3.1 Governing equations

The variation of the drag force with lift is given by the aircraft’s “drag polar.” This depends upon Mach number and Reynolds number and the complete polar covers all phases of flight, including the take-off and landing. However, for the current purposes, attention is restricted to that portion of the “trimmed”Footnote 2 polar that is relevant to cruise, i.e. flight at transonic speeds. Hence, the ranges of lift coefficient and Mach number that need to be considered are greatly reduced and, consequently, it is usually possible to represent the relevant portion, accurately, with an expression of the form

(4) \begin{align}Cd = {D \over {\left( {\gamma /2} \right){p_\infty }M_\infty ^2{S_{ref}}}} \approx {\left( {C{d_o}} \right)_{HS}} + {\left( {C{d_i}} \right)_{HS}}C_L^2 = {\left( {C{d_o}} \right)_{HS}} + \left( {{1 \over {\pi .AR.{e_{HS}}}}} \right)C_L^2 \end{align}

Here, C _L is the lift coefficient, defined as

(5) \begin{align}{C_L} = {{mg} \over {\left( {\gamma /2} \right){p_\infty }M_\infty ^2{S_{ref}}}} \end{align}

In addition, (Cd _o ) _HS is the high-speed, zero-lift drag coefficient, (Cd _i ) _HS is the high-speed, lift dependent drag coefficient, e _HS is the high-speed Oswald efficiency factor and AR is the wing aspect ratio, defined as

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

where s is the wingspan.

The primary source of lift-dependent drag is the induced, or vortex, drag acting on the wings. This mechanism is described in standard aerodynamics texts, e.g. Ref. Reference Shevell2. However, on a complete aircraft configuration, there are other sources of vortex drag, e.g. drag due to the trimming load on the tailplane and drag due to lift generated on the fuselage. In addition, there are lift-dependent drag contributions that are not vortex related. These arise because a change in lift alters the pressure distributions over the various components. This, in turn, changes the boundary layer development and, hence, the component’s profile drag. All these lift-related effects are captured by the Oswald factor.

In general, the high-speed quantities appearing in Equation (4) are Mach number dependent and difficult to estimate. However, their low-speed values can be determined with reasonable accuracy using classical aerodynamic theory. Consequently, it is convenient to recast Equation (4) in the form

(7) \begin{align}Cd \approx C{d_0} + \left( {{1 \over {\pi .AR.{e_{LS}}}}} \right)C_L^2 + C{d_c}, \end{align}

where, Cd ₀, and e _LS have their low-speed, or incompressible, values and Cd _c is the compressibility drag coefficient. Compressibility drag captures all the effects of Mach number. Initially, these come from skin friction variation and changes to the surface pressure distribution, with the consequential impact upon boundary layer development and profile drag. At sufficiently high Mach numbers, the surface pressures change due to the establishment of local regions of supersonic flow. Eventually, shock waves form. These produce further modifications to the wing pressure distribution and involve complex interactions with the boundary layer. At Mach numbers below those at which regions of supersonic flow appear, Cd _c increases slowly as the Mach number increases due to the changing pressure distribution, whilst decreasing slowly due to the reducing surface friction. However, as described in Ref. Reference Shevell2 (Chapter 12), above a Mach number of about 0.5, these opposing effects are usually assumed to cancel each other out and Cd ₀ becomes independent of Mach number. In addition, according to classical aerodynamic theory, e.g. Refs Reference Shevell2 and 3, below the speeds at which shock waves form, the Oswald factor is approximately independent of Mach number. Therefore, for Mach numbers greater than 0.5, Equation (7) may be written as

(8) \begin{align}Cd \approx {\left( {C{d_0}} \right)_{M =0.5}} + \left( {{1 \over {\pi .AR.{e_{LS}}}}} \right)C_L^2 + C{d_w}, \end{align}

where Cd _w is now that part of the drag resulting from the development of significant regions of supersonic flow at the wing surface and, eventually, due to shock wave formation. These high Mach number effects are often referred to collectively as “wave drag”, e.g. see Ref. 2, and, in general, Cd _w depends upon Mach number, lift coefficient and, to a lesser extent, Reynolds number. Since the zero-lift, profile drag coefficient is directly proportional to the mean skin-friction coefficient, it may be linked to the drag coefficient of a flat plate at zero incidence, $C_F^{FP}$ , and a Mach number of 0.5, i.e.

(9) \begin{align}C{d_0} \approx {\psi _0}{\left( {C_F^{FP}} \right)_{M =0.5}} = {\psi _0}\left( {C_F^{ac}} \right), \end{align}

where the constant of proportionality, $\psi$ ₀, depends only on the aircraft geometry and $C_F^{ac}$ depends only on the aircraft’s flight Reynolds number.

If the Reynolds number and, hence, C _F are constantFootnote 3, then, in the absence of wave drag (Cd _w = 0), Equations (8) and (9) may be combined to show that the lift-to-drag ratio (L/D = C _L /C _d ) exhibits a maximum, whose value only depends upon the aircraft geometry, the Oswald factor and the magnitude of the mean skin-friction coefficient, i.e.

(10) \begin{align}{\left( {{L \over D}} \right)_{max}} \approx {1 \over 2}{\left( {{{\pi .AR.{e_{LS}}} \over {{\psi _0}}}} \right)^{1/2}}{\left( {{1 \over {C_F^{ac}}}} \right)^{1/2}}, \end{align}

when

(11) \begin{align}{\left( {{C_L}} \right)_{LDm}} \approx {\left( {\pi .AR.{e_{LS}}{\cdot}{\psi _0}} \right)^{1/2}}{\left( {C_F^{ac}} \right)^{1/2}} \end{align}

As already noted, the Oswald factor contains elements of both vortex and non-vortex drag. However, it depends, primarily, upon the wing form drag. Consequently, it is expected to depend not only upon geometric parameters, but also upon the zero-lift, profile drag coefficient, Cd _o . As described by ShevellFootnote 4 in Ref. Reference Shevell2 (Chapter 11, Fig. 11.8), for this class of aircraft, e _LS depends, primarily, upon wing aspect ratio, wing sweep angle and Cd _o . It is usually expressed in the form

(12) \begin{align}{e_{LS}} \approx {1 \over {\left( {1.04 + \left( {\pi .AR.{k_1}} \right)} \right)}} \end{align}

Figure 1. Examples of the variation of the low-speed, Oswald efficiency factor with the zero-lift, profile drag coefficient and aspect ratio. Full lines are Equation (12) and dashed lines are power-law approximations that are exact when Cd _o is 0.0175. The wing sweep is taken to be 30°.

