Nomenclature

A _e

sum of the engine core and bypass jet exit cross sectional areas

AR

wing aspect ratio

a

constant in the skin friction law – Equation (33)

a _∞

speed of sound = (γℜT _∞ )^1/2

BPR

engine nominal bypass ratio

b

constant in the skin friction law – Equation (33)

b _f

fuselage width

Cd

airframe drag coefficient = D/(q _∞ S _ref )

Cd _o

zero-lift drag coefficient

Cd _w

wave drag coefficient

C _L

overall lift coefficient = L/(q _∞ S _ref )

C _F

mean skin friction coefficient – Equation (33)

C _P

specific heat at constant pressure for air

C _T

total aircraft net thrust coefficient = n.F _n /(q _∞ S _ref )

C _t

engine net thrust coefficient = F _n /(q _∞ A _e )

c _1,2,3

dummy variables – Equations (E11), (E12) and (E13)

D

total drag force

e

aircraft Oswald efficiency factor

FCOM

flight crew operating manual

FL

flight level

F _n

net installed thrust per engine

g

acceleration due to gravity

g _n

net vertical acceleration due to gravity and Coriolis effect

g _ISA

acceleration due to gravity in the International standard atmosphere (9.80665m/sec ² at sea level)

H

geopotential altitude above local sea level – Appendix A

h _0-2

engine dependent functions – Equations (24), (27) and (28)

h

geometric height above local sea level = geodetic height

J _1-3

constants in Equations (42) and (44)

K s

lift-dependent drag factor – Equation (30)

k ₁

miscellaneous lift-dependent drag factor – Equation (37)

L

lift force

LCV

lower calorific value of fuel ( $\approx$ 43×10⁶J/kg for kerosene)

L/D

lift-to-drag ratio

l

characteristic streamwise length = $S_{ref}^{1/2}$

M _∞

flight Mach number = V _∞ /a _∞

M _cc

crest critical Mach number – Equation (40)

M _TF

‘technology level’ constant in Equation (41)

MTOM

maximum permitted take-off mass

m

instantaneous total aircraft mass

${\dot m_f}$

fuel mass flow rate – summed over all engines

n

number of engines on aircraft

p

static pressure

p _i

impact pressure – Equation (59)

q _∞

freestream dynamic pressure = 0.5 $\rho$ _∞ (V _∞ )² = 0.5 $\gamma$ p _∞ (M _∞ )²

R ^ac

characteristic Reynolds number – Equation (34)

ℜ

gas constant for air (287.05J/(kg K))

r

the geocentric radius of the Earth at sea level

r _E

equatorial radius

r _P

polar radius

S

distance travelled through the air

S _ref

aerodynamic reference wing area (Airbus definition)

s

wingspan

T

static temperature

T _R

engine throttle parameter – Equation (E4)

T _o

total temperature – Equation (E5)

TCDS

type certificate data sheet

TET

ratio of total temperature at turbine entry to freestream total temperature

t

time

V _G

speed along flight path relative to the ground

V _w

wind speed

V _∞

true air speed

X

wave drag variable – Equation (40)

$\beta$ _h

aircraft heading angle

$\beta$ _t

aircraft ground track angle

$\beta$ _w

wind direction

$\gamma$

ratio of specific heats for air (=1.4)

$\delta$ ₂

induced drag wing-fuselage interference factor

$\varepsilon$

angle between the thrust line and the flight direction

$\eta$ _o

propulsion system overall efficiency – Equation (1)

$\eta$ ₂

constant in Equation (27)

$\theta$

climb gradient – Equation (4)

$\varLambda$ _w

wing quarter-chord sweep angle

μ

dynamic viscosity

$\phi$

geocentric latitude

$\rho$

air density = p/(ℜT)

$\Sigma$

coefficient in Equation (24)

$\psi$ ₀ , $\psi$ ₆

constant coefficients – Equations (32) and (49)

$\omega$

Earth’s angular velocity

Superscripts

ac

whole aircraft value

Subscripts

AC

at the aerodynamic ceiling

B

best, or local maximum, value

buff

at buffet onset

CAS

calibrated air speed

CO

crossover value

DO

at design optimum conditions

EAS

equivalent air speed

EC

engine characteristic value

FI

flight idle

IS

‘in service’ value

ISA

International Standard Atmosphere

LS

low speed

MU

maximum useable

MCC

maximum continuous climb rating

MTO

maximum take-off rating

MO

maximum permitted operational value

Max

maximum value

Min

minimum value

O

when ( $\eta$ _o L/D) has its absolute maximum value

ref

reference

SC

at the service ceiling

SL

at sea level

SLS

at sea level static conditions

Stall

at the 1-g stall condition

TP

at the tropopause

$\eta$ B

when ( $\eta$ _o L/D) has its best value at a given Mach number

∞

flight, or freestream, value

2.0 Extension to the general case

If the overall propulsive efficiency of the engine, $\eta$ _o , is defined as

(1) \begin{align}{\eta _o} = \frac{{n{F_n}{V_\infty }}}{{{{\dot m}_f}LCV}},\end{align}

where n is the number of engines on the aircraft, F _n is the installedFootnote ³ , net thrust per engine, V _∞ is the true airspeed, $\;{\dot m_f}$ is the total fuel mass flow rate and LCV is the lowerFootnote ⁴ calorific value of the fuel, then the total fuel consumption per unit distance travelled through the air is

(2) \begin{align}\frac{{d{m_f}}}{{dS}} = \frac{{{{\dot m}_f}}}{{{V_\infty }}} = - \frac{{dm}}{{dS}} = \left( {\frac{{n{F_n}}}{D}} \right)\left( {\frac{L}{{\left( {{\eta _o}L/D} \right)LCV}}} \right).\end{align}

Here m is the instantaneous aircraft mass, S is the distance travelled through the air, L is the lift and D is the drag. In general, $\eta$ _o depends upon the net thrust, the altitude and the Mach number, M _∞ , whilst L/D depends upon the drag, the altitude and M _∞ .

Figure 1.

The forces acting upon an aircraft accelerating and climbing in the vertical plane through still air.

In the general flight situation, thrust is not equal to the drag and lift is not equal to the weight. Therefore, consider an aircraft accelerating and climbing in the vertical plane through still air with an instantaneous speed, V _∞ , relative to a stationary, ground-based observer, as shown in Fig. 1. In general, the engine thrust line is set at a fixed angle, known as the thrust setting angle, relative to the aircraft’s longitudinal datum. Hence, the angle, $\varepsilon$ , between the thrust line and the direction of travel is equal to the sum of the thrust setting angle and the angle-of-attack, also measured relative to the aircraft’s longitudinal datum. Since the air is still, V _∞ is also equal to the airspeed and the ground speed, V _G , is

(3) \begin{align}{V_G} = \frac{{{V_\infty }}}{{{\rm{cos\;}}\theta }},\end{align}

where $\theta$ is the climb gradient and, if h is the geometric height above local sea levelFootnote ⁵ , then

(4) \begin{align}\sin \theta = {{dh/dt} \over {{V_\infty }}}{\rm{\;and\;cos\;}}\theta = {\left( {1 - {{\left( {{{dh/dt} \over {{V_\infty }}}} \right)}^2}} \right)^{1/2}}.\end{align}

Relative to the ground-based frame of reference, when an aircraft is following a curved path in the vertical plane, whilst losing mass as fuel is consumed, the acceleration along the flight path is

(5) \begin{align}\frac{1}{{{g_n}}}\frac{{d{V_\infty }}}{{dt}} = \left( {\frac{{n.{F_n}{\rm{cos}}\varepsilon - D}}{{m{g_n}}}} \right) - \sin \theta + \frac{{{V_\infty }{{\dot m}_f}}}{{{g_n}m}}\end{align}

and the acceleration normal to the flight path is

(6) \begin{align}\frac{{{V_\infty }}}{{{g_n}}}\frac{{d\theta }}{{dt}} = \left( {\frac{{L + n.{F_n}{\rm{sin}}\varepsilon }}{{m{g_n}}}} \right) - \cos \theta \approx 0,\end{align}

see for example Bower et al. [Reference Bower, Qi and Bazilevs7].

