11.4.1 Discussion

Considering air as inviscid makes mathematics simple to quickly obtain aerodynamic properties of a body in airflow, lift, moment and so on, in the linear range of operation. However, it is incapable to predict drag, stall and so on. In the earlier days aircraft designers had to rely on experimental data and semi‐empirical relations those matured to sufficient accuracy to send man to the Moon.

With advances made in numerical analyses, today, CFD can approximately solve the exact nonlinear partial differential equations to get comparable results to experimental values (Chapter 23 outlines CFD capabilities). However, unlike getting good results on predicting lift and moment characteristics, CFD capability in predicting drag characteristic accurately and consistently has some way to go. This chapter presents industry standard drag predicting methodology, using well substantiated semi‐empirical relationships.

Figure 11.2 plots the Bizjet drag polar and the aircraft components drag values for the Bizjet, as worked out in Section 11.19. It shows that for the Bizjet cruising at M_crit (LRC at Mach 0.7), about two‐thirds of aircraft drag is viscous dependent, it would exist if air is considered inviscid. There is a small increase in Bizjet drag when flying at M_div (HSC at Mach 0.75), on account of a 20 count of wave drag rise (C_DW = 0.002, ≈ 6%). These percentages of drag breakdown are a good representation of typical transport aircraft drag.

On examining the aircraft‐component drag breakdown, it can be seen that just the wing and fuselage (with belly fairing and canopy) together contribute to about 60% of the C_DPmin. Adding the nacelle drag, together these three components contribute to three‐quarters of the viscous‐dependent parasite drag. Aircraft designers concentrates more on these components to get the best configuration as discussed in Chapters 4–6. Nacelle design is dealt in Section 11.9.3.

11.5.1 Actual Drag Polar

Actual drag polar has the same format as drag polar but not an exact fit to a parabolic equation, especially at low‐ and high‐speed ranges. The methodology for estimating actual drag using semi‐empirical relations is given in this chapter. These semi‐empirical relations are based on experimental data from both wind‐tunnel and flight tests. These are the best available drag polar for industry standard analyses. Drag polar obtained by this methodology is not easily amenable to representation by close‐form equations, especially for high‐subsonic speed aircraft. To generate graphs that fit a parabolic shape, it may require shaping of graphs with the least loss in accuracy. Considerable insight into aircraft performance can be obtained by manipulating the parabolic drag polar equation. The industrial methods presented in this book give all the information but require laborious computational effort.

11.5.2 Parabolic Drag Polar

As mentioned earlier, close‐form solutions from the analytical expressions quickly show important aircraft characteristics and prove useful to make a trend analyses. The readers may refer to [1–4] to study analytical treatments. At LRC, when wave drag is simplified to C_DW = 0, then Eq. 11.2 takes the form as follows

11.3

where k = 1/eπAR

The difference between the actual aircraft drag polar (C_Dpmin + ΔC_Dp + C_L²/πAR) and its corresponding parabolic drag polar (C_Dpmin + kC_L²) is in the approximation of the variable lift‐dependent parasite drag ΔC_Dp to a constant Oswald's efficiency factor ‘e’ associated with the induced drag C_Di. The variable parameter ‘e’ is approximated to a constant value bringing parabolic drag polar close to actual drag polar around the cruise segment and also gives a good match at climb and descent segments.

Figure 11.3 shows the variation of Oswald's efficiency factor ‘e’ with Mach number for the Bizjet aircraft at 18 000 lb weight cruising at a 41 000 ft altitude (C_L ≈ 0.512). It is evaluated using the following equation and plotted in the figure.

where e = (1/kπAR)

Parabolic drag polar incorporates a value of k = 0.0447 (e ≈ 0.95) representing average values. At the design C_L (typically, at mid‐LRC) it approaches 1. This will cause a discrepancy between parabolic drag polar and actual drag polar values at higher speeds. For flying close to the ground, low speed incorporates drag of a ‘dirty configuration’ with values accommodating changes in e. For this reason, this book suggests that, for high‐subsonic aircraft, one should use actual drag polar for accurate industry standard results, for example, for certification substantiation, preparing the pilot manual and so on, and parabolic drag polar may be used for exploring and establishing aircraft characteristics.

11.5.3 Comparison Between Actual and Parabolic Drag Polar

The parabolic representation of aircraft drag polar offers many advantages. Easy mathematical manipulation gives a quick insight to aircraft characteristics at an early design phase to make improvements, if required. Industry needs drag prediction to be as accurate as possible (in the order of few counts) and, therefore, results from approximate analytical expression are not adequate for high‐subsonic aircraft, especially when good prediction of drag polar can be obtained through semi‐empirical relations substantiated by testing.

This section attempts to present approximated analytical expression for high‐subsonic aircraft drag polar. Recall Eq. 11.2, which gives the expression for high‐subsonic aircraft drag as

11.4

where (C_Dpmin + C_L²/πAR) represents parabola in which C_Dpmin is the minimum distance of the drag polar from the y‐axis representing C_L.

Equation 11.4 is not exactly of parabolic shape, depending on the extent to which it is contributed to by the non‐parabolic component of ΔC_Dp and the wave drag term C_DW. In an incompressible flow regime C_DW drops out and ΔC_Dp is integrated with the induced drag term with the suitable coefficient ‘k’ = 1/eπ AR (where e is the Oswald's efficiency factor) to represent the parabola equation as discussed before. A carefully designed aircraft can have ΔC_Dp ≈ 0 at cruise C_L. The simplification brings Eq. 11.4 into a simpler form as in Eq. 11.5, making it easier to handle.

11.5

This form of representation can be applied to high‐speed subsonic aircraft up to LRC, that is, up to M_crit. When the parabolic part of the C_D is plotted against C_L², it is a straight line (Figure 11.16b).

Equation 11.5 can be improved by modifying to more accurate form as shown in Eq. 11.6 (plotted in Figure 11.4a).

11.6

where C_Dpmin is at C_Lm and not at C_L = 0.

In the generalised situation of high‐speed subsonic aircraft, C_Dpmin is not necessarily at C_L = 0. Figure 11.4 typically represents Eqs. 11.4 and 11.5 with drag polar estimated by a semi‐empirical method (as done in Table 11.16 later). The various C_L points shown in the graph appear in analytical equations derived in the subsequent sections using the parabolic drag Eq. 11.3.

