Chapter: 05. Aerodynamic Characteristics

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Chapter: 05. Aerodynamic Characteristics


05.01 High-Speed Aerodynamics

'High speed' aerodynamic calculations cover the normal flight regime, with the aircraft in the 'clean' configuration (i.e. with flaps, slats, and u/c retracted). Starting from the aircraft geometry and other parameters representing technology levels, Piano builds up an estimate of the total drag coefficient (Cd) as a function of the lift coefficient (Cl), Mach number (M) and Reynolds number (Re). A variety of established preliminary-design methodologies are used to calculate the zero-lift, lift-dependent, compressibility and trim drag contributions. The fundamental characteristics can be plotted in the form of a 'Lift/Drag Polar...' curve (see 'Study' menu). You can also examine detailed reports at arbitrary 'spot' conditions through the 'Drag Spot...' feature and tabulations through a 'Drag Table...' (see 'Report' menu).

Several factors and technology adjustments can be used to fine-tune each individual component of drag. (A global user-factor-on-total-drag is also available, but its physical significance is limited; it is a concession to 'management-style' studies).


Sample Aerodynamic Report
 AERODYNAMIC DRAG REPORT at:
 ---------------------------
 MACH                  0.800
 Altitude (pressure)   31000. feet
 KTAS                  469.4
 KEAS                  281.9
 KCAS                  297.4
 Reynolds number       2.202  millions per foot
 Delta-ISA               +0.  deg.C.

 CL                    0.493  based on:
 Reference Area       1207.20 sq.feet  (trapezoidal)

 Drag Coefficients based on ref.area
 -----------------------------------

 Cd Zero-Lift              0.02033    (67.1 %)
 Cd Lift-Induced           0.00905    (29.9 %)
 Cd Compressibility        0.00078    ( 2.6 %)
 Cd Trim                   0.00012    ( 0.4 %)
 Delta Cd (polar-mod)      0.00000    ( 0.0 %)
                           -------    --------
 Cd Total                  0.03028    ( 100 %)

 Aerodynamic Boundaries:
 -----------------------
 Divergence Mach         0.774 {at the given CL 0.493}
 Initial Buffet Mach     0.887 {at the given CL 0.493}
 Initial Buffet CL       0.845 {at the given Mach 0.800}

 Zero-Lift Component Breakdown (Drag Areas,= Cd*S = D/q)
 -----------------------------

 Wing                    7.803  sq.feet    (31.8 %)
 Winglets                0.105  sq.feet    ( 0.4 %)
 Fuselage & fairing     11.225  sq.feet    (45.7 %)
 Stabiliser              1.780  sq.feet    ( 7.3 %)
 Fin                     1.609  sq.feet    ( 6.6 %)  
 Nacelles (total)        2.015  sq.feet    ( 8.2 %)
 User CdS Increment      0.000  sq.feet    ( 0.0 %)
                       -------             --------
 Total Cd0*S            24.536  sq.feet    ( 100 %)

 Overall Lift / Drag Ratio = 16.27 
 =================================

 Total Lift Force      160001. lbf.
 Total Drag Force        9831. lbf. (4916.lbf. per engine) 


05.02 Atmospheric and Airspeed Data

Piano uses the International Standard Atmosphere (ISA). The term 'Delta-ISA' indicates + or - temperature deviations from standard. You can invoke the 'Atmosphere...' dialog under the 'Misc' menu to quickly obtain or plot absolute and relative values of temperature, pressure, density, speed of sound and viscosity at any altitude and Delta-ISA. All references to 'altitude' imply a pressure altitude unless otherwise stated.

For fast airspeed conversions between True Airspeed (KTAS), Equivalent Airspeed (KEAS), Calibrated Airspeed (KCAS) and Mach number at any altitude and Delta-ISA, use the 'Airspeeds...' dialog under the 'Misc' menu.