Strictly speaking, since the Earth is rotating, a co-ordinate system fixed to the ground is not an inertial frame of reference. Consequently, the quantity g _n includes not only the planet’s gravitational acceleration, g, but also the vertical component of the Coriolis acceleration and the centrifugal acceleration due to the aircraft’s speed relative to the ground. As shown in Appendix A, g is the sum of Newtonian gravity and the vertical component of the Earth’s centrifugal acceleration. Both these elements are functions of latitude, $\phi$ and h. On the other hand, the vertical component of the Coriolis acceleration depends upon $\phi$ , V _G , and aircraft’s ground track relative to true North, $\beta$ _t , – see, for example, Menke and Abbott [Reference Menke and Abbott8]. In addition, since an aircraft flying at fixed height is also following a circular path, V _G provides an additional centrifugal acceleration. Combining the two effects gives

(7) \begin{align}{g_n} - g = - 2\omega {V_G}{\rm{sin}}{\beta _t}{\rm{cos}}\phi - \frac{{V_G^2}}{{\left( {r + h} \right)}},\end{align}

where r is the Earth’s geocentric radius at sea level and $\omega$ is Earth’s angular velocity. This acceleration component is sometimes referred to as the “EötvösFootnote ⁶ effect”.

For civil transport aircraft, $\varepsilon$ is usually less than 5 degrees and so its influence is always small. In addition, the climb gradient, $\theta$ , is typically less than 10 degrees and, to ensure passenger comfort, accelerations normal to the flight path are maintained at low levels. Consequently, $\theta$ is always a relatively small angle and any acceleration normal to the flight path can be neglected without causing a significant loss of accuracy.

Therefore, if the instantaneous mass of the aircraft, true airspeed, rate of climb, acceleration along the flight path and lift-to-drag ratio are known, the total required thrust is given by

(8) \begin{align}\frac{{n.{F_n}}}{{m{{\left( {{g_{SL}}} \right)}_{ISA}}}} = \left( {\frac{{{g_n}}}{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}} \right)\left( {\left( {\frac{{{\rm{cos\;}}\theta }}{{L/D}}} \right) + \sin \theta } \right) + \frac{1}{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}\left( {\frac{{d{V_\infty }}}{{dt}} - {V_\infty }\frac{{{{\dot m}_f}}}{m}} \right),\end{align}

whilst the required lift is

(9) \begin{align}\frac{L}{{m{{\left( {{g_{SL}}} \right)}_{ISA}}}} \approx \left( {\frac{{{g_n}}}{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}} \right){\rm{cos\;}}\theta. \end{align}

Here, (g _SL ) _ISA is the (constant) reference value for gravity at sea level in the International Standard Atmosphere (ISA) [9].

As shown in Appendix A, relative to (g _SL ) _ISA , the maximum possible variation of g due to latitude changes is ± 0.25%, whilst, in current operations, the maximum variation of g relative to its value at sea level due to altitude changes is about −0.5%. Hence, at the global level, deviations in g from (g _SL ) _ISA fall in the range +0.25% (sea level at the poles) to −0.75% (15,000m altitude at the equator). By comparison, the vertical component of the Coriolis acceleration has values in the range ± 0.035Footnote ⁷ m/s² (maximum at the equator), i.e. ±0.35% of (g _SL ) _ISA , whilst the maximum centrifugal acceleration due to flight speed is about −0.01m/s², or −0.10% of (g _SL ) _ISA . Therefore, taking g _n to be equal to (g _SL ) _ISA , as is often the case in analytic work, introduces an error somewhere in the range +0.5% to −1.2%.

In general, the air through which the aircraft is travelling will not be still, i.e. there will be a wind. However, if the wind is horizontal with speed, V _w , and direction, $\beta$ _w , measured relative to true North, and both quantities are steady, then according to Galileo’s principle of independence, a frame of reference that moves with the air mass, i.e. with speed, V _w , and direction, $\beta$ _w , relative to the ground, is also inertial. Relative to this ‘wind fixed’ frame of reference, V _∞ is still the airspeed, h is still the geometric height above local sea level and g is unchanged. However, the very small Eötvös acceleration in Equation (7) must always be evaluated relative to ground fixed axes and, if the aircraft heading is $\beta$ _h , V _G and $\beta$ _t are obtained from a solution of the vector wind triangle, see, for example, Huang and Cummings [Reference Huang and Cummings10], i.e.

(10) \begin{align}{V_G} = \sqrt {{{\left( {\frac{{{V_\infty }}}{{{\rm{cos\;}}\theta }}} \right)}^2} + V_w^2 - 2\left( {\frac{{{V_\infty }{V_w}}}{{{\rm{cos\;}}\theta }}} \right){\rm{cos}}\left( {{\beta _h} - {\beta _w}} \right)} \end{align}

and

(11) \begin{align}{\beta _t} = {\rm{\;}}{\beta _h} + {\rm{si}}{{\rm{n}}^{ - 1}}\left( {\frac{{{V_w}}}{{{V_G}}}{\rm{sin}}\left( {{\beta _h} - {\beta _w}} \right)} \right).\end{align}

Consequently, subject only to this small correction, Equations (8) and (9) are valid for a steady, uniform wind. Nevertheless, during normal operations, significant wind velocity variations are to be expected, in which case, a frame of reference moving with the wind speed will no longer be inertial. However, if the horizontal wind has a component in the aircraft’s direction of travel, i.e. a tail wind, V _tw , where

(12) \begin{align}{V_{tw}} = {\rm{\;}} - {V_w}{\rm{cos}}\left( {{\beta _h} - {\beta _w}} \right),\end{align}

by analogy with the Coriolis acceleration, Equation (8) can be extended to include time varying, wind strength by the introduction of an additional, inertial force, i.e.

(13) \begin{align}\frac{{n.{F_n}}}{{m{{\left( {{g_{SL}}} \right)}_{ISA}}}} &= \left( {\frac{{{g_n}}}{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}} \right)\left( {\left( {\frac{{{\rm{cos\;}}\theta }}{{L/D}}} \right) + \sin \theta } \right) + \frac{1}{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}\left( {\frac{{d{V_\infty }}}{{dt}} - {V_\infty }\frac{{{{\dot m}_f}}}{m}} \right)\nonumber\\&\quad + \frac{{{\rm{cos\;}}\theta }}{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}\left( {\frac{{d{V_{tw}}}}{{dt}} - {V_{tw}}\frac{{{{\dot m}_f}}}{m}} \right).\end{align}

As described in Appendix B, the aircraft’s air data system uses impact pressure, p _i , measured with a Pitot tube, the local static pressure, p _∞ , and total temperature, (T ₀ ) _∞ to determine the flight Mach number, M _∞ , local static temperature, T _∞ , and, hence, true airspeed, V _∞ . However, whilst, as shown in Appendix A, the true rate of climb, dh/dt, can be obtained from a knowledge of the variation of p _∞ and T _∞ with time, the air data system alone cannot provide a value for h. Consequently, in operations, when an aircraft is above the ‘transition’ altitude,Footnote ⁸ h is replaced by the non-dimensional flight level, FL. By international agreement, using p _∞ from the air data system, FL is defined as the geopotential altitude, H, measured in feet, that the aircraft would have if it was operating in the International Standard Atmosphere [9] divided by 100. The relationship between flight level and p _∞ is also given in Appendix A.

When the trajectory is obtained from a ground-based source, e.g. an air navigation service provider, the information might be presented as values of ground speed, ground track and flight level at a series of waypoints. In this situation, again using the wind triangle, the airspeed is given by

(14) \begin{align}{V_\infty }{\rm{cos\;}}\theta = {\left( {V_G^2 + V_w^2 + 2{V_G}{V_w}{\rm{cos}}\left( {{\beta _t} - {\beta _w}} \right)} \right)^{1/2}},\end{align}

whilst, as shown in Appendix A, the true rate of climb is given by

(15) \begin{align}\frac{1}{{{{\left( {{a_{SL}}} \right)}_{ISA}}}}\frac{{dh}}{{dt}} = 0.08957\left( {\frac{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}{g}} \right)\left( {\frac{{{T_\infty }}}{{{{\left( {{T_\infty }} \right)}_{ISA}}}}} \right)\frac{{dFL}}{{dt}}{\rm{\;\;}}\left( {{\rm{m}}/{\rm{s}}} \right).\end{align}

Here wind speed, wind direction and temperature information would have to be obtained from a meteorological service provider.

If the aircraft lift and drag coefficients have their usual definitions, i.e.