(The most generalised form of aircraft drag polar can be expressed in a polynomial expression as given in Eq. (11.7).

11.7

For high‐subsonic aircraft, terms above k₃C_L³ contribute very little and can be ignored. Even then, the polynomial form is not amenable to an easy close‐form solution. The supersonic drag expression can be dealt with using a similar rationale (and its details are not dealt with here).

The expression for parabolic drag polar as given in Eq. 11.5 is

11.8

where k = 1/eπAR

Figure 11.4a presents an analytical comparison between actual drag polar with parabolic drag polar. It may be noted that the three graphs are close enough within the operating range below M_crit. Figure 11.4b presents the Bizjet example as worked out in Section 11.19. Section 11.4 represents a typical high speed subsonic aircraft operational segments of an LRC (when C_DW = 0), en‐route climb and descent. At HSC, the non‐linear effects show up. The Bizjet operates in high subsonic speeds at M_crit and, therefore, semi‐empirically determined actual drag polar is the preferred one. This is the standard procedure as practiced in industry, which gives credible output.

11.8.1 Geometric Parameters, Reynolds Number and Basic C_F Determination

The Re (= (ρ_∞L_compV_∞)/μ_∞) has the deciding role in determining the skin friction coefficient, C_F, of a component. First, the Re per unit length, speed and altitude are established. Then, the characteristic lengths of each component are determined. The characteristic length L_comp of each component is as follows.

• Fuselage. Axial length from the tip of the nose cone to the end of the tail cone (L_fus)
• Wing. The wing Mean Aerodynamic Chord (MAC)
• Empennage. The MACs of the V‐tail and the H‐tail
• Nacelle. Axial length from the nacelle highlight plane to the nozzle‐exit plane (L_nac)

Figure 11.24 shows the basic 2D flat plate skin friction coefficient of a fully turbulent flow for local (C_{f_basic}) and average (C_{F_basic}) values. For a partial laminar flow, the C_{F_basic} correction is made using factor f₁, given in Figure 11.25. It has been shown that the compressibility effect increases the boundary layer, thus reducing the local C_F. However, in LRC until the M_crit is reached, there is little sensitivity of the C_F change with Mach number variations, therefore, the incompressible C_F line (i.e. the Mach 0 line in Figure 11.24 (bottom graph) is used. At HSC at the M_crit and above, the appropriate Mach line is used to account for the compressibility effect. The basic C_F changes with changes in the Re, which depends on speed and altitude of the aircraft. Section 11.2 explains that a subsonic aircraft C_Dpmin computed at LRC would cater to the full flight envelope, in this book. For HSC, the wave drag ΔC_Dw is to be added.

11.8.2 Computation of Wetted Areas

Computation of the wetted area, Aw, of the aircraft component is shown herein. Skin friction is generated on that part of the surface over which air flows, the so‐called wetted area. Wetted area Aw computation has to be accurate as parasite drag is directly proportionate to A_w. A 2% error in area estimation will result in about 1% error in overall subsonic aircraft drag.

Three‐dimensional computer aided drawing (CAD) model can give accurate wetted area, A_w. In case CAD data are not available, the component wetted areas can be estimated from manually drawn three‐view diagram in a large sheet (say A0 or A1 size) by draftsmen in a different department and then given to aerodynamics group where drag estimation is done. Planimeters prove useful in making accurate area measurements. Normally, three‐view 2D drawings have the reference areas of the geometry that can help calibrating the planimeter in use.

11.8.3 Stepwise Approach to Compute Minimum Parasite Drag

The following seven steps are carried out to estimate the minimum parasite drag, C_Dpmin:

• Step 1. Dissect and isolate aircraft components such as wing, fuselage nacelle and so on. Determine the geometric parameters of the aircraft components such as the characteristic length and wetted areas.
• Step 2. Compute Re per foot at the LRC condition. Then obtain the component Re by multiplying its characteristic length.
• Step 3. Determine the basic 2D (Figure 11.24, bottom graph) average skin friction coefficient C_{F_basic}, corresponding to the Re for each component.
• Step 4. Estimate the ΔC_F as the increment on account of 3D effects on each component.
• Step 5. Estimate the interference drag of two adjacent components. Avoid duplication.
• Step 6. Add results obtained in steps 3–5 to get the minimum parasite drag of the component in terms of flat plate equivalent area (ft² or m²); that is, C_F = C_{F_2D} + ∑ΔC_F for the component. ( f)_comp = (A_w)_comp × C_F, where (C_Dpmin)_comp = ( f)_comp/S_w.
• Step 7. Add all the component minimum parasite drags. Then add other drags, for example, trim excrescence drag and so on. Finally, add 3% drag due to surface roughness effects. The aircraft minimum parasite drag is expressed in coefficient form, C_Dpmin.

The semi‐empirical formulation for each component is given in the following subsections.

11.9.1 Fuselage

The fuselage characteristic length, L_fus, is the length from the tip of the nose cone to the end of the tail cone. The wetted area, A_wf (↑), and fineness ratio (length/diameter) (↓) of the fuselage are computed. Ensure that cut‐outs at the wing and empennage junctions are subtracted. Obtain the Ref (↓). The corresponding basic C_Ff for the fuselage using (Figure 11.24. bottom graph) is intended for the flat plate at the flight Mach number. Figure 11.24 is accurate and validated over time.

The methodology for the fuselage (denoted by the subscript f) is discussed in this section. The Re_f is calculated first using the fuselage length as the characteristic length. The semi‐empirical formulation is required to correct the 2D skin friction drag for the 3D effects and other influencing parameters, as listed herein. These are incremental values shown by the symbol Δ. There are many incremental effects and it is easy to miss some of them.