Source codes: Functions for atmospheric data are pressure , temperature , density , asound , geopotential-height , pressure-height , mu (viscosity), nu (kinematic viscosity), sigma , delta , theta . Airspeed conversion functions show a '>' symbol to indicate the direction of conversion as 'from>to', for example ktas>keas , mach>kcas , kcas>ktas , etc. for all combinations. The function flight-conditions is used to evaluate arbitrary point flight conditions. It can be called with keyword arguments for mach, altitude and either CL or the equivalent mass. It assigns the global variables flight-mach, flight-alt, flight-tas, flight-q, (the dynamic pressure) flight-cl, and flight-re/m (reynolds number per metre). These are needed by the various routines that determine aerodynamics, engine performance and flight performance.


05.03 Reference Area

All aerodynamic coefficients (Cd, Cl, Cm) are normally based on the trapezoidal wing area, as specified by the wing-area parameter (see Chapter#03section05 ).

For comparison purposes, it is possible to output aerodynamic reports based on other definitions of wing area (also described in Chapter#03section05 ). To do this you must use the 'Re-Size Wing' feature (see 'Plane' menu) and tick the option marked 'Use as ref. area in aero reports'. Note that this only affects the output for your current plane. Also, input parameters such as required-stab-vol-coeff , aerofoil-clmax , delta-cd-due-to-u/c etc are unchanged and still based on the wing-area . Because of the potential for confusion, this option is only recommended in cases where you specifically need to compare Piano's calculations with external data based on different area definitions.


05.04 Zero-Lift Drag Estimation

Zero-lift drag (Cd0) is calculated from scratch for individual aircraft components using a traditional preliminary-design approach, based on wetted areas, body form factors and calibrations with known aircraft data.

Skin friction coefficients can be determined as a function of the Reynolds number by several alternative methods, according to the setting of skin-friction-method . The default 'piano' method derives from industry sources and should normally be used. It includes a correction for the transition point from laminar to turbulent flow and for compressibility effects. The 'boeing' method is fairly similar but always assumes fully turbulent flow. The 'prandtl' and 'karman' methods are classical textbook methods, for fully turbulent flow. The 'blasius' method applies to fully laminar flow and is therefore of academic interest only.

Drag reports sometimes show individual component contributions as the product of a drag coefficient and its reference area, the 'Cd S' or 'drag area'. This is also known as 'D/q' since it equals the drag per unit dynamic pressure. You can specify an addition to the calculated total drag through the parameter user-cds-increment , an arbitrary 'drag area' that defaults to zero, to account for unspecified items such as radomes, external tanks etc.

Various empirical calibration factors have to be calculated prior to any drag estimation. Some of these are based on your initial choice of design conditions, namely the design-cruise-mach and design-cruise-altitude . Once chosen, subsequent adjustments to these parameters are not recommended, since the aerodynamic calibrations and also the mass estimates may change (see also Chapter#04section04 for more details).

Source codes: The skin friction function is skin-cf . Some drag factor calculations are done by find-fixed-aerodynamics .


05.05 - Wing zero-lift drag

The wing is divided into a number of strips across its span to account for differences in local chord and Reynolds number during skin friction calculations. An empirical form factor is applied (a function of the aerofoil thickness/chord ratio, the sweep, and the transition point), followed by the calibration factor. Transition is controlled via the parameter wing-transition (expressed as a fraction of the chord), which defaults to zero. This represents an average value along the entire wing for both upper and lower surfaces. It is best regarded as a 'technology factor': Advanced aerofoils combined with a good surface finish tend to be better represented by values of 0.2 ~ 0.3 or so. The calculated wing zero-lift drag can also be adjusted by the user-factor-on-wing-drag .

Source codes: The function wing-strip-cds is called recursively to find the drag area over the various wing panels set up by find-panel-etas . Other functions are wing-swet-between-etas , wing-section-form-factor , wing-drag-factor , and find-wing-cds which returns the final drag area.


05.06 - Fuselage zero-lift drag

Fuselage zero-lift drag is derived from the wetted area, the skin friction coefficient (with Reynolds number based on fuselage length), and a calibrated body form factor (based on the length/diameter ratio and transition point to turbulent flow). The fuse-transition is expressed as a fraction of the fuselage length (default = 0). An internal correction is included for the effective upsweep angle (if any) of the rear fuselage shape. A nominal correction is also made for the wing-fuselage junction depending on the value of fairing-type (see Chapter#03section15 ). Finally, the drag of the windscreen is added according to the (default) value of windscreen-frontal-cd .

