the vehicle to an attitude to produce climb lift and it will ascend from the ground. The
determination of this safe flying speed or rotation speed, VR, is a critical factor in determining take-off
performance.
For safety reasons VR is usually determined as being 1.2 × VSTALL or 1.1 × VMIN CONTROL ,which ever is
greater. Stall speed, VSTALL, is the lowest speed that the aircraft can be flown before the airflow starts to
separate from wings as the angle of attack becomes too great.
It can be calculated based on knowledge of the aircraft take-off configuration and hence the maximum
achievable lift coefficient CL(max). As shown in the previous section , to maintain level flight the lift produced
must equal the weight, hence the stall speed can be calculated as,
Minimum control speed, VMIN CONTROL is a more complex calculation and requires knowledge of the stall
characteristics of the tailplane and elevator. For conventional aircraft there is only a small difference between
VR calculations based on stall speed or minimum control speed.
As well as rotation speed there are other safety considerations as shown in the following Figure.
•
V1- Abort decision speed. Below this speed the take-off can be safely aborted. After this there will not
be sufficient runway length to allow the aircraft to decelerate to a stop.
•
V2 – Safe climb speed. Below this speed aircraft cannot attain sufficient climb rate.
Aircraft must climb at a minimum gradient to avoid obstacles at the end of the runway. With engine failure on
multi-engined aircraft, this speed should still be achievable.
Thrust
The thrust of gas turbine or turbofan engines will be relatively constant during take-off. A good assumption is
to use the manufacturer's values for maximum static thrust for take-off calculations.
The thrust of a propeller driven aircraft can be found from the given shaft horsepower data for the engine and
the use of the equations using propeller efficiency given in the previous section.
It is critical to correctly estimate the propeller efficiency for the particular aircraft velocity along the runway. At
V=0 the efficiency is 0 so the above equation makes no sense. At V=VR the efficiency will be in the range 50%
to 80% depending on the type of propeller system used and the thrust value at this point will be easy to obtain.
In practice, the thrust obtained throughout the take-off roll is roughly constant so this end point value is a good
approximation from V=0 to V=VR
Drag
The resistance to motion due to the air viscosity will give a drag of
where CD can be considered constant and calculated using the formula shown in the previous section
Although Drag Coefficient is constant, Drag will increase in proportion to the square of velocity.
Rolling Resistance
The friction between aircraft and runway will be proportional to the normal force exerted by the aircraft on the
runway.
The normal force will be the difference between Weight of aircraft and Lift, the friction coefficient will be
typically of a magnitude of 0.02 for a standard tarmac runway.
Average acceleration and distance to rotation
The rate of change of velocity can be predicted at any point on the take-off roll by substituting results for T, D
and F into the initial equation for dV/dt. The subsequent velocity at any point can be found by integrating this
resulting equation and the distance traveled found by then integrating the velocity.
Typically acceleration will be dominated by the drag component as thrust, weight and friction are relatively
constant during this period. This leads to the result shown where acceleration is inversely proportional to
velocity squared.
Due to the quadratic nature of acceleration change, an average value, (dV/dt)avg =
take-off run. This average acceleration can be found at the point where,
, can be used for the
This average acceleration can be used to simplify calculations and the take-off run can be calculated as an
equivalent constant acceleration over the complete period of time (tR) taken to get from 0 to VR. For a constant
acceleration take-off calculation,
and distance travelled
Rearranging leads to a relatively simple calculation to predict distance to rotation point.
Obstacle Clearance Distance
From the rotation point, the end of the runway can be defined by the requirement to clear a 50ft obstacle at the
end. During rotation it can be assumed that any residual excess thrust is absorbed in overcoming the lift induced
drag as the aircraft begins to climb. Acceleration reduces and a constant flight speed during this climb phase
can be assumed. The distance along the ground from rotation point to obstacle clearance point with thus be,
ft.
This distance estimate will require knowledge of the climb gradient which can be calculated by using the
methods in the following section on Climb and Descent.
Take-Off (Balanced) Field Length
The required length of runway will be the sum of the distance required to get to rotation speed and the extra
length required to clear a 50ft obstacle or the extra length required to allow for rapid braking if the pilot decides
to abort take-off at the decision speed V1.