(16) \begin{align}{C_L} = \frac{L}{{\left( {\gamma /2} \right){p_\infty }M_\infty ^2{S_{ref}}}}{\rm{\;and\;}}{C_D} = \frac{D}{{\left( {\gamma /2} \right){p_\infty }M_\infty ^2{S_{ref}}}},\end{align}

where air is taken to be an ideal gas, S _ref is the reference wing area, $\gamma$ is the ratio of specific heats, plus a total thrust coefficient, C _T , defined as

(17) \begin{align}{C_T} = \frac{{n.{F_n}}}{{\left( {\gamma /2} \right){p_\infty }M_\infty ^2{S_{ref}}}}.\end{align}

Then Equation (8) becomes

(18) \begin{align}{C_T} &= {C_L}\left( {\frac{1}{{L/D}} + {\rm{tan}}\theta } \right) + \frac{{{C_L}}}{{{\rm{cos\;}}\theta }}\left( {\frac{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}{{{g_n}}}} \right)\left( {\frac{1}{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}\left( {\frac{{d{V_\infty }}}{{dt}} - {V_\infty }\frac{{{{\dot m}_f}}}{m}} \right) + } \right.\nonumber\\&\quad \left. {\frac{{{\rm{cos\;}}\theta }}{{{{\left( {{g_{SL}}} \right)}_{ISA}}}}\left( {\frac{{d{V_{tw}}}}{{dt}} - {V_{tw}}\frac{{{{\dot m}_f}}}{m}} \right)} \right).\end{align}

This result is applicable to all phases of flight. Consequently, the total fuel consumed per unit time, ${\dot m_f}$ , is given by combining Equations (1) and (18), i.e.

(19) \begin{align}{\dot m_f} = \frac{{n{F_n}{V_\infty }}}{{{\eta _o}LCV}} = \left( {\frac{\gamma }{2}} \right)\left( {\frac{{{C_T}}}{{{\eta _o}}}M_\infty ^3} \right)\left( {\frac{{{p_\infty }{a_\infty }{S_{ref}}}}{{LCV}}} \right).\end{align}

Further information on aircraft performance can be found in standard texts, e.g. Shevell [Reference Shevell11] or Young [Reference Young12].

3.0 Estimating the overall engine propulsive efficiency

The model relating engine overall efficieny to net thrust, altitiude and Mach number has been fully described in Poll and Schumann [Reference Poll and Schumann6]. Therefore, only a summary will be presented here.

The thrust coefficient for an individual engine is given by

(20) \begin{align}{C_t} = \frac{{{F_n}}}{{\left( {\gamma /2} \right){p_\infty }M_\infty ^2{A_e}}}.\end{align}

If the bypass and core flows are mixed at exit, A _e is the jet exit area. However, if the bypass and core flows are separate, A _e is the sum of the engine core and bypass jet exit cross sectional areas. Hence, from Equation (17), the total thrust coefficent is

(21) \begin{align}{C_T} = \left( {\frac{{n.{A_e}}}{{{S_{ref}}}}} \right){C_t}.\end{align}

The typical variation of $\eta$ _o with C _t and M _∞ is shown in Fig. 2. For operation at fixed Mach number, $\eta$ _o goes though a local maximum, ${\left( {{\eta _o}} \right)_B}$ , at a particular value of C _t , i.e. ${\left( {{C_t}} \right)_{\eta B}},$ and the locus of these ‘best’ points is given by the dashed line in Fig. 2. Poll and Schumann [Reference Poll and Schumann6] have shown that, for a given Mach number, if the values of $\eta$ _o are normalised with ${\left( {{\eta _o}} \right)_B}$ and the corresponding values of C _t are normalised with ${\left( {{C_t}} \right)_{\eta B}}$ , the resulting curves are approximately independent of Mach number for Mach numbers greater than 0.4 and almost the same for all engines, irrespective of the overall pressure and bypass ratios.

Figure 2.

The variation of overall efficiency with thrust coefficient and Mach number for a civil aircraft turbofan engine with a nominal bypass ratio of 8. Figure taken from Poll and Schumann [Reference Poll and Schumann6].

Furthermore, since, from Equations (20) and (21),

(22) \begin{align}\frac{{{C_t}}}{{{{\left( {{C_t}} \right)}_{\eta B}}}} = \frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}},\end{align}

this near-universal curve for individual engines is also applicable to the complete aircraft and, for

(23) \begin{align}0.3 \le \frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}} \lt 1.8,\end{align}

it may be represented to a good approximation, by

(24) \begin{align}\frac{{{\eta _o}}}{{{{\left( {{\eta _o}} \right)}_B}}} = {h_0} \approx \left( {1 - 0.43{{\left( {\frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}} - 1} \right)}^2}} \right)\left( {1 + {\rm{\Sigma }}{{\left( {\frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}} - 1} \right)}^2}} \right).\end{align}

For $M_{\infty}$ greater than 0.4, $\Sigma$ is zero and, for

(25) \begin{align}0.2 \le {M_\infty } \le 0.4,\end{align}

(26) \begin{align}{\rm{\Sigma }} \approx 1.30\left( {0.4 - {M_\infty }} \right).\end{align}

If values of normalised overall efficiency are required for normalised thrust coefficients below 0.3, the variation may be represented by a fourth order polynomial as described in Appendix C.

In addition, it has been shown that ( $\eta$ _o) _B and (C _T ) $_{\eta\textrm{B}}$ are functions of the Mach number, such that, for Mach numbers greater than about 0.2,

(27) \begin{align}\frac{{{{\left( {{\eta _o}} \right)}_B}}}{{{{\left( {{\eta _o}} \right)}_{DO}}}} = {h_1} \approx {\left( {\frac{{{M_\infty }}}{{{M_{DO}}}}} \right)^{{\eta _2}}}\end{align}

and

(28) \begin{align}\frac{{{{\left( {{C_t}} \right)}_{\eta B}}}}{{{{\left( {{C_t}} \right)}_{DO}}}} = \frac{{{{\left( {{C_T}} \right)}_{\eta B}}}}{{{{\left( {{C_T}} \right)}_{DO}}}} = {\rm{\;}}{h_2} \approx \left( {\frac{{1 + 0.55{M_\infty }}}{{1 + 0.55{M_{DO}}}}} \right){\left( {\frac{{{M_{DO}}}}{{{M_\infty }}}} \right)^2}.\end{align}

Here, $\eta$ ₂ is a function of the bypass ratio, with

(29) \begin{align}{\eta _2} \approx 0.65\left( {1 - 0.035\left( {BPR} \right)} \right)\end{align}

and ( $\eta$ _o) _DO , M _DO , and (C _T ) _DO are the ‘design optimum’ values.

The design optimum condition is the single combination of aircraft weight, flight level and Mach number at which both the airframe lift-to-drag ratio and the engine’s overall efficiency have local maximum values when operating in a specified atmosphere, i.e. ( $\eta$ _o L/D) is an absolute maximum. As described in Poll and Schumann [Reference Poll and Schumann5], the design weight is set to 80% of the maximum permitted take-off value, i.e. a value close to a typical mid-cruise condition, and the design atmosphere is taken to be the International Standard Atmosphere [9]. The design optimum values are characteristic of the aircraft and engine combination. Therefore, when the ( $\eta$ _o) _DO , M _DO , and (C _T ) _DO , are known, the normalising values ( $\eta$ _o) _B and (C _T ) $_{\eta\textit{B}}$ for any other operating condition are obtained from Equations (27) and (28). These are then used in Equation (24) to give the corresponding value of $\eta$ _o.

Estimates of ( $\eta$ _o) _DO , M _DO , and (C _T ) _DO for 53 different aircraft types and variants have been reported previously by Poll and Schumann [Reference Poll and Schumann4, Reference Poll and Schumann5, Reference Poll and Schumann6]. This list has now been extended to cover 67 aircraft and the complete set of values can be found in Tables 1 and 2.

Table 1.

Principal characteristics of a range of turbofan engines powering civil transport aircraft. The characteristics are averages over all engines appropriate to the aircraft type and the static thrust and fuel flow at flight idle are total aircraft values

Table 2.

Principal characteristics of a range of turbofan powered civil transport aircraft; the design optimum conditions are those at which ( $\eta$ _o L/D) has its maximum value for an aircraft with a mass equal to 80% of the maximum permitted take-off value cruising in the ISA

4.0 Estimating the lift-to-drag ratio for the clean configuration

The variation of an aircraft’s drag with lift is known as its drag polar. During the take-off, very early climb, final approach and landing, the high lift devices are deployed and the undercarriage is lowered. However, above about 3,000 feet, the aircraft is usually in the clean condition with high lift devices and undercarriage fully retracted, but still travelling at low Mach number (M _∞ <0.6). In this situation, when the lift coefficient lies between 0.3 to 0.7, the polar may be represented, approximately, by the classical two-term, low-speed form, i.e.