Piano's estimation of fuselage drag is applicable to the conventional, smooth shapes that are commonly found in modern commercial aircraft. Some designs (for example the Shorts SD 330/360) use instead simple, easy-to-manufacture fuselages that don't fit this description. Such unusual shapes may require substantially increased values for the parameter user-factor-on-fuse-drag (which defaults to 1). Note that the choice of fuselage cross-section (via fuse-xsection-type ) is only used by Piano to correct the wetted area calculations. Any additional aerodynamic penalty imposed by poor cross-sections (such as a square) depends critically on detailed design aspects, such as fairing radii and the use of carefully positioned strakes to control the shedding of vortices. It cannot be estimated in a generic fashion. The necessary factor could be as high as 1.8 in extreme cases.

Source codes: Relevant functions are find-fuse-cds , body-form-factor , fuse-drag-factor , upsweep-drag-factor .


05.07 - Stabiliser and Fin zero-lift drag

Zero-lift drags of the stabiliser and fin are calculated in the same fashion as the wing, except that it is unnecessary to split them into strips and each surface is treated as a single panel. In the case of a fuselage-mounted stabiliser, a proportional drag adjustment is made for the area immersed in the fuselage. The fin area is assumed to be fully exposed to the free stream (see also Chapter#03section09 ). Transition is presumed to occur at the leading edges. Adjustments are possible through user-factor-on-stab-drag and user-factor-on-fin-drag .

Source codes: See functions find-stab-cds , stab-form-factor , find-fin-cds , fin-form-factor .


05.08 - Winglets zero-lift drag

Cd0 calculations for winglets are similar to those for the fin zero-lift drag. The primary effect of winglets is of course on induced drag (which see).

Source codes: See find-winglets-cds .


05.09 - Nacelle zero-lift drag

For turbofan engines, nacelle drag is based on an industry-derived consensus method and refers to the external or 'scrubbing' drag alone. It is normal practice to assume that only the outermost surfaces of a nacelle contribute to pure aerodynamic drag. This is consistent with Piano's requirement that engine performance data should be 'installed'. The drag of any parts immersed in the stream-tube passing through the intake (such as internal cowlings and support structures) is regarded as a thrust loss. Accordingly, the nacelle length (see nac-length/width ) refers to the external cowling only. It excludes any aft protrusion attributable to the gas generator cowling of some turbofan engines.

Interference and pylon drag cannot be predicted from simple geometric considerations. Modern, optimised installations tend to be aerodynamically 'clean' and these contributions are small. Piano simply adds a +10% allowance to the drag of wing-mounted nacelles. It should however be noted that high-bypass, large-diameter nacelles located at the outboard stations of 4-engined aircraft, where the wing chord is proportionately small, can cause additional interference. If reliable data are available, use user-factor-on-nac-drag with an average value between the inboard and outboard stations. Fuselage-mounted nacelles generally require bulkier and more exposed pylons. They also operate in the 'dirty' air of the rear fuselage region, where the boundary layer is thick, and Piano includes a +50% allowance to the nacelle drag for this case.

For turboprop engines, a method similar to the prediction of fuselage drag is used based on the unadjusted nacelle wetted area. The intricate shapes, increased scrubbing, propeller-induced swirl, multiple intakes and generally poor aerodynamics of turboprop installations require a substantial correction, for which Piano adds a +80% allowance to the predicted nacelle drag. Reverse-flow engines (such as the PT-6) with simple stub exhausts can cause even more increases requiring a user-factor-on-nac-drag . It is not realistic to attempt to separate-out various power-related effects. For consistency, the engine thrust and fuel-flow data should once again be representative of a typical installed configuration, and include the jet thrust of the gas generator.

Source codes: See function find-one-nac-cds , and variables nac-drag-calibration.wing , nac-drag-calibration.rear , nac-drag-calibration.prop .


