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13 - 1
13.1 Drag Polar In Section 2 under Step 14 it was a requirement that the flight performance has to be checked to complete the aircraft design process. This is only possible if the drag polar that forms the starting point for checking the flight performance is known. The polar creates the connection between lift and drag (Fig. 13.1). When determining the polar, the main task is to calculate the drag: the required lift is predefined, as a rule (in cruise flight, for example, due to the fact that lift equals the aircraft's weight).
The polar for uncambered and cambered airfoils. The angel of attack α
for CD = f CL is also indicated on the polar
If the polar is stated as an equation, the drag coefficient is written as a function of the lift coefficient. A polar for an uncambered airfoil as in Fig. 13.1 can be written in the following form: CD = CD0 + k ⋅ CL
1 π ⋅ A⋅ e
13 - 2 with the Oswald factor e and the aspect ratio A. Here the effective aspect ratio Aeff according to Section 7 should be used for the aspect ratio A, in order to take account of the influence of endplates or winglets, which increase the effective aspect ratio Aeff compared to the aspect ratio calculated with A = b 2 / S . Irrespective of which aspect ratio is included in the calculation, the symbol A is simply retained in the equations in this case. The final result is 2
CD = CD0 +
CL π ⋅ A⋅e
A polar for an uncambered airfoil as in Fig. 13.1 can be written in the following form:
CD = CD ,min + k CL − CL ,min
• In the case of an uncambered airfoil the drag is minimal if the lift is zero. This is the case with an angle of attack of α = 0. • In the case of a cambered airfoil the drag is minimal for a specific positive lift. This shifts the polar upward. For airfoils with a small camber this shift is minimal. For this reason, a polar according to equation (13.3) can be used for simplification. • In the case of high angles of attack close to the maximum angle of attack α CL , max , the drag
increases more sharply than in the parabolic form according to equations (13.3) and (13.4).
A formulation with a term (C L − C L ,mi n ) is able to represent the correct drag coefficients 4
for both low and high lift coefficients:
C D = C D ,mi n + k1 (C L − C L ,mi n ) + k 2 (C L − C L ,mi n ) 2
In the further course of this section only the simple description of the polar according to equation (13.3) will be used.
13.2 Drag The drag can be classified according to physical causes or according to the drag-inducing elements.
13 - 3
Classification of drag according to physical causes The total drag can be subdivided into (compare with Equation 13.3): 1. zero-lift drag: drag without the presents of lift; 2. induced drag: drag due to lift. In addition wave drag comes into play, caused by a Mach number M that is greater than the critical Mach number M crit . By definition, M crit is the flight Mach number where a flow Mach number M = 1 arises locally on the aircraft for the first time with increasing airspeed. This can, for example, occur on top of the wing. Chapter 7 gives more information on this phenomenon. Very varied suggestions are made in the literature to further classify drag. One version (see Fig. 13.2) subdivides the zero-lift drag into profile drag, interference drag, and miscellaneous drag (trim drag and additional or parasite drag). The profile drag can in turn be subdivided into skin-friction drag and pressure drag.
One of several possibilities to subdivide drag
• Wave drag is caused by shock waves on the airfoil. At high Mach numbers, both the zerolift drag and the induced drag are increased by wave drag. The wave drag is only shown if it is calculated separately (and not as part of zero-lift drag and induced drag). • Skin-friction drag is caused by the shear flow in the thin boundary layer close to the airfoil surface.
13 - 4 • Form drag is dependent upon the boundary layer, which can assume a large thickness especially in areas of separated flow and therefore prevents pressure being regained in the area of the trailing edge. • Interference drag is caused by the mutual influence of the flow around neighboring components. The closer two components are, the greater the interference drag. • Trim drag is caused by the elevator or horizontal stabilizer being deflected to retain the equilibrium around the lateral axis, and lift or negative lift being created on the horizontal tailplane. This results in a change in the induced drag on the horizontal tailplane and on the wing. This change in drag can be individually recorded on the wing and on the horizontal tailplane, but is sometimes also calculated separately and shown as trim drag. • Additional drag (parasite drag) refers to drag components that tend to be recorded approximately and generally. The term "additional drag" is used, for example, with reference to drag caused by: • flaps and slats; • landing gear; • cockpit windows; • leakages in the pressurized fuselage.
