Drag estimation
Each main component of the aircraft (wing, fuselage, tail surfaces, nacelles; and in the low speed flight phases the flaps and undercarriage) must be separately assessed for its contribution to the overall drag of the aircraft.2 It is not sufficient to consider only the wing effects in the estimation of drag.
For subsonic civil aircraft the overall drag of the aircraft can be considered under three categories:
1. profile drag resulting from the pressure field around the shape and from the surface skin friction effects of the boundary layer;
2. lift induced drag resulting from the changes in pressure due to attitude variations resulting from the generation of lift;
3. wave drag from shock waves as parts of the accelerated flow over the surfaces become supersonic.
These effects result in the following formations for aircraft drag coefficient:
where: CDo = estimated total profile drag coefficient (i.e. the summation of the drag from all the aircraft components appropriate to the flight conditions under investigation) CD. = total effect of all the lift dependent components (principally this is a function of CDj; as the design becomes more established this term may be extended to include a direct CL term as well as the square term)
ACDW = additional drag resulting from the shock waves. As civil aircraft are not intended to be flown past the drag divergence Mach number this term may be assumed to be 0.0005 if no other details are available
The total drag coefficient can be plotted against Cl to give the graph shown in
Fig. 8.13 and known as the drag polar. Each of the drag terms will now be assessed.
Profile drag
The profile drag can be estimated using the formula below:
where: Cf = skin friction coefficient which is a function of Reynolds number F = component form (shape) factor Q = interference factor Swet = component wetted area Sref = reference area used for the calculation of CD (normally the wing gross area)
We start by calculating the Reynolds number Re for each component:
Re = (Vl)/u where: V = aircraft forward speed in the flight case under investigation u = kinematic viscosity at the speed and height of operation I = component characteristic length, i.e. fuselage overall length, wing mean chord, tail mean chord, nacelle overall length
The skin friction coefficient for turbulent boundary layer conditions can now be calculated for each component using the PrandtlSchlichting formula:
where: M = Mach number at operational conditions under investigation. Rec = Reynolds number of component
For any component or area with laminar flow the following equation should be used:
For components with both laminar and turbulent flows the value for Cf should be a weighted (by area) average of the two results.
The form factors for each component are calculated from the input geometry using a specific formula for each component.
(i) For the fuselage
If = fuselage overall length Ax = fuselage crosssectional area
(Note: [(4/7t)^4x]0 5 = fuselage diameter for circular fuselage shapes) Q= 1.0
(ii) For the wing
where: A0 5c = sweepback angle at 50% chord
Q = 1.0 for well filleted low/mid wings, = 1.11.4 for small or no fillet (a value of 1.0 to 1.2 seems to work for conventional designs)
(iii) For the tail surfaces. F as for wings with
(iv) For the nacelle. Estimating the drag of a nacelle is complicated by the intricate geometry of many nacelles and the interrelationship with the definition of engine thrust. For initial estimates you may use FQ = 1.25 for wingmounted engines and a 20% higher value for aft fuselagemounted installations (to account for the increased interference on the rear of the aircraft).
(v) For the undercarriage. This is influenced by the size of the undercarriage and the number of wheels. At the preliminary design stage much of this data will not be available. The number of wheels and the general undercarriage size will be primarily influenced by the aircraft maximum landing weight. It is suggested that for aircraft with a multiwheel bogie type undercarriage the following formulae may be used for undercarriage drag:
imperial units AD/q = 0.0025 (HQ073 WL in lb and (AD/q) in sq ft metric units AD/q = 0.00157 (WL)0 73 WL in kg and (AD/q) in sq m where AD/q is the increase in drag area (i.e. S ■ CDl.) where S is the wing reference area and WL is the weight/mass of the undercarriage.
For smaller aircraft in the F100, DC9 and B737 class, which will usually have twinwheel main undercarriages, the following formula is suggested:
imperial units AD/q = 0.006 (Wi.)0 73 WL in lb and (AD/q) in sq ft metric units AD/q = 0.00093 (JFL)°73 WL in kg and (AD/q) in sq m
For a quicker result just assume AD/q = 0.02 S (where S is the wing reference area).
(vi) For the flaps. The drag due to the highlift systems depends upon the types of trailing edge and leading edge flaps envisaged for the aircraft under consideration. As was noted earlier there are several possibilities, from plain flaps to area extending flaps all with one, two or three slots. The method suggested for use at the preliminary design stage covers area extending flaps such as:
• Fowler type with one, two or three slots;
• flaps with offset hinge and a linkage to give some area extension at low flap angles again with one or two slots and plain flaps with offset hinge with one or two slots.
The parameters fundamentally influencing the flap drag increments have been taken as type of flap, flap angle, wing area increase and the sweep angle. The definition of the flap drag increment along with the above parameters is important, particularly the wing area increase. The symbols used are defined in the following list along with the associated figures:
ACD is the flap drag increment = total drag increment for both leading and trailing edge devices extended, or for trailing edge devices only  see Fig. 8.14
wing 1/4 chord sweep extended flap area ratio as defined in Fig. 8.15 number of slots in trailing edge flap systems trailing edge flap angle as defined in Fig. 8.16
effective wing chord with wing trailing edge flaps extended (LE flaps retracted) as defined in Fig. 8.17
effective wing chord with leading edge devices extended (TE flaps retracted)
certified maximum lift coefficient at appropriate flap setting total flap drag increment at C^ /1.44, for TO/second segment climb total flap drag increment at C^ /1.69, for landing/approach
The objective of this method has been to produce a quick estimation of the flap drag at takeoff, second segment and landing approach conditions. Flap drag has therefore been presented at l.2Vs and 1.3K; (where is the aircraft stall speed with flaps extended) at the appropriate flap condition. Figures 8.18 and 8.19 give the drag increment at l.2Vs and 1.3FS respectively. It is important that note is taken of the definition of this drag increment as shown in Fig. 8.14. By its nature this method of assessing flap drag is crude and more rigorous methods should be applied as soon as the flap and wing geometry have been established.