05.10 Lift-Induced Drag

Basic lift-induced drag is represented in the standard parabolic form:

CDi = k CL^2 / ( pi AR )

The calculation of the k-factor comprises both a theoretical element (from a classical lifting-line function of taper ratio and aspect ratio AR), and an empirical calibration derived from actual aircraft data.

If winglets are used (depending on exist-winglets ), a further empirical correction is added to CDi through a curve-fitted (non-parabolic) function of CL. This is based on limited data representative of traditional Whitcomb-style winglets. The correction depends on the relative winglet span via winglet-span/wing-halfspan .

Induced drag is significantly influenced by the precise details of the wing tip shape, even if there are no winglets. Wind-tunnel and CFD work is indispensable for these. The calculated total can be factored via user-factor-on-induced-drag . Well designed modern, sculpted winglets may offer something like a 3% extra reduction on CDi compared to traditional winglets.

Simple adjustments are included for wing twist (see twist-deg ) and fuselage incidence (see incidence-correction ). These are minor theoretical contributions derived from Torenbeek's 'Synthesis of subsonic airplane design'. A nominal Incidence is estimated by assuming a level fuselage attitude at a representative cruise condition.

Strictly speaking, and consistent with the calibration adopted, the term 'induced drag' in this section should be interpreted as covering all lift-dependent drag other than compressibility and trim drag.

Note: For historical reasons, two induced drag methods are implemented, chosen through the parameter induced-drag-method . This should not normally be disturbed from its setting (default is 'revision98'). The 'original' method is only retained for compatibility, to avoid the need to recalibrate some users' aircraft database.

Source codes: See find-induced-drag-factors , induced-cd , winglet-factor-on-cdi , delta-cdi-twist , delta-cdi-incidence .


05.11 Trim Drag

The increase in wing lift needed to balance any tail download, together with the induced drag of the horizontal tail itself, will generate a trim drag penalty. Calculation of trim drag follows standard aircraft balancing considerations. It is generally a small contribution and may even be marginally negative under some circumstances if the tail carries an upload.

The balancing process is explained separately (see Chapter#08section13 ) and depends on various parameters, notably the min-static-margin . Trim drag is also substantially influenced by aerofoil-cm0 , the zero-lift pitching moment coefficient. Modern aft-loaded aerofoils are associated with increases in pitching moment (more negative values of aerofoil-cm0 ) compared to older conventional aerofoils. Piano's default value (-0.1) is fairly high and typical of aft-loaded types. Calculations are based on a midway position for the centre of gravity, or as specified via cruise-cg-position (where 0 = front c.g., 1 = aft c.g. limit). This is merely a representative cruise condition since the c.g. will shift during the flight. For aircraft equipped with a dynamic fuel transfer system (as may be the case when stab-is-wet is 'true'), higher values of cruise-cg-position could be appropriate.

Note: The above description applies to the current method implemented in Piano. An older implementation is invoked if the parameter induced-drag-method is set to 'original'. This uses a simplified trim drag calculation that was retained for compatibility.

Source codes: See functions cd-due-to-trim , tail-cl-at-aircraft-cl .


05.12 Compressibility Drag

High-speed compressibility drag is calculated by Piano as a function of the wing's sweepback angle, thickness/chord ratio, lift coefficient, and one or more user-controlled parameters that reflect the aerofoil technology level. A variety of methods can be used, according to compressibility-method . In addition to this basic wing-related compressibility drag, corrections can be specified to allow for some zero-lift compressibility drag contribution from the fuselage and nacelles, and for any drag 'creep' displayed by older aerofoils.

The various compressibility methods are described below. They all start by deriving a nominal Divergence Mach number (Mdiv) , then calculate compressibility drag from the difference between the flight Mach and Mdiv. Divergence is the point at which the drag rise due to compressibility starts to become 'steep', as defined in various arbitrary ways. It is often set where the slope of the drag-versus-Mach curve (delta Cd/ delta Mach) exceeds a certain value (typically 0.1), or where the drag increment itself exceeds a value (typically 0.002). Due to variations in definition and calibration considerations, results for Mdiv differ between methods. The calculated Mdiv, as shown in drag reports, can be adjusted via a user-factor-on-divergence-mach . This will directly affect drag estimates. Each method uses a representative value of t/c (between t/c-root and t/c-tip/root ) and sweepback (from sweep-deg ).