1. 3D effects [1]: 3D effects are on account of surface curvature resulting in change in local flow speed and associated pressure gradients:
   1. (i) Wrapping:
      11.16
      where k is between 0.022 and 0.025 (take the higher value) and R_e = Reynolds Number of fuselage
   2. (ii) Supervelocity:
      11.17
   3. (iii) Pressure:
      11.18
2. Other effects on the fuselage (increments are given as a percentage of 2D C_Ff) are listed herein. The industry has more accurate values of these incremental ΔC_Ff. Readers in the industry should not use the values given here – they are intended only for coursework using estimates extracted from industrial data.
   1. (i) Canopy drag. There are two types of canopy (Figure 11.6), as follows:
      1. 1. Raised or bubble‐type canopy or its variants. These canopies are mostly associated with military aircraft and smaller aircraft. The canopy drag coefficient C_Dπ is based on the frontal cross‐sectional area shown in the military‐type aircraft in Figure 11.6 (the front view of the raised canopy is shaded). The extent of the raised frontal area contributes to the extent of drag increment and the C_Dπ accounts for the effects of canopy rise. C_Dπ is then converted to C_Ffcanopy = (A_π × C_Dπ)/A_wf, where A_wf is the fuselage wetted area. The dominant types of a raised or bubble‐type canopy and their associated C_Dπ are summarised in Table 11.1.
      2. 2. Windshield‐type canopy for larger aircraft. These canopies are typically associated with payload‐carrying commercial aircraft from a small Bizjet and larger. Flat panes lower the manufacturing cost but result in a kink at the double curvature nose cone of the fuselage. A curved and smooth transparent windshield avoids the kink that would reduce drag at an additional cost. Smoother types have curved panes with a single or double curvature. Single‐curvature panes come in smaller pieces, with a straight side and a curved side. Double curvature panes are the most expensive and considerable attention is required during manufacturing to avoid distortion of vision. The values in square feet in Table 11.2 are used to obtain a sharp‐edged windshield‐type canopy drag.

         

         Raised frontal area (older boxy design – has sharp edges)	CDπ = 0.2
         Raised flat shield (reduced sharp edges)	CDπ = 0.15
         Bubble canopy (partial)	CDπ = 0.12
         Bubble canopy (short)	CDπ = 0.08
         Bubble canopy (long)	CDπ = 0.06

         Adjust the values for the following variations.
         Kinked windshield (less sharp)	reduce the value by 10%
         Smoothed (single‐curvature) windshield	reduce the value by 20–30%
         Smoothed (double‐curvature) windshield	reduce the value by 30–50%

   2. (ii) Body pressurisation – fuselage surface waviness, 4–6%
   3. (iii) Non‐optimum fuselage shape (interpolate the in‐between values)
      1. 1. Front fuselage closure ratio, F_cf (see Section 5.4.3)
         • For F_cf ≤ 1.5: 8%
         • For 1.5 ≤ F_cf ≤ 1.75: 6%
         • For F_cf ≥ 1.75: 4%
         • For military aircraft type with high nose fineness: 3%
      2. 2. Aft fuselage closure angle, (see Section 5.4.3)
         • Less than 10°: 0
         • 11–12° closure: 1%
         • 13–14°: 4%
      3. 3. Upsweep closure (see Section 5.4.3) use in conjunction with (4)
         • No upsweep: 0
         • 4° of upsweep: 0–2%
         • 10° of upsweep: 4–8%
         • 15° of upsweep: 10–15%
         • (interpolate in‐between values)
      4. 4. Aft‐end cross‐sectional shape
         • Circular: 0
         • Shallow keel: 0–1%
         • Deep keel: 1–2%
      5. 5. Rear‐mounted door (with fuselage upsweep): 5–10%
   4. (iv) Cabin‐pressurisation leakage (if unknown, use higher value): 2–5%
   5. (v) Excrescence (non‐manufacturing types such as windows)
      1. Windows and doors (use higher values for larger aircraft): 2–4%
      2. Miscellaneous: 1%
   6. (vi) Wing‐fuselage belly fairing, if any: 1–5% (use higher value if houses undercarriage)
   7. (vii) Undercarriage fairing – typically for high‐wing aircraft (if any fairing): 2–6% (based on fairing protrusion height from fuselage)
3. The interference drag increment with the wing and empennage is included in the calculation of lifting‐surface drag and therefore is not duplicated when computing the fuselage parasite drag. Totalling the C_Ff and ΔC_Ff from the wetted area A_wf of the isolated fuselage, the flat plate equivalent drag, f_fus (see Step 6 in Section 11.8.3), is estimated in square feet.
4. Surface roughness is 2–3%. These effects are from the manufacturing origin, discussed in Section .

Total all the components of parasite drag to obtain C_Dpmin, as follows. It should include the excrescence drag increment. Converted into the fuselage contribution to [C_Dpmin]_f in terms of aircraft wing area, it becomes:

11.19

11.20

11.21

Because surface roughness drag is the same percentage for all components, it is convenient to total them after evaluating all components. In that case, the term ΔC_{Ff_rough} is dropped from Eq. 11.20.

11.9.2 Wing, Empennage, Pylons and Winglets

The wing, empennage, pylon and winglets are treated as lifting surfaces and use identical methodology to estimate their minimum parasite drag. It is similar to the fuselage methodology except that it does not have the wrapping effect. Here, the interference drag with the joining body (e.g. for the wing, it is the fuselage) is taken into account because it is not included in the fuselage f_fus.

The methodology for the wing (denoted by the subscript w) is discussed in this section. The Re_w is calculated first using the wing MAC as the characteristic length. Next, the exposed wing area is computed by subtracting the portion buried in the fuselage and then the wetted area, A_Ww, using the k factors for the t/c as in Section 11.8.2.1. Using the Re_w, the basic C_{Fw_basic} is obtained from the graph in Figure 11.24, bottom graph for the flight Mach number. The incremental parasite drag formulae are as follows:

1. 3D effects (taken from [1]):
   1. Supervelocity:
      11.22
      where K₁ = 1.2–1.5 for supercritical aerofoil

      K₁ = 1.6–2 for conventional aerofoil

   2. Pressure:
      11.23

      where aspect ratio AR ≥ 2 (modified from [1])

      Last term of this expression includes the effect of non‐elliptical lift distribution.