Here, the empirical factor 1.04 represents the vortex drag element, whilst k ₁ captures the miscellaneous, lift-dependent drag effects. Shevell’s data suggest that k ₁ may be represented by the approximate relation

(13) \begin{align}{k_1} \approx 0.80\left( {1 - 0.53cos\left( {{{\rm{\Lambda }}_w}} \right)} \right)C{d_o}, \end{align}

where $\Lambda_{w} $ is the wing sweep angle measured at the $1/4$ chord line. Examples of this variation, based upon typical aircraft characteristics, are plotted in Fig. 1. However, given the form of Equations (10) and (11) and recognising that, in practice, the maximum changes in Cd _o due to Reynolds number variation of an aircraft in flight are only about 10%, it is convenient to replace Equation (12) with a local, power-law approximation of the form

(14) \begin{align}{e_{LS}} \propto \left(\frac{1}{C{d_o}}\right)^\tau \approx \frac{E}{{\left( {C_F^{ac}} \right)}^\tau}\ {\rm{where}}\ \tau = - \left(\frac{C_F^{ac}}{e_{LS}}\right)\frac{d{e_{LS}}}{dC_F^{ac}}, \end{align}

where the coefficients E and τ are constants that depend only upon the aircraft geometry and are determined at a representative value of Cd _o. This form is also given in Fig. 1, where the power-law coefficients have been determined by matching both e _LS and its gradient, as determined from Equation (12), when Cd _o is 0.0175. For values of Cd _o lying in the range 0.015 (−14%) to 0.020 (+14%), the difference between Equations (12) and (14) is found to be less than 0.25%. Therefore, the power-law form is accurate enough for the current purposes. These sample calculations also indicate that, in general, τ lies in the range 0.10 to 0.25. Consequently, variations in e _LS resulting from changes in C _F and, hence, Reynolds number are expected to be significant.

Hence, Equations (10) and (11) become

(15) \begin{align}{\left( {{L \over D}} \right)_{max}} \approx {1 \over 2}{\left( {{{\pi .AR.E} \over {{\psi _0}}}} \right)^{1/2}}{\left( {{1 \over {C_F^{ac}}}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}} \end{align}

and

(16) \begin{align}{\left( {{C_L}} \right)_{LDm}} \approx {\left( {\pi .AR.E.{\psi _0}} \right)^{1/2}}{\left( {C_F^{ac}} \right)^{\left( {{{1 - \tau } \over 2}} \right)}} \end{align}

Now consider a situation in which, as the Mach number increases, the cruise altitude is increased by just enough to keep the Reynolds number and, hence, the mean skin-friction coefficient constant. When the Mach number exceeds the value at which “compressibility drag” becomes significant, the maximum (L/D) begins to drop below the level indicated in Equation (15) and continues to reduce at a rate that increases rapidly with further increases in Mach number. At the same time, the engine overall propulsive efficiency, $\eta$ _o , which, as shown in Appendix B of Ref. Reference Poll1, is a strong function of Mach number, is rising monotonically as M _∞ increases. Therefore, at some point in the high-subsonic speed range, ( $\eta$ _o L/D) must have an absolute maximum and the Mach number at this condition is M _o .

Simple methods for the estimation of wave drag are rare in the open literature. However, one useful example is provided by Shevell in Ref. Reference Shevell2, where Cd _w is presented as a function of Mach number and lift coefficient. As indicated in Fig. 12.13 of Ref. Reference Shevell2, the variation with Mach number is such that the gradient dCd _w /dM _∞ increases very rapidly as Mach number increases and the embedded shock waves become stronger. However, Cd _w itself stays relatively small, having maximum values in the region of 0.0020 ( $\approx$ 10% of Cd _o ).

This being the case, as Mach number increases, the values of (L/D) _max and the lift coefficient at which the maximum L/D occurs, whilst reducing, still remain quite close to the values given in Equations (15) and (16). An example showing this general behaviour is provided in Fig. 15.16 of Ref. Reference Shevell2. The deviations from the low-speed values are due, primarily, to the Mach number dependence of the wave drag. Furthermore, as demonstrated in Appendix B of Ref. Reference Poll1 and also discussed in detail by Cumpsty and Hayes in Chapter 8 of Ref. Reference Cumpsty and Hayes4, application of dimensional analysis to the overall engine flow shows that, whilst $\eta$ _o depends on M _∞ and, to a lesser extent, on the net thrust being produced, it is independent of ${R^{ac}}$ . Therefore, at the optimum condition, both M _o and $\eta$ _o are essentially independent of ${R^{ac}}$ . It follows that, when the Reynolds number is constant everywhere, the predominant dependencies at the optimum condition are

(17) \begin{align}\left( {{\eta _o}L/D} \right)_o^R \approx {\psi _1}{\left( {{1 \over {C_F^{ac}}}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}}, \end{align}

(18) \begin{align}\left( {{C_L}} \right)_o^R \approx {\psi _2}{\left( {C_F^{ac}} \right)^{\left( {{{1 - \tau } \over 2}} \right)}}, \end{align}

(19) \begin{align}\left( {{L \over D}} \right)_o^R \approx {\psi _3}{\left( {{1 \over {C_F^{ac}}}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}} \end{align}

and

(20) \begin{align}M_o^R \approx {\psi _4}, \end{align}

where, from Equation (14),

(21) \begin{align}\tau = - \left( {{{C_F^{ac}} \over {{e_o}}}} \right){{d{e_o}} \over {dC_F^{ac}}} \end{align}

and the superscript R indicates that the quantity is evaluated at a particular value of ${R^{ac}}$ . It is also important to note that $\psi$ ₀, $\psi$ ₁, $\psi$ ₂, $\psi$ ₃ and $\psi$ ₄ are constant and characteristics of the aircraft and engine combination.

In Ref. Reference Poll1, it is shown that, for a given Reynolds number, when the variations of ( $\eta$ _o L/D) with cruise Mach number and lift coefficient are normalised with ( $\eta$ _o L/D) _o , M _o and (C _L ) _o , the resulting distributions are approximately independent of the aircraft type. These universal functions are listed in Appendix A and the result may be summarised in the statement

(22) \begin{align}\frac{\left(\eta _{o}L/D\right)}{\left( {{\eta _o}L/D} \right)_o^R} = function\left(\frac{C_L}{\left(C_L\right)_o^R}, \frac{M_\infty}{M_o^R}\right) \end{align}

That is to say, at a given Reynolds number, ( $\eta$ _o L/D) is a function of the independent variables ${C_L}$ and ${M_\infty }$ and the values of three, aircraft specific, parameters at the optimum condition, i.e. $\left( {{C_L}} \right)_o^R$ , $M_o^R$ , and $\left( {{\eta _o}L/D} \right)_o^R$ . This result can be extended to include Reynolds number variation by retaining the same functional relationships, but with the constants $\left( {{\eta _o}L/D} \right)_o^R,\left( {{C_L}} \right)_o^R$ and $M_o^R$ being replaced by the functions given in Equations (17), (18) and (20), i.e.

(23) \begin{align}{{\left( {{\eta _o}L/D} \right)} \over {{\psi _1}}}{\left( {C_F^{ac}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}} = function\left( {{{{C_L}} \over {{\psi _2}}}{{\left( {{1 \over {C_F^{ac}}}} \right)}^{\left( {{{1 - \tau } \over 2}} \right)}},{{{M_\infty }} \over {{\psi _4}}}} \right). \end{align}

The equation set is closed by specifying the relationship between the flat-plate, skin-friction coefficient and Reynolds number and, assuming that the variation of temperature with pressure in the atmosphere is known, the relationship between dynamic viscosity and temperature.

It is generally accepted that, for incompressible flow (M = 0), the most accurate flat-plate, skin friction relation is that due to von Kármán and Schoenherr⁽⁵⁾. This has been extended to compressible flow over an adiabatic wall by van Driest^{(Reference van Driest6)} and, when the Mach number is 0.5, the resulting relation is

(24) \begin{align}{{0.5482} \over {\left( {C_F^{FP}} \right)_{M =0.5}^{1/2}}} = ln\left( {{{\left( {C_F^{FP}} \right)}_{M =0.5}}{R^{FP}}} \right) - 0.0649, \end{align}

where the flat-plate Reynolds number, R ^FP , is based upon speed, atmospheric density and viscosity and the streamwise length of the plate. It is valid for Reynolds numbers in the range 2 × 10⁵ to 1 × 10⁹ and is based upon a fit to an experimental data base that exhibits a variation of ± 5% relative to the mean line.