05.13 - RAE Compressibility Method

The default setting for compressibility-method is 'rae-modified'. This implements a modification of a method from [data not online].

Aerofoil technology level is simulated by the roof-top-end . This is the most important parameter controlling compressibility drag. It represents the extent of the flat 'roof-top' pressure distribution along the upper surface of the aerofoil. Modern supercritical aerofoils are designed with very flat distributions (to move the shock-wave location associated with pressure recovery at transonic conditions as far aft as possible). The best examples in service at the turn of the century correspond to roof-top-end values in the region of 0.65 . Old conventional (NACA style) aerofoils have 'peaky' pressure distributions, with roof-top-end values around 0.25 or less.

Figure#10rooftop

Compressibility drag is quite sensitive to roof-top-end , indicative of the important role that advanced aerofoils play in high-subsonic designs.

Note: Another option under compressibility-method entitled 'rae-simple' refers to an older implementation of the RAE method. This is only retained to provide backward compatibility for some users.


05.14 - RR Compressibility Method

An alternative choice for compressibility-method is 'rr-adjustable', which implements a method developed within Rolls-Royce plc. This also uses the roof-top-end in its basic calculations, but it allows more adjustments via several dedicated parameters. The increased flexibility makes it easier to alter the shape of certain compressibility curves and to match known lift/drag polars for existing aircraft. The additional parameters used by 'rr-adjustable' are:

rr-adjust-compress.coeff1 . An empirical adjustment to the steepness of the compressibility drag curve at post-divergence conditions. Default is 1; it may be increased to simulate a sharper degradation in aerodynamics at high CL and Mach numbers.

rr-adjust-compress.coeff2 . An empirical adjustment representing the sensitivity of divergence Mach to CL at typical cruise conditions. Generally best left at its default.

rr-adjust-compress.drag-curve . This is a 'data curve' type of parameter. Such parameters accept alternating x and f(x) values along a curve f(x). This curve determines the aerofoil's compressibility Cd (at typical cruise lift coefficients) as a function of delta-Mach, where delta-Mach = Mach - Mdiv. The default curve (used when this parameter is not supplied) corresponds roughly to the following data:

 -0.06  0.0000 
 -0.04  0.0001 
 -0.02  0.0004 
  0.0   0.0020 
  0.02  0.0074 
  0.03  0.0130 
  0.04  0.0210 
  0.05  0.0342 

rr-adjust-compress.mdiv-curve . This is another 'data curve' parameter, for adjusting the variation of divergence Mach with CL. It is a list of numbers representing alternately a CL and a corresponding correction (+ or -) to the Mdiv. Use the 'Drag Table...' (under the 'Report' menu) to find current values. A spline fit is used to interpolate between points.

Note: The remaining options for compressibility-method , namely 'rr-naca', 'rr-supercrit1', and 'rr-supercrit2' refer to older methods. These are only retained for backward compatibility, and apply respectively to NACA-generation aerofoils, 1st-generation supercritical (1970s-80s), and 2nd-generation supercritical (1990s) aerofoils. They do not use roof-top-end nor any of the parameters in 'rr-adjustable'.


05.15 - Additional Compressibility Terms

Zero-lift compressibility drag: The compressibility drag calculated according to compressibility-method is essentially due to the wing. Some additional drag may be contributed by non-lifting bodies such as the fuselage and nacelles, though it tends to be small for modern aircraft operating up to Mach .85 or thereabouts. Such drag can't be predicted in a generic way, but you can allow for it via cd0-compressibility-factor (default = 1). This is defined as the ratio of (zero-lift drag at the design-cruise-mach ) to the (zero-lift drag at the start of compressibility). Nominally, compressibility starts at a Mach number equal to cd0-compress.start-mach , which defaults to 0.6. Piano then applies a representative curve to find the drag at all other Mach numbers.