Classification of drag according to drag-inducing elements In Roskam VI, for example, drag is calculated individually for the following elements: • wing; • fuselage; • empennage; • nacelle and pylon; • flaps and slats; • landing gear; • cockpit windows; • and other elements (lump-sum).
Literature on calculating drag This section contains key parameters and equations for calculating a simple polar for initial comparative studies and flight performance calculations. A selection of basic equations from DATCOM 1978 is presented primarily for subsonic flow. It is important to bear in mind that DATCOM 1978 was not actually created for detailed flight performance calculations, but rather only for calculating aircraft dynamics. In DATCOM 1978 (Section 188.8.131.52) the following comment therefore appears:
13 - 5 It should be pointed out that the basic approach taken here is satisfactory for preliminary design stability studies and that no attempt is made to provide methods suitable for performance estimates.
Despite this statement, DATCOM 1978 contains one of the most detailed publicly available handbook methods for calculating drag. The DATCOM version therefore also forms the basis for aircraft design in textbooks such as Roskam VI and Raymer 1992. Hoerner 1965 is still a central source for answering further detailed questions on the subject of "drag".
Significance of drag calculation in aircraft design In a similar way to forecasting mass, forecasting aircraft drag is of considerable importance for the aircraft project. If it should transpire during flight testing that the aircraft drag is higher than assumed, it may be the case that the specified range of the aircraft cannot be complied with. This may lead to contractual penalties for aircraft which have already been ordered or even to orders being cancelled. Owing to the importance of forecasting aircraft drag, aircraft manufacturers have developed their own detailed computer-aided procedures, which are not accessible to the public. One in-house method that meanwhile appeared in public is Boeing 1970.
Procedures for calculating drag On a somewhat extended scale compared to equation (13.3), the polar is described here by
CD = CD , 0 + ∆CD , flap + ∆CD ,slat + ∆CD , gear
CL 2 + ∆CD ,wave + . π ⋅ A⋅e
In this version C D ,0 is the zero-lift drag of the aircraft when the slats, landing flaps and landing gear are retracted (clean configuration). Their additional drag is taken into account separately by ∆CD , flap , ∆CD ,slat , ∆CD , gear calculated according to Section 5. In addition, ∆CD ,wave
is the drag rise due to wave drag. A typical increase in drag with the Mach
e = 0.7 e = 0.85
number is shown by Fig. 13.6 for four different types of aircraft. An estimation method is derived from Fig. 13.6 with equation (13.25). with extended flaps, slats and landing gear, can be used for retracted flaps, slats and landing gear for simplicity’s sake. If one wishes to go further, a calculation of the Oswald factor e according to equation (13.26) is also possible.
13 - 6
13.3 Zero-lift Drag Two methods are put forward to calculate the zero-lift drag of aircraft C D ,0 : 1. Calculation of zero-lift drag from equivalent skin-fraction drag (equivalent skin-friction method); 2. Calculation of zero-lift drag from the individual drag of components (component build-up method). The first version is simpler and generally less precise, as details of the flow phenomena are not incorporated in the method.
Calculating the zero-lift drag coefficient C D ,0 from the equivalent skinfriction drag coefficient C fe This method uses the aircraft geometry which is now known according to the preceding design steps to estimate the zero-lift drag D0 with the aid of an equivalent skin-friction coefficient C fe . The skin-friction coefficient multiplied by the dynamic pressure and the wetted area gives the zero-lift drag: D0 = q C fe S wet = q C D , 0 SW . In contrast to the skinfriction drag coefficient C f , the equivalent skin-friction coefficient C fe also takes into account the other forms of drag contributing to the zero-lift drag; these are form drag, interference drag, trim drag and additional drag. The equivalent skin-friction coefficient C fe is derived from measured values of the zero-lift drag. For this reason C fe includes all drag contributions as mentioned. CD ,0 = C fe ⋅
S wet SW
Empirical values for C fe are contained in Table 13.1 and Table 13.2. S wet is the wetted area of the whole aircraft. In the case of conventional configurations the
following parts must be taken into account, as a rule: • fuselage; • wings; • horizontal and vertical tailplanes; • nacelles and pylons.
13 - 7 S wet = S wet , F + S wet ,W + S wet , H + S wet ,V + n E ⋅ S wet , N + n E ⋅ S wet , pylons .