In normal atmospheric conditions, the relationship between dynamic viscosity and temperature is adequately represented by Sutherland’s Law, i.e.

(25) \begin{align}{\mu _\infty } = {1.458.10^{ - 6}}{{{{\left( {{T_\infty }} \right)}^{1.5}}} \over {\left( {{T_\infty } + 110.4} \right)}}\left( {{{kg} \over {m.s}}} \right), \end{align}

where the temperature is given in Kelvins.

At any stage in the flight, the aircraft will have a weight that is equal to the take-off value less the weight of the fuel burned up to that point. The crew-controlled variables are speed and altitude, or, since aircraft fly along isobars, speed and static pressure, p _∞ , and the problem is to find the combination of Mach number and altitude that gives the absolute minimum fuel burn. This being the case, the Reynolds number becomes a variable and the “optimum” flight conditions occur when

(26) \begin{align}d\!\left( {{\eta _o}L/D} \right) = {{\partial\!\left( {{\eta _o}L/D} \right)} \over {\partial {p_\infty }}}d{p_\infty } + {{\partial\!\left( {{\eta _o}L/D} \right)} \over {\partial {M_\infty }}}d{M_\infty } =0 \end{align}

This equation is satisfied when

(27) \begin{align}{{\partial\!\left( {{\eta _o}L/D} \right)} \over {\partial {p_\infty }}} = {{\partial\!\left( {{\eta _o}L/D} \right)} \over {\partial {M_\infty }}} =0 \end{align}

Clearly, for a given aircraft and engine combination, i.e. with aircraft mass, m, reference wing area, S _ref , and coefficients $\psi$ ₁, $\psi$ ₂, and $\psi$ ₄ all specified, the optimum pressure level and Mach number can be found by solving Equations (23), (24), (25) and (27) numerically. However, as shown in the next section, a simple and accurate analytic solution can be derived.

4.1 Approximate relations for Reynolds number and $\boldsymbol{\Gamma}$

As can be seen from Equation (30), the temperature dependent quantities that influence Reynolds number are confined to the term $\phi$ and, since $\phi$ involves the product of dynamic viscosity and the speed of sound, it is a function of temperature alone. For air temperatures in the range 175K to 265K, Equation (25) can be used to show that $\phi$ may be represented by the simple power law

(59) \begin{align}{{{\mu _\infty }{a_\infty }} \over {{{\left( {{\mu _{TP}}{a_{TP}}} \right)}_{ISA}}}} = \phi \approx {\left( {{{{T_\infty }} \over {{{\left( {{T_{TP}}} \right)}_{ISA}}}}} \right)^{1.34}}, \end{align}

to an accuracy of better than ±0.5% - see Fig. 5. Hence,

(60) \begin{align}{{{T_\infty }} \over \phi }{{d\phi } \over {d{T_\infty }}} \approx 1.34 \end{align}

and this result is accurate to better than ± 3.5% over the same temperature range.

Figure 5. The variation of $\phi$ with temperature and comparison with the power-law approximation given in Equation (59).

If the static temperature, T _∞ , in a general atmosphere is now written as (T _∞ ) _ISA plus ΔT, where ΔT is a function of altitude and is generally small compared to (T _∞ ) _ISA , then

(61) \begin{align}{{{T_\infty }} \over {{{\left( {{T_{TP}}} \right)}_{ISA}}}}{\rm{ = }}{\left( {{{{T_\infty }} \over {{T_{TP}}}}} \right)_{ISA}}{\rm{ + }}{{{\rm{\Delta }}T} \over {{{\left( {{T_{TP}}} \right)}_{ISA}}}} = \overline{T} + \overline {{\rm{\Delta }}T} \end{align}

Hence,

(62) \begin{align}\phi \approx \left( {1 +1.34{{\overline {{\rm{\Delta }}T} } \over {\overline{T}}}} \right){\overline{T}^{1.34}} \end{align}

and, in the vicinity of the 226.32 hPa isobar (ISA tropopause) where $\overline{T}$ is close to unity, the simplified expression,

(63) \begin{align}\phi \approx \left( {1 +1.34\overline {{\rm{\Delta }}T} } \right){\overline{T}^{1.34}}, \end{align}

may be used. For attitudes above 28,000 feet, Equation (63) is found to be accurate to better than ± 2% for values of $\overline {{\rm{\Delta }}T} $ in the range ± 0.15.

In the International Standard Atmosphere⁽⁸⁾, the troposphere and the stratosphere are regions within which the local, vertical temperature gradient, or “lapse” rate, is constant. This gradient is usually expressed as the derivative of temperature with respect to the geopotential altitudeFootnote 6, h, designated here as L _h and which has units of K/m. In a general atmosphere, temperature, pressure and altitude are linked by the hydrostatic equation, i.e.

(64) \begin{align}dp = - \rho {g_{SL}}dh, \end{align}

where ρ is the local air density and g _SL is the acceleration due to gravity at sea level. Hence, for the ISA,

(65) \begin{align}\overline{T} = {\chi ^\omega }, \end{align}

where the exponent $\omega$ is a constant given by

(66) \begin{align}\omega = {{\Re {L_h}} \over {{g_{SL}}}} \end{align}

and ℜ is the gas constant for air. Therefore, using Equations (5), (31), (63) and (65),

(67) \begin{align}\phi \approx \left( {1 +1.34\overline {{\rm{\Delta }}T} } \right){\chi ^{1.34\omega }} = \left( {1 +1.34\overline {{\rm{\Delta }}T} } \right){\left( {{{{\psi _6}} \over {{C_L}{\zeta ^2}}}\left( {{m \over {MTOM}}} \right)} \right)^{1 - i}}, \end{align}

where

(68) \begin{align}\iota =1 +1.34\omega \end{align}

In the ISA troposphere, where $\chi$ is less than, or equal to, unity, L _h is −0.0065K/m. Hence, $\omega$ is −0.19026 and i is 0.74505. In the ISA stratosphere, $\chi$ is greater than unity and, since both L _h and $\omega$ are zero, i is equal to one.

Hence, from Equations (30), (31) and (67), the general expression for Reynolds number is

(69) \begin{align}{R^{ac}} \approx {{{\psi _5}} \over {\left( {1 +1.34\overline {{\rm{\Delta }}T} } \right)}}{\zeta ^{\left( {1 - 2i} \right)}}{\left( {{{{\psi _6}} \over {{C_L}}}\left( {{m \over {MTOM}}} \right)} \right)^i} = {{{\psi _5}} \over {\left( {1 +1.34\overline {{\rm{\Delta }}T} } \right)}}\left( {{\zeta \over {{\chi ^i}}}} \right) \end{align}

In aircraft operations, to guarantee safe vertical separation of traffic, altitude is replaced by the non-dimensional “Flight Level”, FL, which, by international agreement, is defined as the geopotential altitude, in units of feet, in the International Standard Atmosphere at the same value of the static pressure, p _∞ , divided by 100 feet. This means that the Flight Level only depends upon the atmospheric static pressure. Therefore, using the relations given in Ref. 8, if χ is less than, or equal to, 1,

(70) \begin{align}FL =1454.42\left( {1 - 0.751865{{\left( \chi \right)}^{ - 0.19026}}} \right)\ {\rm{and}}\ i =0.74505, \end{align}

whilst, if χ is greater than 1,

(71) \begin{align}FL =360.8924 +208.058{{ln}}\left( \chi \right)\ {\rm{and}}\ i =1 \end{align}

It is important to note that, if Flight Level is used to describe an aircraft’s altitude, then, by definition, Equations (70) and (71) are true for all atmospheres.