(30) \begin{align}Cd = function{\rm{\;}}\left( {{C_L},{R^{ac}}} \right) \approx C{d_0} + \frac{{C_L^2}}{{\pi .AR.{e_{LS}}}} = C{d_0} + KC_L^2.\end{align}

Here, K is the lift-dependent drag factor, AR is the wing aspect ratio, defined as

(31) \begin{align}AR = \frac{{{s^2}}}{{{S_{ref}}}},\end{align}

where s is the wingspan, e _LS is the low-speed Oswald efficiency factor and Cd ₀ is the zero-lift drag coefficient. As described in Poll and Schumann [Reference Poll and Schumann4], Cd ₀ is related to the mean skin friction coefficient, C _F , by

(32) \begin{align}C{d_0} = {\psi _0}C_F^{ac},\end{align}

where $\psi$ ₀ depends upon the aircraft geometry and, as shown in Poll and Schumann [Reference Poll and Schumann3], the variation of C _F with Reynolds number at Mach 0.5 can be approximated by a simple power law with constant coefficients, i.e.

(33) \begin{align}C_F^{ac} \approx \frac{a}{{{{\left( {{R^{ac}}} \right)}^b}}} = \frac{{0.0269}}{{{{\left( {{R^{ac}}} \right)}^{0.14}}}}.\end{align}

Here, the aircraft flight Reynolds number, $\;{R^{ac}}$ , is defined as

(34) \begin{align}{R^{ac}} = \frac{{l{\rho _\infty }{V_\infty }}}{{{\mu _\infty }}} = S_{ref}^{1/2}\left( {\frac{{{\rho _\infty }{a_\infty }}}{{{\mu _\infty }}}} \right){M_\infty } = S_{ref}^{1/2}\left( {\frac{{\gamma {p_\infty }}}{{{\mu _\infty }{a_\infty }}}} \right){M_\infty },\end{align}

where, l is a typical aircraft reference length, taken to be the square root of the reference wing area, a _∞ the local speed of sound and μ _∞ the dynamic viscosity.

As described in Refs (Reference Poll and Schumann3) and (Reference Poll and Schumann4), the Oswald factor is a function of Cd ₀ and aircraft geometry such that, from Equations (26), (27) and (28) of Ref. (Reference Poll and Schumann4),

(35) \begin{align}\;{e_{LS}} \approx \frac{1}{{1.03 + {\delta _2} + \pi .AR.{k_1}}},\end{align}

where

(36) \begin{align}{\delta _2} \approx 2{\left( {\frac{{{b_f}}}{s}} \right)^2}\end{align}

and

(37) \begin{align}{\rm{\;}}{k_1} \approx 0.80\left( {1 - 0.53{\rm{cos}}\left( {{\varLambda _w}} \right)} \right)C{d_0},\end{align}

with b_f being the fuselage maximum width and ${\varLambda _w}$ the wing quarter-chord sweep angle. Therefore,

(38) \begin{align}{\rm{\;}}K = \frac{1}{{\pi .AR.{e_{LS}}}} \approx {k_1} + \left( {\frac{{1.03 + {\delta _2}}}{{\pi .AR}}} \right).\end{align}

The zero-lift drag coefficient, Cd ₀ , is the sum of the pressure, or form, drag and the surface skin friction drag. At low Mach numbers, the form drag increases with increasing Mach number, whilst the skin friction drag decreases. Above a Mach number of about 0.5 these effects are approximately equal. Consequently, Cd ₀ and, hence, K may be assumed to be independent of Mach number.

At higher Mach numbers, the drag polar is modified by the effects of compressibility and the general form becomes

(39) \begin{align}Cd = function{\rm{\;}}\left( {{C_L},{M_\infty },{R^{ac}}} \right) \approx C{d_0} + KC_L^2 + C{d_w},\end{align}

where Cd _w is the wave drag coefficient. If Cd ₀ and K are independent of Mach number, Cd _w must capture all the compressibility effects, up to and including the development of regions of supersonic flow and, ultimately, the formation of shockwaves. When defined in this way, the wave drag coefficient is a function of Mach number, lift coefficient and, to a lesser extent, Reynolds number – see Shevell [Reference Shevell11].

As the flight speed increases, sonic conditions are eventually reached at the point on the wing where the local static pressure is lowest. Further increases lead to the formation of a region of supersonic flow bounded by the wing surface, a sonic interfaceFootnote ⁹ in the region adjacent to the surface and a terminating shockwave. Once this local supersonic zone is established, the terminating shockwave moves rearwards and strengthens when either the Mach number, or the lift coefficient is increased. Whilst this zone is confined to the front portion of the wing, the associated drag increase is small, being typically less than about 10 drag countsFootnote ¹⁰ and this variation is known as drag creep. However, when the terminating shockwave moves onto the rear part of the wing, it strengthens. Consequently, the drag changes following increases in M_∞ , or C _L , become much larger and this more rapid variation is referred to as drag rise.

According to the Poll and Schumann [Reference Poll and Schumann5] model, which is based upon ideas by Shevell [Reference Shevell11], transition from drag creep to drag rise is governed by the local Mach number at the wing crest. This is the point on the aerofoil where the surface slope is parallel to the undisturbed freestream direction. The drag rise begins when supersonic flow is established downstream of the crest and this occurs when the freestream Mach number exceeds a threshold value known as the crest critical Mach number, M _cc . This quantity depends, primarily, upon wing geometry and the lift coefficient, whilst the magnitude of the wave drag is governed by the ratio of M _∞ to M _cc . Hence, the variable controlling the wave drag is taken to be

(40) \begin{align}X = \frac{{{M_\infty }{\rm{cos}}\left( {{\varLambda _w}} \right)}}{{{{\left( {{M_{CC}}} \right)}^{ac}}}},\end{align}

where

(41) \begin{align}{\left( {{M_{CC}}} \right)^{ac}} = {\left( {{M_{TF}}} \right)^{ac}} - 0.10\left( {\frac{{{C_L}}}{{{\rm{co}}{{\rm{s}}^2}\left( {{\varLambda _w}} \right)}}} \right).\end{align}

Here, ${\left( {{M_{TF}}} \right)^{ac}}$ is a constant aircraft characteristic that captures the aerofoil technology level and the distribution of wing thickness-to-streamwise chord ratio across the span.

In the drag creep region,

(42) \begin{align}{\left( {C{d_w}} \right)_{creep}} \approx {\rm{co}}{{\rm{s}}^3}\left( {{{\rm{\varLambda }}_w}} \right)\left( {{j_1}{{\left( {X - {j_2}} \right)}^2}} \right),\end{align}

where j ₁ and j ₂ are constant aircraft dependent characteristics. If X is less than j ₂ , the wave drag is zero and the drag creep region is deemed to have ended when X reaches the design optimum value, X _DO , where, from Equations (40) and (41),

(43) \begin{align}{X_{DO}} = \frac{{{M_{DO}}{\rm{cos}}\left( {{\varLambda _w}} \right)}}{{{{\left( {{M_{TF}}} \right)}^{ac}} - 0.10\left( {\frac{{{{\left( {{C_L}} \right)}_{DO}}}}{{{\rm{co}}{{\rm{s}}^2}\left( {{\varLambda _w}} \right)}}} \right)}}.\end{align}

At larger values of X, i.e. in the drag rise region, an additional term is introduced to capture the effect of strengthening shock waves. Hence,

(44) \begin{align}{\left( {C{d_w}} \right)_{rise}} \approx {\rm{co}}{{\rm{s}}^3}\left( {{{\rm{\varLambda }}_w}} \right)\left( {{j_1}{{\left( {X - {j_2}} \right)}^2} + {j_3}{{\left( {X - {X_{DO}}} \right)}^4}} \right),\end{align}

where j ₃ is also a constant and is currently taken to be 40. Values of j ₁ and j ₂ are given in Table 2.