2. Interference:
   11.24

   where K₂ = 0.6 for high and low wing designs and C_B is the root chord at fuselage intersection. For mid‐wing, K₂ = 1.2. This is valid for t/c ratio > 0.07. For a t/c ratio below 0.07, take the interference drag

   11.25
   • The same relations apply for the V‐tail and H‐tail.
   • For pylon interference 10–12%
   • Interference drag is not accounted for in fuselage drag, it is accounted for in wing drag.
   • (For the pylon, one interference is at the aircraft side and one is with the nacelle.)
3. Other effects:
   • Excrescence (non‐manufacturing, e.g. control surface gaps etc.):

   Flap gaps	4–5%
   Slat gaps	4–5%
   Others	4–5%

4. Surface roughness (to be added later)

The flat plate equivalent of wing drag contribution is (subscript is self‐explanatory)

11.26

which can be converted into C_Dpmin in terms of aircraft wing area

11.27

(Drop the term ΔC_Fwrough of Eq. 11.20 if it is accounted for at the end after computing  fs for all components as shown in Eq. 11.28.)

The same procedure is used to compute the parasite drag of empennage, pylons and so on, which are considered as being wing‐like lifting surfaces.

11.28

which can be converted into

11.29

As before, it is convenient to total ΔC_Ffrough after evaluating all components. In that case, the term ΔC_Ffrough is dropped from Eq. 11.20 and it is accounted for as shown in Eq. 11.21.

11.9.3 Nacelle Drag

The nacelle requires different treatment, with the special consideration of throttle dependent air flowing through as well as over it, like the fuselage. This section provides the definitions and other considerations needed to estimate nacelle parasite drag (see [2, 9, 15, 16]). The nacelle is described in Section 5.12.

The throttle dependent variable of the internal flow passing through the turbofan engine affects the external flow over the nacelle. The dominant changes in the flow field due to throttle dependency are around the nacelle at the lip and aft end. When the flow field around the nacelle is known, the parasite drag estimation method for the nacelle is the same as for the other components but must also consider the throttle dependent effects.

Civil aircraft nacelles are typically pod‐mounted. In this book, only the long duct is considered. Military aircraft engines are generally buried in the aircraft shell (i.e. fuselage). A podded nacelle may be thought of as a wrapped‐around wing in an axi‐symmetric shape like that of a fuselage. The nacelle section shows aerofoil‐like sections in Figure 11.7; the important sources of nacelle drag are listed here (a short duct nacelle (see Figure 12.17) is similar except for the fan exhaust coming out at high‐speed over the exposed outer surface of the core nozzle, for which its skin friction must be considered):

Nacelles do not have front fuselage like curvature and aft fuselage like upsweep curvatures. Nacelle external flow is throttle dependent and is affected by the internal intake flow when the lip suction effect (Figure 11.7) is considered. Nacelle aft has little separation with the exhaust flow entrainment effect and boat tail (see next) drag takes into account of the pressure drag. Therefore, unlike fuselage drag considerations, the supervelocity and pressure effects are not considered in case of nacelle drag estimation.

• Throttle‐independent drag (external surface) – at cruise thrust
  1. skin friction
  2. wrapping effects of axi‐symmetric body excrescence effects (includes non‐manufacturing types such as cooling ducts.)
• Throttle dependent drag
  • inlet drag (front end of the diffuser)
  • nacelle base drag (zero for an engine operating at cruise settings and higher)
  • boat tail drag (curvature of the nozzle at the aft end of the nacelle)

Definitions and typical considerations for drag estimation associated with flow field around an isolated long‐duct podded nacelle (approximated to circular cross‐section) are shown in Figure 11.7. Although there is internal flow through the nacelle, the external geometry of the nacelle may be treated as a fuselage, except that there is a lip section similar to the LE of an aerofoil. The prevailing engine throttle setting is maintained at a rating for LRC or HSC for the mission profile. The intake drag and the base drag/boat tail drag are explained next.

11.9.3.1 Intake Drag

The intake stream tube flow pattern at cruise is complex that makes intake drag estimation difficult (Figure 11.7). There is spillage during the subcritical operation due to the stream tube being smaller than the cross‐sectional area at the nacelle highlight diameter, where external flow turns around the lip creating suction (i.e. thrust). This develops pre‐compression, ahead of the intake, when the intake velocity is slower compared to the free‐stream velocity expressed in the fraction (V_intake/V_∞). At (V_intake/V_∞) < 0.8. The excess air flow spills over the nacelle lip. The intake lip acts as the LE of a circular aerofoil. The subcritical air flow diffusion ahead of the inlet results in pre‐entry drag called additive drag. The net effect results in spillage drag, as described herein. The spillage drag added to the friction drag at the lip results in the intake drag, which is a form of parasite drag. (For the military aircraft intake, see Sections 12.8.2 and 5.16.1.)

11.30

intake drag = spillage drag + friction drag at the lip (supervelocity effect).

Section 11.8 shows intake drag variations with the mass flow rate for both subsonic and supersonic (i.e. sharp LE) intake.

11.9.3.5 Total Nacelle Drag

By adding all the components, the flat plate equivalent of nacelle drag contribution is given by Eq. 11.32.

11.32

As before, it is convenient to total ΔC_Ffrough after evaluating all components. In that case, the term ΔC_{F_rough} is dropped from Eq. 11.32 and it is accounted for as shown in Eq. 11.34. Converted into the nacelle contribution to C_Dpmin in terms of aircraft wing area it becomes

11.33

In the last three decades, the nacelle drag has been reduced by approximately twice as much as what has been achieved to other aircraft components. This demonstrates the complexity of and unknowns associated with the flow field around nacelles. CFD is important in nacelle design and its integration with aircraft. In this book, nacelle geometry is simplified to the axi‐symmetric shape without loss of methodology.

11.9.4 Recovery Factor

Intake duct loss will deplete some flow energy making fan face total pressure, P_tf, less than free‐stream total pressure, P_t∞. The extent of total loss is expressed as the recovery factor, RF. Thrust loss on account of RF is seen as installation loss and is accounted in installed engine thrust and is dealt in Section 13.2.1.