It follows that the variation of d $\chi$ /dFL with Flight Level is also the same in all atmospheres and from Equation (64),

(72) \begin{align}\left( {{\chi \over {FL}}{{dFL} \over {d\chi }}} \right) = {\left( {{\chi \over {FL}}{{dFL} \over {d\chi }}} \right)_{ISA}} = \left( {{{\Re {{\left( {{T_{TP}}} \right)}_{ISA}}} \over {30.48{g_{SL}}}}} \right)\left( {{{\overline{T}} \over {FL}}} \right) =208.06\left( {{{\overline{T}} \over {FL}}} \right) \end{align}

However, the variation of temperature with pressure and, hence, the variation of dT _∞ /dFL with Flight Level depend upon the actual atmospheric conditions. Therefore, in general, from Equations (37), (60) and (72)

(73) \begin{align}{\rm{\Gamma }} = \left( {{{{T_\infty }} \over \phi }{{d\phi } \over {d{T_\infty }}}} \right)\left( {{\chi \over {FL}}{{dFL} \over {d\chi }}} \right)\left( {{{FL} \over {{T_\infty }}}{{d{T_\infty }} \over {dFL}}} \right) \approx 278.8\left( {{{\overline{T}} \over {FL}}} \right)\left( {{{FL} \over {{T_\infty }}}{{d{T_\infty }} \over {dFL}}} \right) \end{align}

and, from Equation (61),

(74) \begin{align}\left( {{{FL} \over {{T_\infty }}}{{d{T_\infty }} \over {dFL}}} \right) = {{FL} \over {\left( {\overline{T} + \overline {{\rm{\Delta }}T} } \right)}}\left( {{{d\left( {\overline{T} + \overline {{\rm{\Delta }}T} } \right)} \over {dFL}}} \right) = {{\overline{T}} \over {\left( {\overline{T} + \overline {{\rm{\Delta }}T} } \right)}}\left( {{{FL} \over {\overline{T}}}} \right){1 \over {{{\left( {{T_{TP}}} \right)}_{ISA}}}}{{d{T_\infty }} \over {dFL}} \end{align}

Hence,

(75) \begin{align}{\rm{\Gamma }} \approx 278.8\left( {{{\overline{T}} \over {\overline{T} + \overline {{\rm{\Delta }}T} }}} \right)LR, \end{align}

where LR is the non-dimensional, temperature change per Flight Level defined as

(76) \begin{align}LR = {1 \over {{{\left( {{T_{TP}}} \right)}_{ISA}}}}{{d{T_\infty }} \over {dFL}} =0.1407{L_h} \end{align}

This result is valid for all Flight Levels. However, in the vicinity of FL 360, where $\overline{T}$ is close to unity,

(77) \begin{align}{\rm{\Gamma }} \approx 277\left( {1 - \overline {{\rm{\Delta }}T} } \right)LR \end{align}

Here, the constant of proportionality has been adjusted so that, for Flight Levels greater than 280, this expression is accurate to better than ± 3.5%, when $\overline {\Delta T} $ and LR are in the range ± 0.15 and ± 0.0045, respectively.

In the International Standard Atmosphere, $\Gamma$ is approximately −0.254 in the upper troposphere and zero in the stratosphere. However, the ISA is a reference scenario based upon long-time average observations and, on short time scales, the variation of temperature with FL may differ significantly from the ISA values. An example of an “instantaneous” vertical temperature profile is shown in Fig. 6. Since Equation (77) is valid for all atmospheric conditions, $\Gamma$ can be evaluated from the data given in Fig. 6 and the results are shown in Table 1. The first two columns give the observed variation of temperature with pressure, whilst the next eight columns contain the derived quantities and the estimated value of $\Gamma$ is given in the last column.

Figure 6. An example of an observed variation of atmospheric temperature with Flight Level compared with that in the International Standard Atmosphere.

4.2 The optimum conditions

For an aircraft of specified mass, at any given Mach number, the value of the Reynolds number at which ( $\eta$ _o L/D) has its best (local maximum) value is obtained by combining Equations (28), (40) and (69), i.e.

(78) \begin{align}R_B^{ac} \approx {G_1}{\psi _5}{\left( {{\zeta ^{\left( {1 - 2{\iota _B}} \right)}}{{\left( {{1 \over {{\psi _7}{f_2}}}\left( {{m \over {MTOM}}} \right)} \right)}^{{\iota _B}}}} \right)^{{\kappa _B}}}, \end{align}

where

(79) \begin{align}{G_1} = {\left( {\left( {1 +1.34{{\overline {\Delta T} }_B}} \right){{\left( {1 + {{\rm{\Delta }}_B}} \right)}^{{i_B}}}} \right)^{ - {\kappa _B}}}, \end{align}

(80) \begin{align}\hspace*{-27pt}{\kappa _B} = \left( {{2 \over {2 - {i_B}b\left( {1 - \tau } \right)}}} \right) \hspace*{26pt}\end{align}

and

(81) \begin{align}{\psi _7} = {{{\psi _2}} \over {{\psi _6}}}{\left( {{a \over {\psi _5^b}}} \right)^{\left( {{{1 - \tau } \over 2}} \right)}} \end{align}

The value of the Reynolds number when ( $\eta$ _o L/D) has its optimum value depends upon M _o , which comes from a combination of Equations (51) and (57). With the Mach number at this optimum value, approximations for f _1, f ₂ and A can be obtained from Equations (A-5), (A-8) and (A-12), i.e.

(82) \begin{align}\hspace*{-8pt} {\left( {{f_1}} \right)_o} \approx 1 - 5.897{\varepsilon ^2} + \ldots, \hspace*{3pt}\end{align}

(83) \begin{align}{\left( {{f_2}} \right)_o} \approx 1 - \varepsilon - 10.46{\varepsilon ^2} + \ldots \end{align}

and

(84) \begin{align}{A_o} \approx - 2.675\left( {1 +2.243\varepsilon +44.86{\varepsilon ^2} + \ldots } \right) \end{align}

Table 1 An example of a typical variation of atmospheric temperature with pressure and the processing steps to estimate $\Gamma$

Substituting these results into Equations (78) and (79), expanding and neglecting the products of small quantities, the value of the Reynolds number at the optimum condition is given by

(85) \begin{align}R_o^{ac} \approx {G_2}{\psi _5}{\left( {{1 \over {{\psi _7}}}\left( {{m \over {MTOM}}} \right)} \right)^{{\iota _o}{\kappa _o}}}, \end{align}

where

(86) \begin{align}{G_2} \approx {\left( {\left( {1 +1.34{{\overline {\Delta T} }_o}} \right){{\left( {\left( {1 + {{\rm{\Delta }}_o}} \right)\left( {1 - \varepsilon } \right)} \right)}^{{i_o}}}{{\left( {1 + \varepsilon } \right)}^{\left( {2{i_o} - 1} \right)}}} \right)^{ - {\kappa _o}}} \end{align}

and

(87) \begin{align}{\kappa _o} = \left( {{2 \over {2 - {i_o}b\left( {1 - \tau } \right)}}} \right) \end{align}

For maximum accuracy, G ₂ is evaluated directly using Equations (57) and (58). Unfortunately, the form of Equation (86) does not yield any clear insight as to how G ₂ relates to the various aircraft and atmospheric parameters. However, by expansion, approximation and use of Equation (77), (86) may be re-cast in terms of the key parametersFootnote 7, i.e.