(vii) Drag of secondary items. The drag of secondary items may be as high as 10% of the profile drag calculated using the above method. The extra drag is typically due to excrescence, surface imperfections and system installations. For initial project design work the following estimates are suggested.
Wing: 6% of wing profile drag Fuselage and empennage: 7% of fuselage profile drag Engine installation: 15% of nacelle profile drag Systems: 3% of total profile drag
In addition to the above items the cockpit windshield will increase fuselage drag by 23%.
Wing chord line
Wing chord line
Fig. 8.16 Definition of trailing edge flap angle.
AO.25 Sr
AC, AC, d1.21fs DUIK
Fig. 8.16 Definition of trailing edge flap angle.
Trim drag may also be a significant contribution to the total aircraft drag. At the initial project design stage, however, insufficient data is available to accurately estimate it. A typical figure for a well designed aircraft is about 5 drag counts (1 count = 1 x 10"4).
There are several published methods for estimating wetted area of the various components but all rely on the detailed definition of the layout. When you know the shape of the component it is relatively easy to make a detailed estimate of the wetted area of the part using your own initiative.
Some performance calculations are done with one engine failed. In such a condition the aircraft will be subjected to drag increases from the flow blockage of the failed engine and the extra drag from the asymmetric flight attitude of the aircraft. The drag increment from the engine, known as the windmill drag, can be estimated from ACD = 0.3 Af/S (where Af is the area of the fan cross section, and S is the wing reference area). The drag increment from asymmetric flight is difficult to predict quickly. The best that can be done in the project stage is to add a percentage to the overall drag of the aircraft. A value of 5% CDo seems reasonable for conventional design configurations.
Adding all the component drags together with the corrections detailed above gives the aircraft profile drag coefficient:
^^^Dcomponents)
Example (double slotted flap):
Example (double slotted flap):
Number of slots
Fig. 8.18 Example of estimation of flap drag at 1.2 Vs.
Number of slots
Fig. 8.18 Example of estimation of flap drag at 1.2 Vs.
No. of flap slots
Fig. 8.19 Example of estimation of flap drag at 1.3 Vs.
No. of flap slots
Fig. 8.19 Example of estimation of flap drag at 1.3 Vs.
Lift induced drag
Lift dependent drag arises from three principal effects:
1. a component from the wing planform geometry
2. a contribution from nonoptimum wing twist
3. a component due to viscous flow forces
All of these effects are associated with the distribution of lift along the wing span (sometimes called span loading). The best (lowest induced drag) loading consists of a smooth elliptical distribution from wing tip to tip with no discontinuities due to fuselage, nacelles, flaps, etc. Obviously, except for high performance sailplanes, it is not feasible to arrange the aircraft layout to get such a spanwise load distribution. The comments below will help you to estimate the lift dependent drag coefficient for civil aircraft.
The component arising from the planform geometry is derived from classical lifting line theory details of which can be found in good aerodynamic textbooks. In this theory the wing is represented by a series of horseshoe vortices which generate the aerodynamic circulation around and along the wing shape. Figure 8.20 shows the theoretical distribution of the induced drag factor relative to the wing aspect ratio and taper ratio. These values are corrected by the application of an empirically derived factor (C2) derived from previous aircraft designs (Fig. 8.21).
As might be expected, aircraft with older wing sections are inferior at high aspect ratios. Modern aircraft tend to adopt higher aspect ratios partly because the wings are designed using advanced technology threedimensional aerodynamic analysis methods.
1.10
Old wing design and sections c2 = 1.2350.0245 A
1.10
1.05
1.05
0.90
0.90
0.85
Wing aspect ratio (A)
Fig. 8.21 Empirical correction c2 to planform factor.
Induced drag coefficient is estimated by the equation below:
where: Cx is found from Fig. 8.20 C2 is found from Fig. 8.21 A is the wing aspect ratio
CL is the lift coefficient of the aircraft in the flight condition under investigation (i.e. aircraft mass, speed and altitude)
The contribution from the effect of nonoptimum wing twist requires a knowledge of the distribution of aerofoil section twist and changes of the sectional lift curve shape, along the span. In the early stages of the project design such intricacies in wing geometry will not have been decided; however, a contribution is appropriate as the final wing shape will include such distributions. The addition of CD, for such effects will be between 0.0003 to 0.0005 with a value of 0.0004 being suitable for conventional civil turbofan layouts.
The viscous flow effects are significant. These forces manifest themselves mainly in the boundary layer growth arising from changes in wing incidence. Without the assistance of powerful computer fluid flow analysis it is difficult to predict these effects accurately. An empirical analysis of conventional civil aircraft geometry and operating conditions shows that the contribution to dCD/dC]_ is proportional to aircraft profile drag. The following relationship is appropriate for current wing geometry.
The older technology value seems to work well for B737300, B757 and B767, the A320 lies in between the two values and the A330/340 and B777 match the advanced technology value.
dCp/dCf, = 0.35 CDo (for older technology designs) dCD/dCL = 0.15CDo (for advanced technology designs)
Hence the total lift dependent drag coefficient for the aircraft is the sum of the three effects:
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