Although the primary adjustment is cd0-compressibility-factor , the steepness of the drag rise clearly also depends on the other two parameters ( design-cruise-mach and cd0-compress.start-mach ). If the cd0-compress.start-mach alone is raised, drag will reduce at Mach numbers below design-cruise-mach but also increase more rapidly above it. This reflects the behaviour of 'highly tuned' designs. Recall however that design-cruise-mach also influences many other calculations (see Chapter#04section04 ) and is therefore best left alone during these adjustments.

Figure#11cd0comp

Typical values for cd0-compressibility-factor are in the region of 1 to 1.05 for modern commercial aircraft. Some very high-speed transonic designs (such as the Cessna X) may require higher values, say 1.1 to 1.15. At these Mach numbers (.88 ~ .90) area ruling becomes an issue and careful contouring of the rear fuselage, pylons and nacelle shapes is critical. This is a matter beyond preliminary design.

Drag Creep: Another, rarely-used, correction is 'drag creep'. This can happen with some aerofoils, particularly older supercritical types. It is a small and roughly constant positive slope in the Cd-versus-Mach curve starting well before divergence is reached. The relevant parameter is drag-creep-slope (default = 0). This is unlikely to exceed values in the region of 0.005 to 0.01. Creep starts at drag-creep-start , defined as the ratio of (Mach at creep start) to Mdiv.

Figure#12cdcreep

Creep corrections are only included to provide backward compatibility for older users. If the compressibility-method is 'rr-adjustable', creep can be included in rr-adjust-compress.drag-curve .

Source codes: Basic functions for compressibility calculations include divergence-mach , divergence-mach-rae , divergence-mach-rr-adjust , compressibility-cd , compressibility-cd-rae , compressibility-cd-rr-adjust , creep-cd , delta-cd0-compress .


05.16 Low-Speed Aerodynamics

Low-speed aerodynamic characteristics are inherently difficult to predict accurately, even with the most sophisticated of analytical/computational techniques. Piano generates its own 'preliminary design' estimates for both the maximum lift coefficient (CLmax) and the Lift/Drag ratio in the low-speed configuration. However, it is often necessary to adjust these predictions through user-factors when modelling existing aircraft. Flight-verified data are the only reliable basis for such adjustments.


05.17 - Estimation of CLmax

Piano can estimate a maximum lift coefficient (CLmax) at any flap deflection, based on the flap geometry and flap type. The underlying methods are essentially similar to Torenbeek's approach ('Synthesis of Subsonic Airplane Design' Appendix G, Delft University Press), with elements from B.W.McCormick (Aerodynamics, Aeronautics, and Flight Mechanics - Wiley) and other sources.

Initially, a Clmax for the 'clean' aerofoil is specified via aerofoil-clmax . It defaults to 1.5, which is typical of sections found in high-subsonic commercial aircraft. Aerofoils for low-speed commuter planes tend to have somewhat higher values, say 1.7 ~ 1.8. Piano then calculates the increase in CLmax due to flap, depending on flap-type and flap geometry ( eta-flap , flap-chord-fraction ). For performance calculations, flap deflection is taken from takeoff-flap-deg or landing-flap-deg . Finally, if the setting of exist-slats is 'true', an increment is added for any leading-edge devices via delta-clmax-due-to-slat . This defaults to 0.5, but may be as high as 0.9 or so for wings tailored to low-speed performance. The resulting total CLmax for the aircraft can be factored separately for the takeoff and landing, via either user-factor-on-takeoff-clmax or user-factor-on-landing-clmax . Calculated values of CLmax are included in the output of 'Field Lengths' (Command- I, see 'Report' menu).

Flap characteristics depend largely on the extension of geometric chord that occurs during flap deployment, the so-called 'Fowler' movement. Piano uses representative curves for this extension depending on flap-type and deflection. Inevitably, any choice of flap-type is somewhat subjective and individual designs can vary widely. There are no known systematic methods for differentiating the performance of various types of leading edge devices. Their contribution is simply set by the value of delta-clmax-due-to-slat , factored by the slat-exp-span-fraction (exposed span fraction covered by slats, default = 1). The final CLmax calculation includes adjustments for 3-D effects, sweepback, and for a reduction caused by the tail download needed to trim the flaps' pitching moment.