(88) \begin{align}{G_2} \approx & \left( {1 +0.033\left( {1 +0.89\tau } \right)} \right) \times \left( 1 - 0.040\left( {1 - {i_o}} \right) - 1.47{{\overline {\Delta T} }_o}\left( {1 - 1.40{{\overline {\Delta T} }_o}} \right)\right.\nonumber\\[5pt] & \left. + 16.35\left( {{i_0} - 0.235} \right)\left( {1 - {{\overline {\Delta T} }_o}} \right)L{R_o} \right) \end{align}

Here, the first bracketed term contains the result for cruise in the ISA stratosphere, i.e. where FL > 360.9, p _∞  < 226.32 hPa, i _o is unity and ΔT and LR are both zero, whilst the second gives a correction to allow for the actual atmospheric conditions being encountered. The accuracy of the correction term deceases monotonically as both $\overline {\Delta T} $ and LR get either larger, or smaller, with the maximum deviation from Equation (86) being ± 2% for values of τ between 0.1 and 0.3 and ${\overline {\Delta T} _o}$ and LR _o lying in the ranges ± 0.15 and ± 0.0045, respectively.

Having found the Reynolds number, the mean skin-friction coefficient at the optimum condition follows from Equation (28), i.e.

(89) \begin{align}{\left( {C_F^{ac}} \right)_o} = {a \over {{{\left( {R_o^{ac}} \right)}^b}}} \approx {G_3}\left( {{a \over {\psi _5^b}}} \right){\left( {{\psi _7}\left( {{{MTOM} \over m}} \right)} \right)^{b{\iota _o}{\kappa _o}}}, \end{align}

where, using Equation (88) and neglecting small quantities,

(90) \begin{align}{G_3} = {\left( {{G_2}} \right)^{ - b}} \approx 0.995\left( {1 +0.20{{\overline {\Delta T} }_o} - 1.75\left( {1 - {{\overline {\Delta T} }_o}} \right)L{R_o}} \right) \end{align}

Therefore, the lift coefficient at the optimum condition is given by combining Equations (40) and (89), i.e.

(91) \begin{align}{\left( {{C_L}} \right)_o} = \left( {1 + {{\rm{\Delta }}_o}} \right){\left( {{f_2}} \right)_o}{\psi _2}\left( {C_F^{ac}} \right)_o^{\left( {{{1 - \tau } \over 2}} \right)} = {G_4}{\psi _2}{\left( {\left( {{a \over {\psi _5^b}}} \right){{\left( {{\psi _7}\left( {{{MTOM} \over m}} \right)} \right)}^{b{\iota _o}{\kappa _o}}}} \right)^{\left( {{{1 - \tau } \over 2}} \right)}}, \end{align}

where

(92) \begin{align}{G_4} = \left( {1 + {{\rm{\Delta }}_o}} \right){\left( {{f_2}} \right)_o}{\left( {{G_3}} \right)^{\left( {{{1 - \tau } \over 2}} \right)}} \end{align}

The approximate form of this relation is

(93) \begin{align}{G_4} \approx 0.9685\left( {1 - 0.027\tau } \right)\left( {1 +0.080{{\overline {\Delta T} }_o} - 9.70\left( {1 - {{\overline {\Delta T} }_o}} \right)L{R_o}} \right), \end{align}

which is within ±1% of the full solution for the same parameter range as Equation (88).

Finally, using Equation (43) and (89), the optimum value of ( $\eta$ _o L/D) is

(94) \begin{align}{\left( {{\eta _o}L/D} \right)_o} = {\left( {{f_1}} \right)_o}{\psi _1}\left( {C_F^{ac}} \right)_o^{ - \left( {{{1 + \tau } \over 2}} \right)} = {G_5}{\psi _1}{\left( {\left( {{{\psi _5^b} \over a}} \right){{\left( {{1 \over {{\psi _7}}}\left( {{m \over {MTOM}}} \right)} \right)}^{b{\iota _o}{\kappa _o}}}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}}, \end{align}

where

(95) \begin{align}{G_5} = {\left( {{f_1}} \right)_o}\left( {1 + {{{A_o}} \over 2}{\rm{\Delta }}_o^2 + {{{B_o}} \over 6}{\rm{\Delta }}_o^3} \right){\left( {{1 \over {{G_3}}}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}} \end{align}

and

(96) \begin{align}{G_5} \approx 1 - 0.13{\overline {\Delta T} _o}, \end{align}

which is found to be accurate to ±0.5%. It is important to note that the influence of lapse rate on ( $\eta$ _o L/D)_o is very small and negligible in most cases of practical interest.

The value of $\chi$ for optimum ( $\eta$ _o L/D) comes from Equations (5), (31), (51) and (91), i.e.

(97) \begin{align}{\chi _o} \approx {{{{\left( {{C_L}} \right)}_o}} \over {{\psi _6}}}{\left( {1 + \varepsilon } \right)^2}\left( {{{MTOM} \over m}} \right) = {G_6}{\left( {{\psi _7}\left( {{{MTOM} \over m}} \right)} \right)^{{\kappa _o}}}, \end{align}

where

(98) \begin{align}{G_6} = {G_4}{\left( {1 + \varepsilon } \right)^2} \end{align}

or, in approximate form,

(99) \begin{align}{G_6} \approx 0.968\left( {1 - 0.0285\tau } \right)\left( {1 +0.079{{\overline {\Delta T} }_o} - 13.0\left( {1 +0.99\tau } \right)\left( {1 - {{\overline {\Delta T} }_o}} \right)L{R_o}} \right) \end{align}

Since both $\overline {\Delta T} $ and LR depend upon χ, Equation (97) is implicit. However, for a given atmosphere, the variations of $\iota$ , $\overline {\Delta T} $ , LR and $\Gamma$ with χ are known; see the example given in Table 1. Therefore, Equation (97) can be inverted to give an explicit relation for the mass ratio that gives the optimum fuel burn at a specified value of χ or, via Equations (70) and (71), at a specified Flight Level, i.e.