At all flight conditions, the lift-to-drag ratio is given by

(45) \begin{align}\frac{L}{D} = \frac{{{C_L}}}{{Cd}} = \frac{{{C_L}}}{{\left( {C{d_0} + KC_L^2 + C{d_w}} \right)}}.\end{align}

By way of illustration, the variation of L/D with drag coefficient and Mach number at a fixed total mass for the A320 has been estimated using the Equations (30)–(45) with the input parameters given in Table 2. The results are presented in Fig. 3. As expected, the maximum value of L/D and the corresponding drag coefficient decrease as the flight Mach number increases.

Figure 3.

Approximate variation of lift-to-drag ratio for the A320 aircraft with drag coefficient and Mach number operating at a total mass of 58,800kg.

5.0 The operational limits

Whilst the model is valid over a wide range of conditions, there are four principal performance factors that can limit the achievable combinations of Mach number and flight level in normal operations.

5.1. Maximum lift limit

The first constraint is the lift limit, sometimes referred to as the ‘manoeuvre’ limit, which is linked to the conditions at which wing buffetingFootnote ¹¹ is first encountered during a specified manoeuvre – usually a level turn. For a given aircraft weight, the onset of buffeting defines an envelope of maximum achievable flight level for straight and level flight as a function of Mach number. This envelope also has a local maximum, i.e. at a given weight, there is an absolute maximum flight level and this is referred to as the “aerodynamic” ceiling.

Starting immediately after take-off, the aircraft must stay above the minimum speed dictated by maximum usable wing lift. In straight and level flight at low speed, the wing stalls at a particular value of the lift coefficient, (C _L ) _stall , known as the 1g stall condition. However, the stall is preceded by buffeting beginning at (C _L ) _buff , which is about 90% of (C _L ) _stall . In addition, to give the aircraft some margin for safe manoeuvre in an emergency, it is a regulatory requirement that the maximum useable lift coefficient, (C _L ) _mu , in straight and level flight be set at a value that allows a 1.3g manoeuvre, e.g. a 40° level banked turn, to be executed before buffet onset, i.e.

(46) \begin{align}{\left( {{C_L}} \right)_{MU}} = \frac{{{{\left( {{C_L}} \right)}_{buff}}}}{{1.3}} \approx \frac{{{{\left( {{C_L}} \right)}_{stall}}}}{{1.5}},\end{align}

Data given in Obert [Reference Obert13] suggests that, for aircraft in the clean configuration at low Mach number, (C _L ) _stall is about 1.4 ±0.3, whilst, from Table 2, the average value of (C _L ) _DO is seen to be about 0.55. Hence, at low speed, the ratio of (C _L ) _mu to (C _L ) _DO is about 1.8 ±0.4. Clean condition values are applicable for flight above 3,000 feet (FL>30), since at lower altitudes high-lift devices are likely to be used and, consequently, (C _L ) _mu will be much higher.

At high speeds, as Mach number increases, the shock waves on the wing eventually become strong enough to cause local, boundary-layer separation and the shock wave system itself may become unsteady. This also produces time varying forces that present as buffeting. Further increases in speed produce major changes in the wing flow, eventually leading to a situation in which the aircraft can no longer be controlled. This phenomenon is known as the high-speed stall and, for safety reasons, at each Mach number, a maximum usable, lift coefficient is imposed such that, relative to straight and level flight, and consistent with the low-speed stall, the aircraft must be able to execute a 1.3g level turn without encountering any buffeting.

Typical 1g buffet onset and 1.3g manoeuvre boundaries are shown in Fig. 4. Here, the solid line is a smoothed, empirical estimate of the typical manoeuvre boundary linking the low-speed limit of (C _L ) _mu /(C _L ) _DO equal to approximately 1.8 and the high-speed limit based upon data taken from the charts given in the Flight Crew Operating Manual (FCOM) of a typical civil transport aircraft. In general, the boundary depends upon the location of the aircraft’s centre of gravity and an average value of 35% of the mean aerodynamic chord has been assumed. More information can be found in Obert [Reference Obert13] Chapter 26 for the low-speed stall and Chapter 28 for buffet onset at high speed. The data are normalised using (C _L ) _DO and M _DO and, in this form, the curves are expected to be broadly similar for all the aircraft considered here. The solid diamond symbol shows the flight condition for ( $\eta$ _o L/D) _DO and the manoeuvre boundary loops around it, leaving a clear margin of safety before the stall.

Figure 4.

An estimate of the normalised 1g buffet onset and 1.3g manoeuvre boundaries for a typical civil transport aircraft in the clean condition. Data are taken from an FCOM and the centre of gravity is 35% of the mean aerodynamic chord.

Using the definition of lift coefficient given in Equation (16), the variation of (p _∞ )_min with Mach number at the manoeuvre boundary is given by

(47) \begin{align}\frac{{{{\left( {{p_\infty }} \right)}_{DO}}}}{{{{\left( {{p_\infty }} \right)}_{{\rm{min}}}}}} = 0.8\left( {\frac{{{{\left( {{C_L}} \right)}_{MU}}}}{{{{\left( {{C_L}} \right)}_{DO}}}}} \right)\left( {\frac{{MTOM}}{m}} \right){\left( {\frac{{{M_\infty }}}{{{M_{DO}}}}} \right)^2},\end{align}

whilst the static pressure at the design optimum condition is,

(48) \begin{align}\frac{{{{\left( {{p_\infty }} \right)}_{DO}}}}{{{{\left( {{p_{TP}}} \right)}_{ISA}}}} = \frac{{0.80}}{{{{\left( {{C_L}} \right)}_{DO}}}}{\psi _6},\end{align}

where (p _TP ) _ISA is the static pressure at the tropopause in the International Standard Atmosphere [9] and, using the notation from Poll and Schumann [Reference Poll and Schumann4],

(49) \begin{align}{\psi _6} = \left( {\frac{{MTOM.g}}{{\left( {\gamma /2} \right){{\left( {{p_{TP}}} \right)}_{ISA}}M_{DO}^2{S_{ref}}}}} \right).\end{align}

Values of $\psi$ ₆ are given in Poll and Schumann [Reference Poll and Schumann5] and are also listed in Table 2.

Figure 5.

Variation of the minimum static pressure (maximum flight level) with Mach number for the approximate manoeuvre boundary for two values of aircraft mass. Also shown is a typical cabin pressure limit (FL 420), together with a typical maximum operational Mach number limit and an approximate structural strength boundary ((V_EAS)_MO of 360 kt), plus an alternative ATM limit of 250 kt CAS below 10,000 feet.

Static pressure is linked directly to the flight level through Equations (A13) and (A14) in Appendix B, i.e., if (p _∞ /(p _SL ) _ISA ) is greater than 0.223363,

(50) \begin{align}FL = 1454.4302\left( {1 - {{\left( {\frac{{{p_\infty }}}{{{{\left( {{p_{SL}}} \right)}_{ISA}}}}} \right)}^{0.190263}}} \right),\end{align}

otherwise

(51) \begin{align}FL = 49.02022\left( {1 - 4.24436ln\left( {\frac{{{p_\infty }}}{{{{\left( {{p_{SL}}} \right)}_{ISA}}}}} \right)} \right).\end{align}

Relations for the approximate variation of normalised (C _L ) _mu with normalised Mach number are given in Appendix D and the form of the manoeuvre boundary, expressed as a normalised pressure versus normalised Mach number, is given in Fig. 5. As the aircraft climbs, the Mach number for the onset of the low-speed stall increases, whilst that for high-speed stall decreases. Eventually, a condition is reached at which both begin at the same Mach number. The resulting flight level is an absolute maximum and this is the aircraft’s aerodynamic ceiling. From Equation (D4), it is found that the Mach number at this condition, (M _∞ ) _AC , is given by

(52) \begin{align}{\left( {{M_\infty }} \right)_{AC}} \approx 1.035{M_{DO}}\end{align}

and (p _∞ ) _AC is

(53) \begin{align}\frac{{{{\left( {{p_\infty }} \right)}_{DO}}}}{{{{\left( {{p_\infty }} \right)}_{AC}}}} \approx 0.544{\left( {\frac{{{{\left( {{C_L}} \right)}_{mu}}}}{{{{\left( {{C_L}} \right)}_{DO}}}}} \right)_{LS}}\left( {\frac{{MTOM}}{m}} \right) \approx 0.98\left( {\frac{{MTOM}}{m}} \right).\end{align}

Clearly, the aerodynamic ceiling is strongly dependent upon the aircraft mass.