This length will typically be considerably longer than the distance required to achieve rotation (flying) speed. A
rough approximation is that runway total length is around twice s1
Distance to V1 can be calculated in a manner similar to that shown for VR. The calculation of braking distance
will require knowledge of the maximum braking friction coefficient that can be generated by the aircraft. This
information should be available from manufacturer's data. Braking distance calculations should also be done
without any assumption of reverse thrust from engines as during a take-off abort, engine power may not be
available.
Landing
The landing run can be calculated in a similar fashion to the take off distance. The aim is again to minimise the
distance.
The touch down velocity should be approximately the stall speed of the aircraft in landing configuration. This
will be achieved by a pitch manoeuvre during the flare portion of the approach which which increase drag an
decelerate the aircraft to minimum flying speed.
The deceleration on the landing roll from VTD to V0 will be accomplished by braking and reverse thrust. This
can be solved by the average acceleration approach that was used to estimate the take-off roll.
The negative acceleration or deceleration value will be based on friction coefficient for maximum braking and
the value of reverse thrust (if available).
Climb Performance
Once the aircraft has left the ground and a constant speed climb established, then climb performance can be
simply calculated using a balance of the forces acting on the climbing vehicle.
The vehicle is assumed to be climbing at a constant angle (
)
) and at a constant forward velocity (V) and rate (
The balance of forces in the direction perpendicular to aircraft flight (z-wind-axis) will give :
The balance of forces in the direction parallel to the aircraft flight path (x-wind-axis) will give :
Normal aircraft climb at relatively small angles so making the small angle assumption that
allows the following prediction of climb angle,
,
Then by observing the relationship between climb angle, climb rate and flight velocity,
hence
The use of these equations in a numerical scheme is relatively straight forward. At a given aircraft weight,
altitude and flight speed it should be possible to estimate the current Thrust (T) being produced and the current
Drag (D) (see previous section on calculation of Thrust and Drag).
The difference between thrust and drag at the specified flight condition can hence be used to calculate climb
angle and climb rate for the aircraft. Clearly if T=D then the aircraft is not climbing or descending but is in
level flight. Also a obvious consequence is that if T<D then the aircraft must be descending.
This combination of terms :
is quite important in the evaluation of aircraft performance.
It is given the name Specific Excess Power ( Ps ). The fact that the terms have velocity times force units
involved makes this a term in power; excess power, because of the assumption that T × V, power available
from engines, is greater than D × V, power required to overcome energy loss due to motion. The Ps term is
denoted as Specific since the power terms are divided by the weight of the vehicle so that it is not being used in
absolute terms but relative to the weight of the vehicle.
For this steady flight analysis Specific Excess Power will create a climb rate but for dynamic motion of the
aircraft it can be used to climb, accelerate or turn. This will be covered in later chapters.
Investigation of the Specific Excess Power term over a range of flight speeds in the initial climb at sea level
conditions can be used to find important results. Using the above simple equations in a spreadsheet for example
can be used to predict the flight speed to use that will maximise the climb rate or the climb angle (Note: These
two optima may not occur at the same flight speed).
A Sea Level Climb Performance Spreadsheet can be obtained here to help with the estimations.
Descent
Descent from altitude to sea level is a similar but not identical problem to climb. It is most easily analysed by
looking at a force balance for a steady decent but the optimum solution fro descent is not the same as that for
climb so the equation will be used to find different solutions in this application.
Taking the balance of forces along the flight path gives,
Taking the balance at right angles to the flight path gives,
.
For descent the objective is to firstly minimise fuel usage and at the same time maximise the distance travelled.
This is a different approach compared to climb where the aim was to optimise climb rate. To minimise fuel, the
Thrust should be reduced to a minimum, so in this case we can assume T=0. To maximise distance travelled
then the aim will be to have the smallest descent angle with no thrust, that is, to find the best glide angle.
If the descent angle is assumed to be small then,
and
so that
and
Hence the optimum for descent will be to use a velocity that maximises the L/D of the vehicle.
Range and Endurance Predictions
The range of the aircraft is a function of the rate at which fuel is being burnt, the duration of cruise, the flight
speed and the aerodynamic performance during the flight. The mode of flight can also have a measurable effect
on the distance traveled. Of major importance is the fact that the fuel used to supply the engines must be carried
onboard the aircraft. This fuel weight must be supported by the aerodynamic lift along with the aircraft
structural weight and the payload. While the structure and payload weights are fixed, the fuel weight changes
significantly during the flight, leading to changing performance during long range cruise.