(100) \begin{align}{\bigg( {\frac{m}{{MTOM}}} \bigg)_o} = {\psi _7}{\left( {{{{G_6}} \over \chi }} \right)^{{1 \over \kappa }}} \approx {\psi _7}{\left( {{{1 +0.079\overline {\Delta T} - 13.0\left( {1 +0.99\tau } \right)\left( {1 - \overline {\Delta T} } \right)LR} \over {1.033\left( {1 +0.0285\tau } \right)\!\chi }}} \right)^{{1 \over \kappa }}}, \end{align}

where

(101) \begin{align}\kappa = \kappa\!\left( \chi \right) = \left( {{2 \over {2 - ib\left( {1 - \tau } \right)}}} \right) \end{align}

In general, Equation (100) is single valued. However, at a point of lapse rate discontinuity, there are two solutions. This may be illustrated by using the example of the International Standard Atmosphere, which has a lapse rate discontinuity at the tropopause (χ = 1). In this case, if χ approaches unity from below, i.e. the aircraft is climbing though the troposphere, where i is 0.74505 and LR is −0.0009145 (L _h is −0.0065 K/m), at the tropopause,

(102) \begin{align}{\bigg( {{\frac{m}{MTOM}}} \bigg)_{{\chi _o} =1}} \approx 0.980\left( {1 - 0.016\tau } \right)\!{\psi _7} \end{align}

Conversely, if the same point is approached from above, i.e. by descending through the stratosphere, where i is unity and LR is zero,

(103) \begin{align}{\bigg( {{\frac{m}{MTOM}}} \bigg)_{{\chi _o} =1}} \approx 0.970\left( {1 - 0.027\tau } \right)\!{\psi _7} \end{align}

Between these two mass values, the optimum Flight Level is constant, but the optimum Mach number and ( $\eta$ _o L/D)_o both vary. The change in M _o is small and depends only upon the discontinuous change in $\gamma$ . However, as seen in Equations (94) and (96), ( $\eta$ _o L/D)_o varies continuously with mass and, since the atmospheric temperature is also continuous, it may be treated as a continuous function to a very good approximation. For flight operations, the change in Mach number between the two mass values would follow a particular profile. Whist this cannot be determined from the theory and in the absence of any other information, for the purposes of this paper, it will be assumed that it is directly proportional to the aircraft mass. In practise, viscosity prevents true discontinuities. Nevertheless, in those regions of the atmosphere where the lapse rate varies very rapidly with altitude, the optimum Flight Level will be almost independent of the aircraft mass.

An important consequence of the existence of two solutions at a point of lapse rate discontinuity is that, if the vertical temperature profile contains more than one discontinuity, Equation (97) may have multiple solutions at a given mass ratio. If this is the case, whilst each solution corresponds to a local maximum, one will have the highest, or optimum, value of ( $\eta$ _o L/D). This optimum is found by substituting the resulting χ_o and (m/MTOM) solution pairs into Equation (94) and comparing the results. Therefore, some care needs to be exercised when using Equations (97) and (100) to make sure that all the solutions have been found and the true optimum has been identified.

Continuing with the ISA example, if the mass ratio is greater than, or equal to, the value given by Equation (102), the optimum altitude lies in the troposphere and ${{\rm{\chi }}_{\rm{o}}}$ is found from Equations (97) and (99), i.e.

(104) \begin{align}{\chi _o} \approx 0.980\left( {1 - 0.017\tau } \right){\left( {{\psi _7}\left( {{{MTOM} \over m}} \right)} \right)^{1.052\left( {1 - 0.050\tau } \right)}} \end{align}

The optimum Flight Level follows directly from Equation (70), with the optimum Mach number coming from Equations (51), (57) and (77), i.e.

(105) \begin{align}{M_o} \approx 1.0016\left( {1 +0.0032\tau } \right){\psi _4} \end{align}

The optimum lift coefficient comes from Equations (91) and (93), i.e.

(106) \begin{align}{\left( {{C_L}} \right)_o} \approx 0.977\left( {1 - 0.027\tau } \right){\psi _2}{\left( {{a \over {\psi _5^b}}} \right)^{\left( {{{1 - \tau } \over 2}} \right)}}{\left( {{\psi _7}\left( {{{MTOM} \over m}} \right)} \right)^{0.055\left( {1 - 1.050\tau } \right)}} \end{align}

and, finally, the optimum value of ( $\eta$ _o L/D) is obtained from Equations (94) and (96)

(107) \begin{align}{\left( {{\eta _o}L/D} \right)_o} \approx {\psi _1}{\left( {{{\psi _5^b} \over a}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}}{\left( {{1 \over {{\psi _7}}}\left( {{m \over {MTOM}}} \right)} \right)^{0.055\left( {1 +0.950\tau } \right)}} \end{align}

Conversely, if the mass ratio is less than the value from Equation (102), the optimum altitude must lie in the stratosphere. Hence,

(108) \begin{align}{\chi _o} \approx 0.968\left( {1 - 0.0285\tau } \right){\left( {{\psi _7}\left( {{{MTOM} \over m}} \right)} \right)^{1.070\left( {1 - 0.065\tau } \right)}} \end{align}

and the optimum Flight level comes from Equation (71). The optimum Mach number is

(109) \begin{align}{M_o} \approx 0.9997\left( {1 - 0.0007\tau } \right){\psi _4}, \end{align}

the optimum lift coefficient is

(110) \begin{align}{\left( {{C_L}} \right)_o} \approx 0.9685\left( {1 - 0.027\tau } \right){\psi _2}{\left( {{a \over {\psi _5^b}}} \right)^{\left( {{{1 - \tau } \over 2}} \right)}}{\left( {{\psi _7}\left( {{{MTOM} \over m}} \right)} \right)^{0.075\left( {1 - 1.065\tau } \right)}} \end{align}

and the optimum value of ( $\eta$ _o L/D) is

(111) \begin{align}{\left( {{\eta _o}L/D} \right)_o} \approx {\psi _1}{\left( {{{\psi _5^b} \over a}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}}{\left( {{1 \over {{\psi _7}}}\left( {{m \over {MTOM}}} \right)} \right)^{0.075\left( {1 +0.935\tau } \right)}} \end{align}

In summary, Equations (85)–(100) provide a complete solution to the problem of determining the optimum ( $\eta$ _o L/D) and the values of Mach number and Flight Level at which it occurs for an aircraft at a given mass flying in a general atmosphere. For highest accuracy, the coefficients G ₂ to G ₆ are evaluated from Equations (86), (90), (92), (95) and (98) using the results from Equations (57) and (58). However, the approximate relations given in Equations (88), (90), (93), (96) and (99) will be sufficient for most practical purposes.

When using the method in situations where the vertical temperature profile is complex, e.g. as in the example given in Fig. 6, the first step is to characterise the atmospheric temperature variation, i.e. to construct the corresponding Table 1. The second step is to use Equations (100), (70) and (71) to determine how the mass ratio for optimum ( $\eta$ _o L/D) varies with the Flight Level. If this reveals that there is more than one value of χ for a given value of (m/MTOM)_o, i.e. Equation (97) has multiple solutions, Equation (94) is used to identify the Flight Level that has the largest ( $\eta$ _o L/D) _o for that mass ratio and the other solutions are discarded. Therefore, for a specified mass ratio, with (FL) _o and, hence, $\iota$ _o , κ _o , LR _o , ${\overline {\Delta T} _o}$ and $\Gamma$ _o now established, ( $\eta$ _o L/D)_o follows from Equation (94), whilst the corresponding values of the cruise Mach number, M _o , and the lift coefficient, (C _L ) _o , and come from Equations (51), (57), (58) and (91).

4.3 Comparison with numerical solutions

In order to assess the accuracy of the approximate method, sample cases are compared with accurate numerical solutions to Equations (23) to (27) for an aircraft where $\psi$ ₁, $\psi$ ₂, $\psi$ ₄, $\psi$ ₅, $\psi$ ₆ and τ have values of 0.17, 6.56, 0.812, 1.27 × 10⁸, 0.57 and 0.19, respectively,. This parameter set is representative of a typical, modern, long-range, wide-bodied aircraft. The results are presented in Figs 7–10.