5.2. Maximum thrust limit

The next constraint is the thrust limit, sometimes referred to as the minimum available climb rate. This is governed by the engine’s net thrust when it is operating at its maximum continuous climb rating. The engine rating is usually determined by specifying a maximum value for the total temperature of the mixture of air and the products of combustion at the entry to the engine’s turbine section, i.e. the turbine entry temperature, or TET. As described in Ref. (Reference Poll and Schumann6), the maximum continuous climb value, (TET) _MCC , is about 0.92 times the maximum at take-off value, (TET) _MTO . With the engine at maximum continuous climb, the achievable rate of climb at a given Mach number decreases as the altitude increases and the maximum useful operational altitude is deemed to have been reached when it drops to about 300 feet/minute. This is called the service ceiling for that Mach number.

From Equations (8) and (9), assuming small angles and neglecting small terms, when the airspeed is constant, the residual available rate of climb is

(54) \begin{align}\frac{{dh}}{{dt}} \approx {V_\infty }\frac{{\left( {n.{F_n} - D} \right)}}{{m{g_n}}} = {M_\infty }{a_\infty }\frac{{\left( {{C_T} - Cd} \right)}}{{{C_L}}},\end{align}

With the engines at maximum continuous climb, the thrust coefficient is (C _T ) _MCC and when the rate of climb is 300 feet/min, the corresponding service-ceiling, lift coefficient, (C _L ) _SC , is obtained using Equations (39) and (54),

(55) \begin{align}K\left( {{C_L}} \right)_{SC}^2 + {{0.00516} \over {{M_\infty }}}{\left( {{{{{\left( {{T_{TP}}} \right)}_{ISA}}} \over {{T_\infty }}}} \right)^{1/2}}{\left( {{C_L}} \right)_{SC}} - \left( {{{\left( {{C_T}} \right)}_{MCC}} - \left( {C{d_0} + C{d_w}} \right)} \right) = 0,\end{align}

where (T _TP ) _ISA is the temperature at the tropopause in the International Standard Atmosphere. An exact solution of this equation requires several iterations.

Sample calculations have been performed for values of m/MTOM equal to 0.9 and 0.7, with ambient temperatures of ISA and ISA +20 $^{\circ}$ C. The results are given Figs 6 and 7. As expected, the thrust limited boundaries exhibit maxima corresponding to the highest achievable service ceilings. Changing weight has a major effect and increasing the ambient temperature reduces the service ceiling significantly.

Figure 6.

Variation of the minimum static pressure with Mach number for the thrust limited boundary when m/MTOM is equal to 0.9 and for ISA ambient temperature and ISA+20^°C. Also shown is manoeuvre limit for m/MTOM equal to 0.9, plus the cabin pressure limit (FL 420), together with a typical maximum operational Mach number.

Figure 7.

The same Fig. 6, but for m/MTOM is equal to 0.7.

Whilst the full calculation is complex, the exact solutions show that the static pressure at the service ceiling is not particularly sensitive to the Mach number. Consequently, a reasonable approximate solution is given by determining (C _L ) _SC at the design optimum Mach number, M _DO . As shown in Appendix E, this has the form

(56) \begin{align}{{{{\left( {{C_L}} \right)}_{SC}}} \over {{{\left( {{C_L}} \right)}_{DO}}}} \approx {\left( {{c_1}\left( {{{{{\left( {TET} \right)}_{MCC}}} \over {{T_\infty }}}} \right) + {c_2}} \right)^{1/2}} - {c_3}{\left( {{{{{\left( {{T_{TP}}} \right)}_{ISA}}} \over {{T_\infty }}}} \right)^{1/2}},\end{align}

where the coefficients c ₁ , c ₂ and c ₃ are constant and functions of the aircraft parameters at the design optimum condition, as shown in Equations (E11), (E12) and (E13). The resulting values of static pressure ratio are obtained from Equation (47), i.e.

(57) \begin{align}\frac{{{{\left( {{p_\infty }} \right)}_{DO}}}}{{{{\left( {{p_\infty }} \right)}_{SC}}}} \approx 0.8\left( {\frac{{{{\left( {{C_L}} \right)}_{SC}}}}{{{{\left( {{C_L}} \right)}_{DO}}}}} \right)\left( {\frac{{MTOM}}{m}} \right).\end{align}

The service ceiling static pressure is proportional to the aircraft mass and, since, as indicated in Equation (56), (C _L ) _SC decreases as atmospheric temperature increases, it also increases as the atmospheric temperature increases. Hence, the service ceiling flight level decreases as both mass and atmospheric temperature increase.

5.3. Maximum permitted flight level

The third restriction is the maximum cabin pressure limit. This is a structural strength constraint that fixes the maximum allowable difference between the internal cabin pressure and the external atmospheric pressure. It takes the form of a maximum permitted flight level, FL _max , given in the Type Certificate Data Sheet (TCDS), and, for civil transport aircraft, it is, typically, about FL 410. Once again, this can be expressed as a ratio of static pressures, i.e. using Equations (47), (48) and (53),

(58) \begin{align}\frac{{{{\left( {{p_\infty }} \right)}_{DO}}}}{{{{\left( {{p_\infty }} \right)}_{{\rm{min}}}}}} = 0.8\frac{{{\psi _6}}}{{{{\left( {{C_L}} \right)}_{DO{\rm{\;}}}}}}\left( {\frac{{{{\left( {{p_{TP}}} \right)}_{ISA}}}}{{{{\left( {{p_\infty }} \right)}_{{\rm{min}}}}}}} \right) = 0.1412\frac{{{\psi _6}}}{{{{\left( {{C_L}} \right)}_{DO{\rm{\;}}}}}}EXP\left( {\frac{{F{L_{{\rm{max}}}}}}{{208.06}}} \right).\end{align}

Results for this limit are included in Figs 5, 6 and 7.

In these examples, when the ambient temperature has the ISA value, the service ceiling and the aerodynamic ceiling are close together. At the higher aircraft weight, as shown in Fig. 6, both ceilings are below the cabin pressure limit and so the service ceiling is the greatest height that the aircraft can reach. Conversely, at the lower aircraft weight given in Fig. 7, both the service and the aerodynamic ceilings lie above the cabin pressure limit and so the maximum achievable flight level is determined by the cabin pressure limit. However, when the ambient temperature is increased to ISA +20 $^{\circ}$ C, the service ceiling determines the maximum achievable altitude at both weights.

5.4. Maximum speed limit

Finally, the fourth constraint is the maximum operational speed. This is also a structural strength constraint. At low altitudes, this is usually expressed as a maximum permitted value of the dynamic pressure, (q _∞ ) _MO , which, as shown in Appendix B, is equivalent to a maximum value of the equivalent airspeed, (V _EAS ) _MO , i.e.

(59) \begin{align}{\left( {{q_\infty }} \right)_{MO}} = \frac{\gamma }{2}{\left( {{p_{SL}}} \right)_{ISA}}{\left( {\frac{{{{\left( {{V_{EAS}}} \right)}_{MO}}}}{{{{\left( {{a_{SL}}} \right)}_{ISA}}}}} \right)^2} = \frac{\gamma }{2}{p_\infty }M_\infty ^2.\end{align}

Therefore, at a given Mach number, there is a maximum permitted freestream static pressure, corresponding to a minimum permitted flight level, where

(60) \begin{align}{\left( {{p_\infty }} \right)_{{\rm{max}}}} = \frac{{2{{\left( {{q_\infty }} \right)}_{MO}}}}{{\gamma M_\infty ^2}} = {\left( {{p_{SL}}} \right)_{ISA}}{\left( {\frac{{{{\left( {{V_{EAS}}} \right)}_{MO}}}}{{{M_\infty }{{\left( {{a_{SL}}} \right)}_{ISA}}}}} \right)^2},\end{align}

or, using Equation (48),

(61) \begin{align}\frac{{{{\left( {{p_\infty }} \right)}_{DO}}}}{{{{\left( {{p_\infty }} \right)}_{{\rm{max}}}}}} \approx 0.8\left( {\frac{{{\psi _6}M_{DO}^2}}{{{{\left( {{C_L}} \right)}_{DO{\rm{\;}}}}}}} \right){\left( {\frac{{{p_{TP}}}}{{{p_{SL}}}}} \right)_{ISA}}{\left( {\frac{{{{\left( {{a_{SL}}} \right)}_{ISA}}}}{{{{\left( {{V_{EAS}}} \right)}_{MO}}}}} \right)^2}{\left( {\frac{{{M_\infty }}}{{{M_{DO}}}}} \right)^2}.\end{align}

At high altitude, (V _EAS ) _MO is replaced by a maximum operational Mach number, M _MO . Constant equivalent airspeed gives way to constant Mach number at the crossover flight level, (FL) _CO , which is obtained from Equation (61), i.e.