For cruise performance in level flight the excess power goes to zero. It is then a problem of analyzing the
aircraft internal energy balance. The balance of energy being created by the combustion of fuel and the energy
dissipated in overcoming the resistance to motion through the air.
For a short period of time, dt, while the aircraft moves a distance, ds, at a velocity V,so that, ds = V.dt, the
energy balance is,
that is
Ein=Eout
Ein=Energy obtained from fuel burnt =
where
•
dWf is weight of fuel burnt during time dt,
• Hc is energy content (calorific value) of fuel and
•
is the efficiency of the propulsion system.
This is also equivalent to the Energy produced by the propulsion system over the time step
Ein = T.V.dt
where,
• Tis the thrust produced,
• V is the flight velocity, hence
• T.V is the supplied power over time dt.
Thus
Rearranging this term into parameters that are readily able to be measured gives
where, TSFC is thrust specific fuel consumption, and is normally available from engine manufacturer tests.
Eout = Energy dissipated due to aircraft motion = D.V. dt
where
D is drag on the vehicle, V is flight velocity,
hence D.V is required power to be overcome during time dt.
This balance of energy components Ein = Eout gives,
.
In terms of specific energy,
Rearranging gives,
.
Based on the assumption of level flight, L = W, the assumption of roughly constant altitude and velocity cruise,
,and the substitution for change in weight as a loss of vehicle weight rather than fuel burn weight,
, then
or
Assuming approximate steady conditions where velocity and propulsion performance is kept constant and that
the aerodynamic parameter L/D is maintained at a constant value then integration of this equation gives a
prediction of range,
.
This is the classic Breuget Range equation. It can be used to give reasonable estimates of range in still air.
However the assumption of constant L/D and constant V/TSFC may not be too accurate in practice. To
maintain constant L/D with the changing weight the aircraft would need to drift up in altitude so that a constant
angle of attack was maintained. It may also be required that the aircraft change speed to maintain a constant
V/TSFC. So to get detailed estimates over long range cruise conditions it may be necessary to do a numerical
summation over short segments where the assumptions are accurate.
Range with Headwind or Tailwind
The above calculations are based on still air conditions. If the aircraft is flying into a headwind or with a
tailwind, then the distance travelled through the air is the same but the ground distance covered will change.
The energy dissipated in overcoming drag is unchanged but the effective energy output relative to the ground
will be based on the relative velocity, (V - Vw) where Vw is the velocity of the wind, positive meaning a
headwind.
The energy balance becomes,
and the final range equation becomes,
Endurance
This is the total time taken during flight. It is directly related to the rate of fuel consumption.
If the aircraft is flown using a fuel flow rate so that T = D then
Optimum Range or Endurance
In order to maximise range for a given load of fuel, a balance of V/TSFC and L/D must be found. Increasing V
will increase range up to a point where the increasing drag will start to reduce L/D so that the range, which is
the product of these two terms, again starts to reduce.
In order to maximise endurance, fuel flow rate must be reduced. For engines with roughly constant TSFC this
means reducing thrust by flying slowly at minimum drag speed or minimum power speed.
The optimum flight speed for maximum endurance will be quite low and may be approaching the stall speed,
whereas the optimum flight speed for range will be high.
Manoeuvre Performance
In order to manoeuvre or turn an aircraft, additional energy must be expended. For the case of a steady level
turn, an equlibrium of forces can be used to analyse the situation and determine relevant turn parameters.
If the aircraft is in a balanced turn at a constant airspeed then the following forces are applied to the aircraft.
Lift will act at right angles to the fuselage reference line, weight will act vertically down. So to maintain a level
turn the lift will need to be increased so that is vertical component balances the weight.
thus
where n is called the load factor and
The unbalanced horizontal component of the Lift (
) will cause an acceleration of the aircraft in a
direction at right angles to the flight path. As the flight path will be tangent to a circle based on the turn radius
(R), this acceleration will be the angular acceleration of the turn.
where m is the mass of the aircraft, (
Hence the bank angle (
).
) and the aircraft velocity (V) will determine the rate and size of turn, since
and
,
.