Figure 7. The variation of normalised R _o , (C _F ) _o , (FL) _o and M _o with aircraft mass under ISA conditions. Solid lines indicate numerical values, whilst dashed lines are the approximate solution.

Figure 8. The variation of normalised (C _L )_o and ( $\eta$ _o L/D) _o with aircraft mass under ISA conditions. Solid lines indicate numerical values, whilst dashed lines are the approximate solution.

Figure 9. The variation of normalised R _o with temperature deviation from ISA. Solid lines indicate numerical values, whilst dashed lines indicate the approximate solution. Solid symbols are for an (m/MTOM) of 1.0, whilst, for open symbols, (m/MTOM)  =  0.8.

Figure 10. The variation of normalised (C _F ) _o , (C _L ) _o , ( $\eta$ _o L/D) _o , and (FL) _o with temperature deviation from ISA conditions. Solid lines indicate numerical values, whilst dashed lines indicate the approximate solution. Solid symbols are for an (m/MTOM) of 1.0, whilst, for open symbols, (m/MTOM)  =  0.8.

The variation of Reynolds number, skin-friction coefficient, Flight Level and Mach number with mass ratio when ( $\eta$ _o L/D) has its optimum value and the aircraft is flying in ISA conditions is shown in Fig. 7. The change in slope between an (m/MTOM) of 0.91 and 0.885 is due to the discontinuity in the lapse rate at the tropopause, as already described. The strongest dependency is exhibited by the Reynolds number, which decreases as the aircraft mass reduces. This means that the skin friction must rise slowly as mass reduces, whilst the optimum cruise altitude increases. The optimum Mach number decreases by 0.25% as the aircraft passes through the tropopause. In all cases, the approximate solutions are found to be very close to the numerical results with maximum differences being less than about 0.2%.

Figure 8 gives the corresponding results for lift coefficient and ( $\eta$ _o L/D) at the optimum condition. The variation of (C _L )_o with mass is very small and the effect of the discontinuous lapse rate is clearly visible. The difference between the numerical and the approximate solutions is less than about 0.25%. The optimum value of ( $\eta$ _o L/D) also decreases as the mass decreases. The reduction is small, but, in terms of aircraft operations, it is significant. For example, if, on a long route, the fuel to be consumed is 20% of the take-off mass, ( $\eta$ _o L/D)_o will decrease by 2% over the course of the flight and this will have a corresponding adverse effect on the fuel burn rate, see Equation (1).

The variation of optimum Reynolds number with static temperature deviation from the ISA values, for two values of the mass ratio, is shown in Fig. 9. This effect is driven by the relationship between static temperature and viscosity, with Reynolds number decreasing as the local temperature increases. A rise of 10K relative to ISA reduces R _o by 6.5%. The variations of the skin-friction coefficient, lift coefficient, ( $\eta$ _o L/D) and Flight Level at the optimum condition with ΔT are given in Fig. 10. Compared to the influence of temperature on the optimum Reynolds number, these effects are small. The skin-friction coefficient, the lift coefficient and the Flight Level all increase with increasing temperature, with skin friction showing the greatest sensitivity. By contrast, ( $\eta$ _o L/D)_o decreases as temperature rises with a 20K increase reducing ( $\eta$ _o L/D)_o by about 1%. The optimum Mach number is not shown in Fig. 10, since it has no significant dependency upon temperature. In all cases, the approximate solution is in excellent agreement with the numerical results.

4.4 Non-optimum conditions

In order to determine the cruise fuel burn under non-optimum conditions, the relations for the optimum conditions are combined with Equation (32) to give a single expression that enables ( $\eta$ _o L/D) for an aircraft of fixed mass to be obtained for any combination of Mach number and altitude within the range of validity of the basic expressions.

In the constant Reynolds number case, i.e. from Equation (A-6) of Appendix A,

(112) \begin{align}{{\left( {{\eta _o}L/D} \right)} \over {\left( {{\eta _o}L/D} \right)_o^R}} \approx {f_1}\left( {1 + {A \over 2}{{\left( {\left( {{1 \over {{f_2}}}{{{C_L}} \over {\left( {{C_L}} \right)_o^R}}} \right) - 1} \right)}^2}} \right. + \left. {{B \over 6}{{\left( {\left( {{1 \over {{f_2}}}{{{C_L}} \over {\left( {{C_L}} \right)_o^R}}} \right) - 1} \right)}^3}} \right) \end{align}

Here, f ₁, f ₂, A and B, as given in Appendix A, are all functions of ζ. These can be expressed in terms of M/M _o by recalling that, from Equations (21), (29) and (51)

(113) \begin{align}\zeta = {M \over {{\psi _4}}} = {M \over {M_o^R}} = {M \over {{M_o}}}\left( {1 + {\rm{\varepsilon }}} \right) \end{align}

For an aircraft of fixed mass, if both the altitude and the speed are variable, then, from Equation (5),

(114) \begin{align}{{{C_L}} \over {{{\left( {{C_L}} \right)}_o}}} = \left( {{\chi \over {{\chi _o}}}} \right){\left( {{{{M_o}} \over M}} \right)^2} \end{align}

Therefore, when the Reynolds number is allowed to vary, using the results from Equations (91) and (94),

(115) \begin{align}{{\left( {{\eta _o}L/D} \right)} \over {{{\left( {{\eta _o}L/D} \right)}_o}}} \approx\ & {f_1}{\left( {{{{{\left( {C_F^{ac}} \right)}_o}} \over {C_F^{ac}}}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}}\left( {1 + {A \over 2}{{\left( {\left( {\left( {{{1 + {{\rm{\Delta }}_o} - \varepsilon } \over {{f_2}}}} \right){{\left( {{{{{\left( {C_F^{ac}} \right)}_o}} \over {C_F^{ac}}}} \right)}^{\left( {{{1 - \tau } \over 2}} \right)}}{{{C_L}} \over {{{\left( {{C_L}} \right)}_o}}}} \right) - 1} \right)}^2}}\right.\nonumber\\[8pt] & \left. { + {B \over 6}{{\left( {\left( {\left( {{{1 + {{\rm{\Delta }}_o} - \varepsilon } \over {{f_2}}}} \right){{\left( {{{{{\left( {C_F^{ac}} \right)}_o}} \over {C_F^{ac}}}} \right)}^{\left( {{{1 - \tau } \over 2}} \right)}}{{{C_L}} \over {{{\left( {{C_L}} \right)}_o}}}} \right) - 1} \right)}^3}} \right) \end{align}

From Equations (28) and (69),

(116) \begin{align}{{{{\left( {C_F^{ac}} \right)}_o}} \over {C_F^{ac}}} = {\left( {\left( {{{1 +1.34{{\overline {\Delta T} }_o}} \over {1 +1.34\overline {\Delta T} }}} \right)\chi _o^{\left( {{i_o} - i} \right)}{{\left( {{M \over {{M_o}}}} \right)}^{\left( {1 - 2i} \right)}}} \right)^b}{\left( {{{{{\left( {{C_L}} \right)}_o}} \over {{C_L}}}} \right)^{ib}} \end{align}