(62) \begin{align}\frac{{{{\left( {{p_\infty }} \right)}_{DO}}}}{{{{\left( {{p_\infty }} \right)}_{CO}}}} \approx 0.8\frac{{{\psi _6}}}{{{{\left( {{C_L}} \right)}_{DO{\rm{\;}}}}}}{\left( {\frac{{{p_{TP}}}}{{{p_{SL}}}}} \right)_{ISA}}{\left( {\frac{{{M_{MO}}{{\left( {{a_{SL}}} \right)}_{ISA}}}}{{{{\left( {{V_{EAS}}} \right)}_{MO}}}}} \right)^2}.\end{align}

Values of (V _EAS ) _MO and M _MO usually appear in the aircraft’s TCDS and, whilst there is a relationship between the two, it is complex and beyond the scope of a simple representation. Nevertheless, data presented by Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes14] suggest that there is a rough correlation, with an uncertainty of about ± 10%,

(63) \begin{align}\frac{{{{\left( {{V_{EAS}}} \right)}_{MO}}}}{{{{\left( {{a_{SL}}} \right)}_{ISA}}}} \approx 0.57\left( {{M_{MO}} + 0.10)} \right).\end{align}

Therefore, approximate values of (V _EAS ) _MO can been obtained from Equation (63) using the values of M _MO given in Table 2.

Finally, below 10,000 feet, there may be a limit on the calibrated airspeed, (V _CAS ) _MO , imposed by air traffic control. Typically, this would be 250 kt. The relation between calibrated airspeed and Mach number is obtained by combining Equations (B1) and (B5) from Appendix B, which together with Equation (48) gives the relationship between maximum static pressure (minimum flight level) and Mach number to be

(64) \begin{align}\frac{{{{\left( {{p_\infty }} \right)}_{DO}}}}{{{{\left( {{p_\infty }} \right)}_{{\rm{max}}}}}} \approx 0.8\frac{{{\psi _6}}}{{{{\left( {{C_L}} \right)}_{DO{\rm{\;}}}}}}{\left( {\frac{{{p_{TP}}}}{{{p_{SL}}}}} \right)_{ISA}}\left( {\frac{{{{\left( {1 + \frac{{\left( {\gamma - 1} \right)}}{2}M_{DO}^2{{\left( {\frac{{{M_\infty }}}{{{M_{DO}}}}} \right)}^2}} \right)}^{\frac{\gamma }{{\left( {\gamma - 1} \right)}}}} - 1}}{{{{\left( {1 + \frac{{\left( {\gamma - 1} \right)}}{2}{{\left( {\frac{{{{\left( {{V_{CAS}}} \right)}_{MO}}}}{{{{\left( {{a_{SL}}} \right)}_{ISA}}}}} \right)}^2}} \right)}^{\frac{\gamma }{{\left( {\gamma - 1} \right)}}}} - 1}}} \right).\end{align}

A sample maximum operational speed boundary is included in Fig. 5 and it is independent of both the aircraft weight and the ambient temperature.

6.0 Determination of the fuel flow rate and engine overall efficiency in climb and cruise

For an aircraft of given total weight, the method’s principal output parameters are the total fuel mass-flow rate and the engine overall efficiency. These quantities are obtained by applying the following computational scheme.

The input parameters are aircraft type, mass, Mach number, flight level, rate of climb, time rate of change of true airspeed and ambient temperature. With p _∞ determined from the definition of flight level and T _∞ specified, sound speed, a _∞ , true airspeed, V _∞ , and dynamic viscosity, ${\mu}$ _∞ , follow. Lift coefficient and Reynolds number are then given by Equations (16) and (34). Values of AR, ${\delta}$ ₂, C _F , Cd ₀ and k ₁ are obtained from Equations (31) to (37), followed by e _LS and K from Equations (35) and (38).The value of the crest critical Mach number, ${\left( {{M_{CC}}} \right)^{ac}}$ is found from Equation (41) and X from Equation (40), whilst the wave drag coefficient comes from Equation (44). Values of the geometric parameters S _ref , s, b _f , and $\varLambda$ _w, are listed in Table 2 together with the derived coefficients $\psi$ 0, ${\left( {{M_{TF}}} \right)^{ac}}{\rm{\;}},{\rm{\;}}$ (C _L ) _DO , j ₁ and j ₂ . Having obtained Cd _w , the estimate for L/D is obtained from Equation (45).

The estimation of $\eta$ _o begins with the calculation of the total, net engine thrust using Equations (4) and (13), with the thrust coefficient, C _T , following from Equation (17). Using M _∞ , M _DO , ( $\eta$ _o) _DO , (C _T ) _DO and $\eta$ ₂ from Equations (29), (27) and (28) give ( $\eta$ _o) _B and (C _T ) $_{\eta\textrm{B}}$ . These quantities are then used in Equation (24) to give $\eta$ _o. Values for the engine constants ( $\eta$ _o) _DO , (C _T ) _DO and BPR are given in Table 1 and M _DO is given in Table 2.

With the engine overall efficiency and the lift-to-drag ratio now determined, the fuel flow rate is given by Equation (19).

7.0 Fuel flow rate estimate for ‘flight idle’ operation in descent

During the descent phase, the throttles may be set to the flight idle position. In this mode, the net thrust is very low, possibly negative, and fuel is being used to keep the engine running at a safe minimum speed. Under these conditions, the compressor characteristics may be very different from those in climb and cruise and there may be variations in the various blade and engine bleed settings.

When developing the engine model, Poll and Schumann [Reference Poll and Schumann6] made use of information given in Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes14], which includes results for the descent phase. These indicate that, when the flight idle fuel flow rate is normalised with respect to its value at sea-level, static conditions, the result has a strong dependence upon altitude, but only a weak dependence upon Mach number and nominal bypass ratio. Since the engine is producing close to zero net thrust at this condition, unlike the other phases of flight, there is no dependence upon aircraft weight. Curve fitting the data gives

(65) \begin{align}{\left( {\frac{{{{\dot m}_f}}}{{{{\left( {{{\dot m}_f}} \right)}_{SLS}}}}} \right)_{FI}} \approx 1 - 0.178\left( {\frac{{FL}}{{100}}} \right) + 0.0085{\left( {\frac{{FL}}{{100}}} \right)^2}.\end{align}

Information on descent phase fuel flow is given in the aircraft’s FCOM and, whilst these documents are not generally available in the public domain, some manuals and several extracts can be found on the internet. Fortunately, the ICAO Aircraft Engine Emissions Data Bank [15] lists fuel flow rates for the flight idle throttle setting at sea-level, static conditions for a large number of engines and these values are listed in Table 1. The combination of these data with Equation (65) yields an estimate of the flight idle fuel burn rate at any flight level.

Comparisons of FCOM data for three aircraft, one single aisle (aircraft 1) and two twin aisle (aircraft 2 and 3) are presented in Fig. 8, together with the estimates based upon Equation (64) and the ICAO data. The results exhibit a large degree of residual scatter, some of which is due to the difficulty of estimating instantaneous fuel flow rates from the FCOM tables. Nevertheless, the altitude dependence is clearly visible, as is the collapse achieved with the chosen normalising parameter.

Figure 8.

Variation of normalised, flight-idle, fuel flow rate with altitude for three aircraft. Open symbols are FCOM data. The solid line is the estimate from Equation (65) and the dashed lines show the ± 30% variation.

Despite the complexity of the situation, this simplified expression produces estimates that are generally within 30% of the values derived from the limited FCOM data. This rather large uncertainty is offset by the fact that, as indicted by the data given in Table 1, the fuel burn rate at flight idle is an order of magnitude lower than that at take-off. Flight idle values should be used when the aircraft is descending and the estimated fuel flow rate drops below that given by Equation (65).

8.0 Fuel flow rate estimates for take-off, initial climb-out and approach and landing

Below 3,000 feet, the aircraft is likely to have its flaps fully, or partially, deployed and the undercarriage might be lowered. Clearly, this has a major effect on the drag and, since the method, in its current form, only addresses the clean configuration, for completeness, some means of estimating fuel flow rate in these other situations is required.