11.9.5 Excrescence Drag

An aircraft body is not smooth; located all over the body are probes, blisters, bumps, protrusions, surface‐protection mats for steps, small ducts (e.g. for cooling) and exhausts (e.g. environmental control and cooling air) – these are unavoidable features. In addition, there are mismatches at subassembly joints – for example, steps, gaps and waviness originating during manufacture and treated as discreet roughness. Pressurisation also causes the fuselage‐skin waviness (i.e. areas ballooning up). In this book, excrescence drag is addressed separately as two types:

1. Manufacturing origin [16]. This includes aerodynamic mismatches as discreet roughness resulting from tolerance allocation. Aerodynamicists must specify surface smoothness requirements to minimise excrescence drag resulting from the discrete roughness, within the manufacturing‐tolerance allocation.

2. Non‐manufacturing origin. This includes aerials, flap tracks and gaps, cooling ducts and exhausts, bumps, blisters and protrusions.

Excrescence drag due to surface roughness drag is accounted for by using 2–3% of component parasite drag as roughness drag ([1, 7]). As indicated in Step 7 of Section 11.8.3, f_{comp total} is increased by 3% using a factor of 1.03 after computing all component parasite drags, as follows

11.34

The difficulty in understanding the physics of excrescence drag was summarised by Haines [12] in his review by stating ‘…one realises that the analysis of some of these early data seems somewhat confused, because three major factors controlling the level of drag were not immediately recognised as being separate effects’. These factors are as follows:

1. how skin friction is affected by the position of the boundary‐layer transition
2. how surface roughness affects skin friction in a fully turbulent flow
3. how geometric shape (non‐planar) affects skin friction

Haines's study showed that a small but significant amount of excrescence drag results from manufacturing origin and was difficult to understand.

11.9.6 Miscellaneous Parasite Drags

In addition to excrescence drag, there are other drag increments such as from the intake drag of the environmental control system (ECS) (e.g. air‐conditioning), which is a fixed value depending on the number of passengers) and aerials and trim drag, which are included to obtain the minimum parasite drag of the aircraft.

11.9.6.3 Aerials

Navigational and communication systems require aerials that extend from an aircraft body, generating parasite drag on the order 0.06–0.1 ft², depending on the size and number of aerials installed. For midsized transport aircraft, 0.075 ft² is typically used. Therefore:

11.35

11.15.1 High‐Lift Device Drag

High‐lift devices are typically flaps and slats, which can be deployed independently of each other. Some aircraft have flaps but no slats (described in Section 4.15). Flaps and slats conform to the aerofoil shape in the retracted position. The function of a high‐lift device is to increase the aerofoil camber when it is deflected relative to the baseline aerofoil. If it extends beyond the wing LE and trailing edge, then the wing area is increased. A camber increase causes an increase in lift for the same angle of attack at the expense of drag increase. Slats are nearly full span, but flaps can be anywhere from part to full span (i.e. flaperon). Typically, flaps are sized up to about two‐thirds from the wing root. The flap chord to aerofoil chord (c_c/c) ratio is in the order of 0.2–0.3. The main contribution to drag from high‐lift devices is proportional to their projected area normal to free‐stream air. The associated parameters affecting drag contributions are as follows:

• type of flap or slat
• extent of flap or slat chord to aerofoil chord (typically, the flap is 20–30% of the wing chord)
• extent of deflection (flap at takeoff is from 7 to 15°; at landing, it is from 25 to 60°)
• gaps between the wing and flap or slat (depends on the construction)
• extent of flap or slat span
• fuselage width fraction of wing span
• wing sweep, t/c, twist and AR

The myriad variables make formulation of semi‐empirical relations difficult. References [1, 4, 5] offer different methodologies. It is recommended that practitioners use CFD and test data. Reference [17] gives detailed test results of a double‐slotted flap (0.309c) NACA 632‐118 aerofoil (Figure 11.11). Both elements of a double‐slotted flap move together and the deflection of the last element is the overall deflection. For wing application, this requires an aspect ratio correction, as described in Section 4.9.4.

Figure 11.12 is generated from various sources giving averaged typical values of C_L and C_{D_flap} versus flap deflection. It does not represent any particular aerofoil and is intended only for coursework to be familiar with the order of magnitude involved without loss of overall accuracy. The methodology is approximate; practising engineers should use data generated by tests and CFD.

The simple semi‐empirical relation for flap drag given in Eq. 11.39 is generated from flap drag data shown in Figure 11.12. The methodology starts by working on a straight wing (Λ₀) with an aspect ratio of 8, flap‐span‐to‐wing‐span ratio (b_f/b) of two‐thirds and a fuselage‐width‐to‐wing‐span ratio of less than a quarter. Total flap drag on a straight wing (Λ₀) is seen as composed of two‐dimensional parasite drag of the flap (C_{Dp_flap_2D}), change in induced drag due to flap deployment (ΔC_{Di_flap}), and interference generated on deflection (ΔC_{Dint flap}). Equation 11.40 is intended for a swept wing. The basic expressions are corrected for other geometries, as given in Eqs. 11.41 and 11.42.

11.39

11.40

The empirical form of the second term of Eq. 11.39 is given by

11.41

where AR is the wing aspect ratio and (b_f /b) is the flap‐to‐wing‐span ratio.

The empirical form of the third term of Eq. is given by

11.42

k is 0.1 for single slotted, 0.2 for double slotted, 0.25–0.3 for single Fowler and 0.3–0.4 for double Fowler flap.

Lower values may be taken at lower settings.

Figure 11.12 shows the C_{D_ flap_2D} for various flap types at various deflection angles with the corresponding maximum C_L gain given in Figure 11.12. Aircraft fly well below C_Lmax, keeping a safe margin. Increase C_{Di flap} by 0.002 if the slats are deployed.

Worked-Out Example Worked-Out Example

An aircraft has an aspect ratio, AR = 7.5, 1/4 = 20°,

(b_f/b) = 2/3, and fuselage‐to‐wing‐span ratio less than 1/4. The flap type is a single‐slotted Fowler flap and there is a slat. The aircraft has C_Dpmin = 0.019. Construct its drag polar.

At 20° deflection:

It is typical for takeoff with C_L = 2.2 (approximate) but can be used at landing.