Therefore, if

(117) \begin{align}{f_3} \approx {f_1}{\left( {{{{{\left( {C_F^{ac}} \right)}_o}} \over {C_F^{ac}}}{{\left( {{{{C_L}} \over {{{\left( {{C_L}} \right)}_o}}}} \right)}^{ib}}} \right)^{\left( {{{1 + \tau } \over 2}} \right)}} \end{align}

and

(118) \begin{align}{f_4} \approx {{\left( {1 + {{\rm{\Delta }}_o} - \varepsilon } \right)} \over {{f_2}}}{\left( {{{{{\left( {C_F^{ac}} \right)}_o}} \over {C_F^{ac}}}{{\left( {{{{C_L}} \over {{{\left( {{C_L}} \right)}_o}}}} \right)}^{ib}}} \right)^{\left( {{{1 - \tau } \over 2}} \right)}}, \end{align}

then

(119) \begin{align}{{\left( {{\eta _o}L/D} \right)} \over {{{\left( {{\eta _o}L/D} \right)}_o}}} \approx\ & {f_3}{\left( {{{{{\left( {{C_L}} \right)}_o}} \over {{C_L}}}} \right)^{ib\left( {{{1 + \tau } \over 2}} \right)}}\left( {1 + {A \over 2}{{\left( {\left( {{f_4}{{\left( {{{{C_L}} \over {{{\left( {{C_L}} \right)}_o}}}} \right)}^\upsilon }} \right) - 1} \right)}^2}} \right.\nonumber\\[8pt] & + \left. {{B \over 6}{{\left( {\left( {{f_4}{{\left( {{{{C_L}} \over {{{\left( {{C_L}} \right)}_o}}}} \right)}^\upsilon }} \right) - 1} \right)}^3}} \right), \end{align}

where

(120) \begin{align}\upsilon =1 - ib\left( {{{1 - \tau } \over 2}} \right) \end{align}

Equation (119) is a completely general relation that can be used to estimate ( $\eta$ _o L/D) for an aircraft of specified weight at any combination of speed and altitude, provided that the corresponding Mach numbers and lift coefficients fall within the range of validity of the expressions given in Appendix A. Whilst the extension to include Reynolds number has introduced greater complexity, the basic simplicity of the original relation, i.e. Equation (112), is preserved, with Equations (51), (91) and (94) providing the values of the three fundamental normalising quantities M _o , (C _L )_o and ( $\eta$ _o L/D) _o identified in Ref. Reference Poll1.

Figure 11. The variation of normalised ( $\eta$ _o L/D) with normalised Mach number in the ISA for two values of m/MTOM when χ is fixed at the optimum value, χ_o.

Figure 12. The variation of normalised ( $\eta$ _o L/D) with normalised lift coefficient in the ISA for two values of m/MTOM when M is equal to M _o .

In order to illustrate the effects of Reynolds number and to assess the accuracy of the approximations, estimates from Equation (119) are compared with numerical solutions of Equation (23), for the same example aircraft used for Figs. 7–10 flying in the ISA. These results are presented in Figs. 11 and 12. Figure 11 shows the effect of changing the Mach number relative to M _o , whilst the aircraft follows the isobar for optimum ( $\eta$ _o L/D). Results are shown for two values of the mass ratio. When m/MTOM is equal to one, the aircraft is in the tropopause and in the stratosphere when it is 0.8. Results for constant Reynolds number, i.e. from Equation (112), are also shown. In this case, the influence of mass is very weak and the change in normalised ( $\eta$ _o L/D) for a given Mach number change is reduced slightly relative to the constant Reynolds number case. Results from Equation (119) are in good agreement with the numerical solutions. Figure 12 shows the situation in which the Mach number is fixed at M _o and the aircraft is flying in the ISA along a range of isobars. Away from the optimum isobar, ( $\eta$ _o L/D) is reduced. Once again, when Reynolds number is allowed to vary, the reduction is lower than for the constant Reynolds number case and the effect of variations in mass are small. There is seen to be excellent agreement between the approximate and numerical results. Therefore, the Reynolds number affects both the optimum values themselves and the variations relative to the optimum when speed and altitude are changed.

4.5 The effect of temperature profile

When Reynolds number effects are accounted for, the optimum flight level depends on the variation of atmospheric temperature with altitude. As indicated in Fig. 6, actual temperature profiles usually deviate from the International Standard Atmosphere and there are systematic profile changes with both latitude and season. For example, the tropopause occurs at about 10km in the northern latitudes, but nearer to 16km in the tropics. In addition, there are weather dependent temperature variations and changes in the tropopause height that may be relevant for flight optimisation. Locally, temperature deviations from ISA in excess of ±20K are not uncommon. Real temperature profiles can also have vertical temperature gradients that differ considerably from the ISA. Non-monotonic temperature changes, with large lapse rate variations occurring over just a few meters vertically, can occur anywhere at any altitude. Typically, vertical temperature changes stay above –0.3K/FL because for smaller values the atmosphere is unstably layered. However, large positive gradients are possible at inversion layers, with an important example being the tropopause inversion layer. Here there is a region, just above the tropopause, about 1km deep, with increasing temperature (Birner et al.^{(Reference Birner and Schumann9)}). Since temperature profiles are predictable, at least to some degree, the actual profile, or, at least, an average along the flight path, could be considered for optimised flight planning.

An example of the variation of the optimum FL with mass ratio for the example aircraft used in the previous sections operating in the two atmospheric conditions shown in Fig. 6 is given in Fig. 13. These results are obtained using Equation (100). For ISA conditions, there is only one optimum Flight Level for each value of the mass ratio. However, in the case of the example “real” atmosphere in Fig. 6, for some values of the mass ratio there are multiple solutions, e.g. between 0.86 and 0.88, there are five values of (FL)_o for each mass ratio. However, as previously described, each Flight Level and mass ratio pair has a corresponding value of ( $\eta$ _o L/D) _o and, in the case of multiple solutions at a particular mass ratio, one will have a value that is higher than the rest. These values are found from Equation (94) and selecting the points with the largest ( $\eta$ _o L/D) _o at each value of the mass ratio allows the “optimum climb” profile to be determined. This is shown as the solid line in the figure and it is found to be in good agreement with the numerical results.

Figure 13. Optimum Flight Level as a function of aircraft mass for the atmospheric temperature profiles given in Fig. 6. The aircraft characteristics are the same as those used for Figs. 7–12.

The corresponding values of the optimum Mach number are given in Fig. 14. These come from Equations (51) and (57). It is interesting to note that, in those parts of the atmosphere where the lapse rate is constant, M _o is almost constant, whilst (FL) _o increases as the mass decreases. However, as previously noted, when the optimum flight level is independent of the mass, i.e. when there is a lapse rate discontinuity, the aircraft must adjust its speed progressively as the mass reduces. The numerical results suggest that the linear variation assumed in the approximate solution is reasonable. In this example, the maximum change required is about 2%.

Figure 14. Optimum Mach number as a function of aircraft mass for the atmospheric temperature profiles given in Fig. 6.

Finally, the percentage difference between ( $\eta$ _o L/D) _o in the example “real” atmosphere and the ISA for a given aircraft mass ratio is shown in Fig. 15. This is a very severe test of the accuracy of the approximate solution and, since there are discontinuities, the accuracy of the numerical method. However, the agreement between the two is good. In this example, the differences between the two atmospheric temperature profiles produce changes in ( $\eta$ _o L/D) _o approaching 1%. Whilst this is small, in aircraft operations fuel use is managed to about 1%, making this variation significant and worthy of closer investigation.

Figure 15. The percentage difference between ( $\eta$ _o L/D) _o in the example “real” atmosphere in Fig. 6 and in the ISA as a function of the mass ratio.