Source codes: The function for chord extension is fowler-factor . Based on thin aerofoil theory, ideal-flap-lift-factor yields a theoretical lift increment . Empirical corrections are made by flap-lift-correction-factor , flap-lift-sweep-correction , flap-dcl0-at and dclmax/dcl0-mccormick . The wing-clmax-at is corrected for trim depending on flap-dcm0 and the final value is returned by aircraft-clmax-at .


05.18 - Estimation of Low-Speed L/D

Drag with high-lift devices deployed is just as difficult to estimate as the CLmax. The increase in 2-dimensional profile drag is derived from empirical data matrices as a function of flap type, chord fraction and deflection, then adjusted for 3-D according to the proportion of the wing covered by the flaps. Induced drag is treated in a way similar to [data not online] using a simplified parabolic polar with corrections dependent on the spanwise extent of the flaps and the aspect ratio. There are no reliable data for the effects of leading edge devices. You can examine the complete calculated aerodynamics through the 'Low-Speed Polar...' (see 'Study' menu). This polar is primarily applicable to takeoff/climbout or approach conditions (CLmax/1.44 and CLmax/1.69 respectively, depending on v2-speed-ratio and approach-speed-ratio ). During performance calculations, the calculated L/D ratios at these two specific conditions can be adjusted via user-factor-on-takeoff-l/d and user-factor-on-landing-l/d .

Undercarriage drag is not part of the low-speed polar itself. It is added as necessary during takeoff and landing calculations, according to delta-cd-due-to-u/c .

Notes on flap types: At low flap deflections, a single-slotted flap-type often has better characteristics than a multi-slotted one. This is due to differences in fowler movement and the need to deploy the vanes in multi-slot flaps. For aircraft that have a mixture of flap types across their span, choosing the flap-type means making a judgement, normally favouring the type prevalent in the outboard stations.

Sample Low-Speed Polar
 Low-Speed Polar (Ref.area 1207.20 sq.feet trapezoidal)
 ---------------------------------------------------------------------- 
                 Climbout at CLmax/1.44      Approach at CLmax/1.69
                 -----------------------     -------------------------- 
 Flap  CLmax     CL(v2)  CD(v2)  L/D(v2)     CL(app)  CD(app)  L/D(app)

   0.  2.09      1.45    0.0959  15.17        ----    ------   ----
   5.  2.25      1.56    0.1071  14.56        ----    ------   ----
  10.  2.40      1.67    0.1247  13.37        ----    ------   ----
  15.  2.55      1.77    0.1516  11.70        ----    ------   ----
  25.  2.85      1.98    0.2153   9.20        1.69    0.1821   9.27
  35.  3.11       ---    ------  -----        1.84    0.2448   7.52
  45.  3.28       ---    ------  -----        1.94    0.2962   6.54

Source codes: The profile drag function is flap-dcd0-at , which uses flap-dcdp-matrix . Induced drag depends on flap-induced-factor-k . The high-level function is find-flapped-cd .


05.19 One-Engine-Inoperative Drag

With one engine inoperative, additional drag is generated by the asymmetric flight conditions and by the windmilling of the failed engine. The asymmetric contribution depends on flight technique as a pilot can trim the aircraft with varying degrees of yaw and roll. It is estimated as a function of the thrust of the remaining engines and assuming failure of an outboard (critical) engine. (The approach is similar to [data not online]). Windmilling drag for turbofans is a function of Mach number and bypass-ratio (similar to [data not online]). The turboprop case is calculated per [data not online], depending on the propeller-diameter and blades-per-propeller .

The asymmetric and windmilling drag contributions can be adjusted separately via user-factor-on-asymmetric-drag and user-factor-on-windmill-drag . You can find the values of these drag items at any flight condition in the output of the 'in-flight manoeuvre' feature described in Chapter#16section02 onwards.

Source codes: See delta-cd-asymmetric-flight , find-windmill-drag .


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