From Figure 11.12, ΔC_{D_flap_2D} = 0.045 and ΔC_L = 1.46

From Eq. 11.41, ΔC_{Di_flap} = 0.025 × (8/7.5)^0.3 × [(2/3)/(3/2)]^0.5 × (1.46)² = 0.025 × 1.02 × 2.13 = 0.054

From Eq. 11.42, ΔC_{Dint_flap} = 0.25 × 0.045 = 0.01 125

C_{D_flap_Λ0} = 0.045 + 0.054 + 0.01 125 = 0.11

With slat on C_{D_highlift} = 0.112.

For the aircraft wing C_{D_flap_Λ¼} = C_{D_flap_Λ0} × cosΛ₀ = 0.112 × cos20 = 0.105

Induced drag C_Di = (C_L²)/(πAR) = (2.2)²/(3.14 × 7.5) = 4.48/23.55 = 0.21

Total aircraft drag, C_D = 0.019 + 0.105 + 0.21 = 0.334

At 45° deflection:

It is typical for landing with C_L = 2.7 (approximate).

From Figure 11.12, ΔC_{D_flap_2D} = 0.08 and ΔC_L = 2.1

From Eq. 11.41, ΔC_{Di_flap} = 0.025 × (8/7.5)^0.3 × [(2/3)/(3/2)]^0.5 × (2.1)² = 0.025 × 1.02 × 4.41 = 0.112

From Eq. 11.42, ΔC_{Dint_flap} = 0.3 × 0.08 = 0.024

C_{Dp_flap_Λ0} = 0.08 + 0.112 + 0.024 = 0.216

With slat on C_{Dp_highlift} = 0.218

For the aircraft wing C_{D_flap_Λ¼} = C_{D_flap_Λ0} × cosΛ₀ = 0.218 × cos20 = 0.201 × 0.94 = 0.205

Induced drag C_Di = (C_L²)/(πAR) = (2.7)²/(3.14 × 7.5) = 7.29/23.55 = 0.31

Total aircraft drag, C_D = 0.019 + 0.205 + 0.31 = 0.534.

Drag polar with a high‐lift device (Fowler flap) extended is plotted as shown in Figure 11.13 at various deflections. It is cautioned that this graph is intended only for coursework; practising industry‐based engineers must use data generated by tests and CFD.

A typical value of C_L/C_D for high‐subsonic commercial‐transport aircraft at takeoff with flaps deployed is on the order of 10–12; at landing, it is reduced to 6–11.

A more convenient way would that as given in Figure 11.14 and is used for the classroom example (civil aircraft) worked out in Section 11.19.

11.15.2 Dive Brakes and Spoiler Drag

To decrease aircraft speed, whether in combat action or at landing, flat plates – which are attached to the fuselage and shaped to its geometric contour when retracted – are used. They could be placed symmetrically on both sides of the wing or on the upper fuselage (i.e. for military aircraft). The flat plates are deployed during subsonic flight. Use C_{Dπ_brake} = 1.2–2.0 (average 1.6) based on the projected frontal area of the brake to air stream. The force level encountered is high and controlled by the level of deflection. The best position for the dive brake is where the aircraft moment change is the least (i.e. close to the aircraft CG line).

11.15.3 Undercarriage Drag

Undercarriages, fixed or extended (i.e. retractable type), cause considerable drag on smaller, low‐speed aircraft. A fixed undercarriage (not streamlined) can cause up to about a third of aircraft parasite drag. When the undercarriage is covered by a streamlined wheel fairing, the drag level can be halved. It is essential for high‐speed aircraft to retract the undercarriage as soon as it is safe to do so (like birds). Below a 200 ft altitude from takeoff and landing, an aircraft undercarriage is kept extended.

The drag of an undercarriage wheel is computed based on its frontal area: A_{π_wheel} (product of wheel diameter and width). For twin side‐by‐side wheels, the gap between them is ignored and the wheel drag is increased by 50% from a single‐wheel drag. For the bogey type, the drag also would increase – it is assumed by 10% for each bogey, gradually decreasing to a total maximum 50% increase for a large bogey. Finally, interference effects (e.g. due to undercarriage doors and tubing) would double the total of wheel drag. The drag of struts is computed separately. The bare single‐wheel C_{D_wheel} based on the frontal area is in Table 11.6 (wheel aspect ratio = D/W_b).

For the smooth side, reduce by half. In terms of an aircraft:

A circular strut has nearly twice the amount of drag compared to a streamlined strut in a fixed undercarriage. For example, the drag coefficient of a circular strut based on its cross‐sectional area per unit length is C_{Dπ_strut} = 1.0 as it operates at a low Re during takeoff and landing. For streamlined struts with fairings, it decreases to 0.5–0.6, depending on the type.

Toreenbeck [10] suggests using an empirical formula if details of undercarriage sizes are not known at an early conceptual design phase. This formula is given in the FPS system as follows:

11.43

Understandably, it could result in a slightly higher value (see the following example). If aircraft geometry is available then it is advisable to accept the computed drag values as worked out next.

11.15.4 One‐Engine Inoperative Drag

Mandatory requirements by certifying agencies (e.g. Federal Aviation Administration, FAA and Civil Aeronautics Agency, CAA) specify that multiengine commercial aircraft must be able to climb at a minimum specified gradient with one engine inoperative at a ‘dirty’ configuration. This immediately safeguards an aircraft in the rare event of an engine failure; and in certain cases, after lift‐off. Certifying agencies require backup for mission‐critical failures to provide safety regardless of the probability of an event occurring.

Asymmetric drag produced by the loss of an engine would make an aircraft yaw, requiring a rudder to fly straight by compensating for the yawing moment caused by the inoperative engine. Both the failed engine and rudder deflection substantially increase drag, expressed by C_{Dengine out + rudder}. Typical values for coursework are in Table 11.7.

11.19.1 Geometric and Performance Data

The geometric and performance parameters discussed herein were used in previous chapters. Figure 11.15 illustrates the dissected anatomy of the coursework baseline aircraft.

Aircraft cruise performance for the basic drag polar is computed as follows:

Cruise altitude = 40 000 ft., LRC Mach = 0.65 (630 ft s⁻¹) to 0.7 (678.5 ft s⁻¹) at high altitudes. The Bizjet C_Dpmin is evaluated at 0.7 Mach as Bizjets cruises at high altitudes. The 2D flat plate C_{F_basic} decreases as Re increases.