In the case of unconventional configurations the wetted area of the aircraft has to be determined with the other corresponding components of the aircraft. Table 13.1
The equivalent skin-friction drag coefficient C fe on the basis of general experience (Roskam I) aircraft type
A simple approximation formula should be found for the wetted area of the pylon S wet , pylons according to the geometry.
Calculation of the zero-lift drag coefficient C D ,0 from the individual drag of components When calculating the polar from the individual drag of components (component build-up method), the zero-lift drag is calculated separately for each component. For simplification, the induced drag can be determined as in Section 5 with the aid of an assumed Oswald factor e. The zero-lift drag of each component derives from: 1. the skin friction drag coefficient C f ; 2. a form factor FF , which takes into account the pressure drag of the component; 3. an interference factor Q , which takes into account the interference drag; 4. the factor S wet / S ref , which serves to relate the drag coefficient of the component to the reference wing area. Thus, the zero-lift drag would then be n
S wet ,c
C D 0 = ∑ C f ,c ⋅ FFc ⋅ Qc ⋅
This takes account of the individual drag of all n components (referred to as "c"). However, the zero-lift drag of some components and some effects are difficult to deal with in the way described above. Therefore two terms are added to the above equation. It is then
13 - 10
S wet ,c
C D 0 = ∑ C f ,c ⋅ FFc ⋅ Qc ⋅
+ C D ,misc + C D ,L + P .
C D ,misc contains the zero-lift drag of all other components, such as the landing gear. C D ,L + P represents all the additional drag due to leakages in the pressure cabin, e.g. at doors, and smaller drag, e.g. caused by aerials.
Re 1.) The skin-friction coefficient C f describes the drag of a longitudinal flow along a flat
plate. In the case of laminar flow the following applies: C f ,laminar = 1.328 / Re .
In the case of turbulent flow the skin-friction drag according to DATCOM 1978 (184.108.40.206-26) and Raymer 1992 equation (12.27) is C f ,turbulent =
( log Re)
⋅ (1 + 0144 . ⋅ M 2)
Equation (3.17) is illustrated in Fig. 13.4. The Reynold's number is known to be Re =
V ⋅l ν
In the case of a wing or empennage, the characteristic length l is the mean aerodynamic chord (MAC). In the case of the fuselage, the characteristic length is the total length of the fuselage. ν is the kinematic viscosity, which is a function of aircraft altitude. If the surface is relatively rough, the skin-friction drag C f ,turbulent will be higher than calculated by equation (13.16). This effect is taken account of according to DATCOM 1978 (220.127.116.11-27) and Raymer 1992 by means of a so-called cut-off Reynold's number (Fig. 13.5): ⎛l⎞ Re cut-off = 38.21⋅ ⎜ ⎟ ⎝ k⎠
für M < 0.9:
für M ≥ 0.9:
⎛l⎞ = 44.62 ⋅ ⎜ ⎟ ⎝ k⎠
. . ⋅ M 116
13 - 11 In this l is the characteristic length and k is the surface roughness according to Table 13.3.
V ⋅l > Recut − off ν V ⋅l ≤ Recut − off ν
C f ,turbulent is calculated with
Re = R ecut − off ,
C f ,turbulent
is calculated with
Skin-friction coefficient for turbulent flow DATCOM 1978 (18.104.22.168-26)
Surface-roughness height k from DATCOM 1978 (22.214.171.124-A) Type of Surface k [mm] aerodynamically smooth 0,00000 polished metal 0,00127 natural sheet metal 0,00406 smooth paint 0,00635 camouflage paint 0,01016
V ⋅l ν
13 - 12
"Cutoff Reynolds Number" from DATCOM 1978 (126.96.36.199-26)
In the case of most aircraft the flow over the total wetted surface of the fuselage is turbulent. Laminar flow may exist on the front 10% to 20% of the wing. A carefully designed composite aircraft, such as the Piaggio GP 180, may exhibit laminar flow over 50% of the wing and over
13 - 13 20% to 35% of the fuselage. Raymer 1992 suggests estimating the proportion of laminar flow k laminar for the aircraft in question, in order to thus calculate a mean skin-friction drag C f = klaminar ⋅ C f ,laminar + (1 − klaminar ) ⋅ C f ,turbulent .
Re 2.) The form factor FF is designated FFW for wings and FFH or FFV for empennages.