For all but the shortest flights, most of the fuel is consumed in the climb and cruise. Hence, high accuracy in other phases of flight is less important and, perhaps, the simplest approach is the use the average landing and take-off (LTO) cycle specified for use with the ICAO Aircraft Engine Emissions Data Bank [15]. This gives the following recommendations.

1. 1. At take-off, the aircraft operates at the maximum available thrust for 0.7 minutes.

2. 2. During the initial climb out, the engine thrust is reduced to 85% of the maximum available for 2.2 minutes.

and

1. 3. During the final approach for landing, the engine thrust is set to 30% of the maximum available thrust for 4.0 minutes.

If the engine type is known, the fuel flow rates can be read directly from the ICAO data base. However, in many situations, the engine model will not be known. Therefore, average values for maximum take-off thrust and corresponding fuel flow rates have been obtained for each aircraft type. This was done by taking the mean values for all the engine types fitted to each aircraft. These results are listed in Table 1.

In addition, it was found that, relative to maximum take-off values, when the thrust level is reduced to 85% and 30%, the fuel flow rate drops to 0.82±0.02% and 0.28±0.02%. Applying these factors gives the approximate fuel for climb-out and approach and landing.

The relative importance of the fuel used in these stages of flight depends upon how the aircraft is operated. For flights close to the maximum possible range, they represent about 5% of the total fuel used. However, for very short flights this can increase to over 25%.

9.0 Engine deterioration in service

The performance of both airframe and engines varies with time due to in-service deterioration – see, for example, Arrieta, Botez and Lasne [Reference Arrieta, Botez and Lasne16]. In the case of the airframe, this is primarily the result of dirt accumulation on the outer surface and erosion, or damage, to the paint covering. There is also the possibility of rigging errors when major components, e.g. ailerons, are refitted following removal for inspection, or repair. All these can result in increased drag. However, this penalty is expected to be small. In the case of the engines, dirt is also a major issue, leading to a progressive reduction of component polytropic efficiencies through surface contamination (fouling) and surface erosion. There is also the progressive wearing of seals and bearings. These effects all lead to a steady increase in the rate at which fuel is burned for a given thrust.

As discussed in Ref. Reference Arrieta, Botez and Lasne16, engine degradation rate depends upon the operational cycle and the operating environment. Consequently, results for short haul and long haul are different. Airlines can take low-level remedial action, such as core washing, whilst the engine is still in service, or the engine can have a major overhaul. During its lifetime, an engine might undergo two, or three, complete overhauls and during its lifetime an aircraft may need two, or more, sets of new engines. This whole process is driven by airline economics. However, irrespective of the operational cycle, as indicated in Ref. Reference Arrieta, Botez and Lasne16, the maximum acceptable increase in fuel consumption before a maintenance intervention is likely to be in the region of 5%.

When the method is applied to a particular aircraft, it is unlikely that the degree of in-service deterioration will be known. However, as noted above, this extra fuel use is likely to be between 0 to 5%. Therefore, it is assumed that the method gives estimates that are representative of a nearly new aircraft and that the mean, in-service deterioration is taken to be 2.5%. To capture this, a multiplier of 0.975 is applied to the engine overall efficiency, i.e. the in-service overall efficiency is

(66) \begin{align}{\left( {{\eta _o}} \right)_{IS}} \approx 0.975\left( {{\eta _o}} \right).\end{align}

This corrected value is used in Equation (19) to obtain the final estimate of the fuel consumption.

10.0 Comparison with flight data recorder information

The type of in-flight information required to test any performance model is commercially sensitive and, consequently, examples rarely appear in the public domain. However, some years ago Swiss International Airlines released a set of flight recorder data (FRD) from operations in 2008 and this has been used in a number of studies, e.g. Randle et al. [Reference Randle, Hall and Vera-Morales17], Simone et al. [Reference Simone, Stettler and Barrett18] and, more recently, Hall et al. [Reference Hall, Burnell and Deshpande19]. The aircraft involved are single examples of the Airbus A320-200, A330-200 and A340-300 and the Boeing B757-300, B767-300 and B777-300 types, operating on different routes. There are 885 complete flights covering the North Atlantic, the tropics and the Indian Ocean, with a wide range of operational conditions, masses ranging from empty and to the maximum take-off values and ambient temperatures between ±20 K relative to ISA.

The measurement accuracy of the data has been discussed by Vera-Morales and Hall [Reference Vera-Moreles and Hall20], Randle et al. [Reference Randle, Hall and Vera-Morales17] and Simone et al. [Reference Simone, Stettler and Barrett18]. They estimate that the fuel flow data are accurate to ±1 %, whilst the estimated accuracies of FL, M _∞ , and T _∞ to be ±0.3, ±0.01, and ±2K, respectively. In addition, the instantaneous aircraft mass depends upon the input value take-off mass, whose accuracy is not known. The data are also very noisy, exhibiting fluctuations in excess of ±10%. There could be many reasons for this, including errors introduced during data processing, oscillations due to dynamic response to turbulence and other perturbations such as minor pilot, or autopilot, control inputs.

Figures 9–14 show comparisons between the estimates for instantaneous fuel flow rate and the values from the flight data recorder. In each case, 20 complete flights are shown and they have been filtered using the performance boundaries described in Section 5Footnote ¹² . The results arrange themselves into broad groups representing the climb, cruise and descent phases. It is also clear that, whilst the noise level is significant, the majority of the data fall within ±15% of the mean line.

Figure 9.

Comparison between estimates for fuel flow rate and values obtained from the flight data recorder on an Airbus A320-200 aircraft. Data are taken from 20 complete flights $\approx$ 2,200 points. The solid line shows the mean variation, which has a slope of 0.91 and the dashed lines show the ± 15% variation.

Figure 10.

As for Fig. 9, but for an Airbus A330-200, with $\approx$ 2,850 points and a slope of 1.00.

Figure 11.

As for Fig. 9, but for an Airbus A340-300, with $\approx$ 3,850 points and a slope of 0.94.

Figure 12.

As for Fig. 9, but for a Boeing B757-300, with $\approx$ 2,100 points and a slope of 0.97.

Figure 13.

As for Fig. 9, but for a Boeing B767-300, with $\approx$ 1,900 points and a slope of 0.91.

Figure 14.

As for Fig. 9, but for a Boeing B777-300, with $\approx$ 2,700 points and a slope of 1.00.

The slope of the mean line depends, primarily, upon a combination of two factors. The first is the accuracy of the model parameter values listed in Tables 1 and 2, whilst the second is the accuracy of the assumed aircraft take-off mass. As can be seen from Equation (2), in steady, straight and level cruise, ( $\eta$ _o L/D) is constant and, hence, the fuel flow rate is directly proportional to the aircraft’s instantaneous mass. There is a stronger dependency on mass in the climb and a weaker dependency in the descent. Nevertheless, numerical experiments on the data for complete flights show that the slope of the mean line is almost directly proportional to the assumed value of the take-off mass.

Whilst the FDR information includes take-off mass, its origin is unknown and its accuracy does not appear to have been assessed. Take-off mass is an important parameter for aircraft performance. However, in normal operations, aircraft weight is not measured directly before departure. Consequently, the signed-off, take-off mass may not be exactly equal to the true take-off mass. There may be several reasons for this mismatch, for example the passenger weight is estimated and the taxi-out fuel may not all be used prior to take-off.

Notwithstanding the potential uncertainty relating to take-off mass, the overall agreement between the estimates and the FDR values is good, demonstrating that, within reasonable uncertainty bounds, the method works in all phases of flight.

When the estimates for the total amount of fuel used on each flight are compared, the mean deviations relative to the FDR values are −10% for the A320, +1% for the A330, −2% for the A340, −1% for the B757, −3% for the B767 and 0% for the B777. Once again, the total fuel used is directly proportional to the assumed take-off massFootnote ¹³ and so these discrepancies are also due to a combination of errors in the model parameter values and in the assumed take-off mass. Figure 15 shows the comparisons for the total amount of fuel used on each of the 885 flights. The results have been normalised for each aircraft so that deviations from the mean values can be determined. The residual RMS error is found to be less than 2%, with just one point exhibiting an error of more than 10%.

Figure 15.

Comparison of the estimated total trip fuel and the value obtained from the FDR for 885 flights. The data are normalised to remove any differences in the mean values. The dashed lines show the ±5% variation.

Further improvements in the model will be possible if more flight data, with well characterised accuracy, become available.