Re/ft. = 1.325 × 10⁶ (use incompressible zero Mach line as explained in Section 11.8.1).

C_L at LRC (Mach 0.7) ≈ 0.5, C_L at HSC (Mach 0.75) ≈ 0.4

11.19.2 Computation of Wetted Areas, Re and Basic C_F

The aircraft is dissected in to isolated components as shown in Figure 11.15. The Re, wetted area, and basic 2D flat plate C_{F_basic} of each component are worked‐out herein. Bizjet C_Dpmin is evaluated at 0.7 Mach.

Table 11.9 summarises the Bizjet component Re and 2D basic skin friction C_Fbasic.

11.19.3 Computation of 3D and Other Effects to Estimate Component C_Dpmin

A component‐by‐component example of estimating C_Dpmin is provided in this section. The corrected C_F for each component at LRC (i.e. Mach 0.7) is computed in the previous section.

• Fuselage

The basic C_Ff = 0.0022

3D effects (Eqs. 11.16–11.18).

Wrapping: ΔC_Ff = C_Ff × 0.025 × (length/diameter) × Re^−0.2

• = [0.0022 × 0.025 × (8.633)] × (6.625 × 107)−0.2
• = 0.000 475 × 0.0273 = 0.000 013 (5.9% of basic C_Ff)

Supervelocity: ΔC_Ff = C_Ff × (diameter/length)^1.5 = 0.0022 × (1/8.633)^1.5

• = 0.0022 × (0.1158)^1.5 = 0.0 000 867 (3.94% of basic C_Ff)

Pressure: ΔC_Ff = C_Ff × 7 × (diameter/length)³ = 0.0022 × 7 × (0.1158)³

• = 0.0154 × 0.001 553 = 0.0 000 239 (1.09% of basic C_Ff)

Table 11.10 gives the Bizjet fuselage C_Ff components.

Add the canopy drag for two‐abreast seating f = 0.1 ft² (see Section 11.9.1).

Therefore, the equivalent flat plate area, f, becomes = C_Ff × A_wF + canopy drag.

f_f = 1.34 × 0.0022 × 670 + 0.1 = 1.975 + 0.1 = 2.075 ft²

Surface roughness (to be added later): 3%

• Wing

Basic C_FW = 0.003

3D effects (Eqs. 11.15–11.17):

Supervelocity: ΔC_Fw = C_Fw × 1.4 × (aerofoil thickness/chord ratio)

• = 0.003 × 1.4 × 0.1 = 0.00 042 (14% of basic C_Fw)

Pressure: ΔC_Fw = C_Fw × 60 × (aerofoil thickness/chord ratio)⁴ ×

• = (0.003 × 60) × (0.1)⁴ × (6/7.5)^0.125
• = 0.18 × 0.0001 × 0.9725 = 0.0 000 175 (0.58% of basic C_Fw)

Interference: ΔC_Fw = C_B²× 0.6 ×

• = 9² × 0.6 × [{0.75 × (0.1)³ − 0.0003}/552.3]
• = 81 × 0.6 × (0.00 075 − 0.0003)/552.3
• = 0.0219/552.3 = 0.00 004 (1.32% of basic C_Fw)

Other effects:

Excrescence (non‐manufacturing, e.g. control surface gaps etc.):

Table 11.11 summarises Bizjet wing ΔC_Fw correction (3D and other shape effects).

Therefore, in terms of equivalent flat plate area, f, it becomes = C_Fw × A_ww

Surface roughness (to be added later) 3%.

• Empennage

Since it follows the same procedure as in the case of wing, this is not repeated. The same percentage increment as the case of wing is taken as a classroom exercise. It may be emphasised here that in industry, one must compute systematically as shown in the case of the wing.

• Nacelle

Fineness ratio = 2.44, Nacelle Re = 1.23 × 10⁷.

Wetted area of two nacelles, A_wn = 197.3 ft², Basic C_Fnac = 0.0029

3D effects (Eqs. 11.15–11.17):

Wrapping (Eq. 11.10): ΔC_Fn = C_Fn × 0.025 × (length/diameter) × R_e^−0.2

• = (0.025 × 0.003 × 2.44) × (1.23 × 10⁷)^−0.2
• = 0.000 183 × 0.0382 = 0.000 007 (0.24% of basic C_Ff)

Other increments are shown in Table 11.12 for one nacelle.

• Pylon

Since it follows the same procedure as in the case of wing, this is not repeated. The same percentage increment as the case of wing is taken as a classroom exercise. It has interference at both sides of pylon.

Each pylon exposed reference area = 14 ft².

11.19.4 Summary of Parasite Drag

Table 11.13 gives the aircraft parasite drag build‐up summary in a tabular form. Surface roughness effect as 3% increase (Eq. 11.28) in ‘f’ is added in this table for all surfaces. Wing reference area S_w = 323 ft², C_Dpmin = f/S_w. ISA day, 40 000 ft. altitude and Mach 0.7.

11.19.5 ΔC_Dp Estimation

Use Figure 11.16 for typical data (requires CFD/testing). The ΔC_Dp values used in the Bizjet are given in Table 11.14.

11.19.6 Induced Drag

Formula used C_Di = C_L²/(3.14 × 11.5) = C_L²/23.55. The ΔC_Di values used in Bizjet are given in Table 11.15.

11.19.7 Total Aircraft Drag at LRC

Drag polar at LRC can be summed up as shown in Table 11.16.

Drag polar at HSC (Mach 0.75) would require adding wave drag.

This drag polar is plotted in Figure 11.2. C_L² versus C_D is plotted in Figure 11.16. Note the non‐linearity at low and high C_L..

11.20.1 AJT Details

Drag polar at Mach 0.78 and at Mach 0.85 is tabulated in Table 11.17 and plotted in Figure 11.17. From this, AJT parabolic drag can approximated as C_D = 0.0212 + 0.07 C_L² with S_W = 183 ft² (gives Oswald's efficiency factor, e = 0.85).

11.21.1 TPT Details

ISA day, 25 000 ft. altitude. Wing reference area S_w = 16.5 m² (177.38 m²). C_Dpmin = f/S_w. Cruise C_L = 0.3.