According to DATCOM 1978 (188.8.131.52) – in a notation pursuant to Raymer 1992 – the form factor is
4 ⎡ 0.6 ⎛ t ⎞ 0.28 ⎛t⎞ ⎤ . ⋅ M 0.18 ⋅ ( cos ϕ m ) FF = ⎢1 + ⎜ ⎟ + 100⎜ ⎟ ⎥ ⋅ 134 ⎝ c⎠ ⎦ ⎣ xt ⎝ c ⎠
x t is the position of maximum thickness according to Fig. 7.2. ϕ m is the sweep angle of the %-line of maximum relative thickness. According to DATCOM 1978 (184.108.40.206) the form factor for the fuselage is calculated according to
FFF = 1 +
/ dF )
/ dF ) 400
According to RAYMER 1992 the form factor for nacelles is calculated from FFN = 1 +
0.35 . (l N / d N )
l N and d N are the length and the diameter of the nacelle respectively.
Re 3.) The interference factor Q is selected according to Table 13.4.
No interference factor is given for the fuselage, because the interference of the components of a conventional aircraft design exists with the fuselage. On the other hand, the fuselage does not exhibit any interference with itself!
13 - 14 Table 13.4
Interference factor with respect to ...
engine mounted directly on the wing or fuselage distance of engine to wing respectively fuselage
is smaller than engine diameter
fuselage horizontal or vertical tailplane
high-wing, mid-wing or low-wing position with aerodynamically optimized wing-fuselage fairing low-wing position without aerodynamically optimized wing-fuselage fairing conventional empenage H-tail V-tail
distance of engine to wing respectively fuselage is greater than engine diameter
1..0 1.10 ... 1.40 1.0 1.04 1.08 1.03
13.4 Wave drag In this section an attempt is to be made to estimate the wave drag ∆C D ,wave . As the flow configurations approaching sonic speed cannot be ascertained by using simple methods, we shall work on the basis of measured wave drag ∆C D ,wave and try to generalize these measurements somewhat. A glance at Fig. 13.6 shows that the drag increase can be expressed by an equation in the following form:
⎛ M ⎞ = a ⋅⎜ − 1⎟ ⎝ M crit ⎠
M crit is the critical Mach number of the (project) aircraft to be examined. For a and b the values from Table 13.5 have to be inserted. Parameters a and b have been ascertained mathematically, with the aim of reproducing the curve from Fig. 13.6 as precisely as possible. To do this, the critical Mach number M crit was read off directly from Fig. 13.6. It is connected to M DD in a certain way – by definition the Mach number where the wave drag reaches a value of 0.002 or 20 drag counts.
Boeing 727-200. Passenger aircraft, jet, mMTO = 95000 kg, VCR = 254 m/s, M = 0.82. Convair F-106A Delta Dart, the primary all-weather interceptor aircraft for the United States Air Force from the 1960s through the 1980s. Delta wing with NACA 0004-65 mod airfoil at root and tip. The fuselage was designed according to area ruling.
13 - 16 The practical procedure for estimating ∆C D ,wave would therefore be as follows: 1. Select one of the four stated aircraft that looks as similar as possible to the project aircraft; 2. Determine M crit of the project aircraft or estimate M crit of the project aircraft from difference or ratio of M crit and M DD with the aid of Table 13.5. Note: M DD of the project aircraft is known from the wing design; 3. Take parameters a and b from Table 13.6; 4. Estimate ∆C D ,wave with the aid of equation (13.25) as a function of the Mach number.
13.5 Induced drag and Oswald factor The Oswald factor e is required by equation (13.3) to calculate induced drag. 2
CL C D = C D 0 + C Di = C D 0 + π ⋅ A⋅e
According to Howe 2000 (equation 6.14a), the Oswald factor e can be estimated for the subsonic range and for transonic flows (M < 0.95) for aircraft with a wing aspect ratio of A > 5 from
1 ⎧ 0.142 + f (λ ) A (10 t / c) 0.33 0.1 (3 N e + 1) ⎫ (1 + 0.12 M ) ⎨1 + + ⎬ (cosϕ 25 )2 (4 + A)0.8 ⎭ ⎩
In this equation M is the flight Mach number A is the (effective) aspect ratio t/c is the relative airfoil thickness ϕ 25 is the wing sweep of the 25% line Ne
is the number of engines on the wing (if none, then N e = 0). f (λ ) = 0.005 (1 + 1.5 (λ − 0.6) 2 ) .
In this equation λ is the wing taper. A typical value for f (λ ) is 0.0062 .