While these predictions are relatively simple, there are some hidden limiting conditions that must be accounted
for. As a lift increase is required for a turn, without change in speed or altitude, the only way to do this is to
increase the angle of attack (
) . Hence CL increases but
must be kept less than
CL(max).
As lift increases, lift induced drag (
) will increase, so the excess power balance must be maintained
with T=D (Ps = 0). Thrust will also need to be increased compared to level flight and required thrust must not
exceed the maximum available at the altitude, otherwise a minimum Ps = 0 will not be able to be maintained. If
the maximum thrust requirement is exceeded then Ps will become negative and only descending or lower rate
turns will be possible.
The load factor information can be evaluated along with specific excess power to determine the maximum load
factor possible at a given speed and altitude whilst still maintaining at least Ps=0. This is shown in the
following graph.
Separately it is possible to determine the turn rate available at these load factors. By mapping, contours of
maximum turn rate for different speeds and altitudes can be obtained.
Note that the stall speed becomes significantly larger at higher load factors and this will produce significant
limitation for the aircraft's operation at lower speed.
These are not simple calculations. For each altitude and velocity point, different load factors produce different
values of Ps and turn rates. An interpolation technique as shown in the MATLAB script shown here will be
required.
Performance Envelopes
The variation of aircraft performance parameters such as climb rate, acceleration capability, range vary
significantly based on aircraft speed and altitude. To correctly determine the optimum operation of the aircraft,
the performance measures should be mapped over the full operating range of the aircraft.
As seen in the climb section one important performance parameter is Specific Excess power (Ps). Another is
specific Energy (he). The energy (E) of an aircraft is made up of a potential component (mgh) and a dynamic
component (1/2mV2).
Specific energy (he) is the energy per unit weight,
The energy and power of the vehicle are related by the fact that power is the rate of change of energy. Hence
specific excess power is the rate of change of specific energy for a vehicle.
hence
This is an extension of what was found for a constant speed climb and shows that power is also available for
aircraft acceleration.
To truly determine the capability of the vehicle, specific excess power needs to be mapped over the full range
of velocity and altitude for a vehicle. This produces a specific excess power envelope that is effectively
contours of constant specific excess power for a range of velocities and altitudes.
A sample envelope is shown for the DHC-6 twin engine aircraft. The method of obtaining the excess power
envelopes is numerically based and a simple MATLAB/OCTAVE code to perform these contouring functions
is given here.
Similarly, an envelope for specific energy can be produced that maps lines along which the aircraft can move
with any change of thrust setting and hence with no requirement for excess power. The aircraft can trade
potential energy (height) for kinetic energy (speed).
As well , a maximum range envelope can be produced that maps the value of best achievable range (km) over
the operational space.
And finally an endurance envelope,
•
By inspection of these graphical results the best flight positions can be determined and the paths with in the
envelope to reach these can be estimated. In the above example the location for best endurance flight is 60 m/s
at sea level, whereas the location for best range flight is 85 m/s at the pressurisation ceiling of 10,000ft. The
best flight path can be determined by analysing the power envelope in a manner to identify time or fuel used to
climb. For example, assuming a constant maximum throttle setting and then looking for a minimum time climb
flight path would be one method of optimising climb.
As excess power (Ps) is the rate of change of specific energy (he) then
or
and
Hence time taken to climb is the area under a graph of specific energy (he) versus (1/Ps) for points along the
path.
A simple numerical method of finding the path and optimising it so as to minimise climb is to choose a
selection of different paths and to look at the limiting cases which minimise the area.
Shown in the above figure, are paths of 1/Ps versus he for 80 flight paths; that is, all possible horizontal
acceleration paths.
If a flight path is chosen so that it lies on the perimeter of this map then a minimum time will be taken. From
take-off at (A) the aircraft should accelerate to the velocity giving maximum climb rate (B). It should then
climb at the best rates (maximum Ps) until it reaches the cruise altitude (C). The options are then to accelerate
at this altitude to cruise speed (D) or to keep climbing to reach the specific energy height for the cruise point
(E). At (E), the vehicle can then revert to cruise thrust and descent to criuse altitude (D) at the required higher
velocity by trading potential energy for kinetic. This alternate end path will minimise the time at high thrust
settings.