Outstream velocity = 405.2 ft s⁻¹ (≈ Mach 0.4)

Intstream velocity = 567.28 ft s⁻¹ (increase 40%).

Outstream Re/ft. @ 25 000 ft at 405.2 ft s⁻¹ = 1.3355 × 10⁶

Intstream Re/ft. @ 25 000 ft and at 567.28 ft s⁻¹ = 1.87 × 10⁶

Drag estimation follows the Bizjet methodology incorporating slipstream effects. Note that a TPT does not have nacelles. Intake drag is to be estimated accordingly.

Quick component drag parasite drag estimation: The short‐cut method uses Bizjet ΔC_F percentage increases. No credit is taken for partial laminar flow.

• Fuselage

The wetted surface of the fuselage is entirely in slipstream; in short, instream.

Fuselage instream Re is 5.6 × 10⁷ and corresponding basic C_Ff is 0.0022. Wetted area A_wf = 280 ft².

Unlike Bizjet, the TPT does not have (i) fuselage pressurisation, (ii) windows and doors, (iii) wing‐fuselage‐belly fairing and (iv) upsweep closure reducing ΔC_Ff by 20%. Take (i) wrapping, (ii) supervelocity and (iii) pressure together as 5%.

The basic C_Ff = 0.0022 (see Table 11.18).

Take the 3D effects, for example, (i) wrapping, (ii) supervelocity and (iii) pressure together as 5% of the basic C_Ff.

Other effects on fuselage (see Section 11.9.1)

1. (a) nose fineness – for 1.5 ≤ F_cf ≤ 1.75 : 6%
2. (b) aft‐end cross‐sectional shape near circular: 0%

excrescence (non‐manufacturing types): 3%

1. (c) miscellaneous: 1%

Total ΔC_Ff increment: ≈15%

Fuselage blockage factors of f_b and f_h are included as engine installation losses reducing thrust by about 3%. Therefore, fuselage blockage factors are not for accounted here.

Add the canopy drag for two‐abreast seating f = 0.1 ft² (see Section 11.9.1).

Therefore, the equivalent flat plate area, f, becomes = C_Ff × A_wF + canopy drag.

f_fus = 1.15 × 0.0022 × 280 + 0.1 = 0.708 + 0.1 = 0.81 ft².

Surface roughness (to be added later): 3%.

Table 11.19 summarises the TPT fuselage C_Ff.

• Wing:

Of the wing the wetted A_wf surface, approximately 20% is instream and 80% is outstream.

Wetted area A_ww = 320 ft². Instream = A_{ww_in} = 70 ft² and outstream A_{ww_out} = 250 ft² and

Instream Re_in = 1.03 × 10⁷ giving basic C_{FW_in} = 0.0029 and outstream Re_out = 1.135 × 10⁷ giving basic C_{FW_out} = 0.003.

Take the 3D effects, for example, (i) interference, (ii) supervelocity and (iii) pressure together as 16% of the basic C_Fw. Other effects on wing:

Excrescence (non‐manufacturing, e.g. control surface gaps etc.):

• Empennage

Since it follows the same procedure as in the case of wing, this is not repeated. The same percentage increment as in the case of wing is taken as a classroom exercise. It may be emphasised here that in industry, one must compute systematically as shown in the case of the wing at 3% surface roughness (to be added later).

H‐tail: (of the H‐tail the wetted A_wf surface, approximately 45% is instream and 55% is outstream).

Wetted area A_wHT = 90.6 ft². Instream = A_{wHT_in} = 40.6 ft² and outstream A_{wHT_out} = 50 ft²

Instream Re_in = 9.1 × 10⁶ giving basic C_{FW_in} = 0.00 304

Outstream Re_out = 4.4 × 10⁶ giving basic C_{FW_out} = 0.0035.

Strakes + dorsal: f_{strake + dorsal} = 0.05 ft²

V‐tail: (fully immersed in slipstream (strictly speaking a small portion is outside the slipstream).

Wetted area, A_wVT = 45 ft², Basic C_{F_H‐tail} = 0.00 298

Engine intake: There is no nacelle, it is integrated with the fuselage. The TPT has one turboprop engine with low intake mass flow. Only spillage drag and friction drag at the lip are to be estimated. Every aircraft needs to generate a graph of spillage drag versus mass flow for its engine intake like that given in Figure 11.8. TPT intake is sized for takeoff mass flow, which is higher compared with cruise demand. Hence there will be spillage drag (= additive drag + lip suction).

Here, f_intake is given as follows (from similar industrial data), f_intake = 0.1 ft²

Table 11.19 gives the aircraft parasite drag build‐up summary in a tabular form. Surface roughness effect as a 3% increase (Eq. 11.28) in ‘f’ is added in this table for all surfaces.

Table 11.20 tabulates TPT total aircraft drag coefficient, C_D.

The actual drag polar matches closes with the parabolic drag polar expressed as C_D = 0.0226 + 0.052 C_L². The TPT drag polar is plotted in Figure 11.23, later.

11.22.1 Geometric and Performance Data of Vigilante – RA C5 Aircraft

The following pertinent geometric and performance parameters are taken from [3]. The RA‐C5 has two crew, engine: two turbo‐jets GE J‐79‐8(N), 75.6 kN, wing span: 16.2 m, length: 22.3 m, height: 5.9 m, wing area: 65.0 m², start mass: 27 300 kg, max speed: Mach 2.0, ceiling: 18 300 m, range: 3700 km, armament: nuclear bombs and missiles (only clean a configuration is evaluated).

• Fuselage

Fuselage length = 73.25 ft, Average diameter at the maximum cross‐section = 9.785 ft (see Figure 11.21).

Fuselage length/diameter = 9.66 (fineness ratio)

Fuselage upsweep angle = 0°, Fuselage closure angle ≈ 0°.

• Wing

Planform reference area, S_W = 65.03 m² (700 ft²), Span = 19.2 m (53.14 ft).

AR = 3.73, t/c = 5%, Taper ratio λ = 0.19, Camber = 0.

Wing MAC = 4.63 m (15.19 ft), Λ_¼ = 39.5° and Λ_LE = 43°.

Root chord at centreline = 9.65 m (20 ft) and Tip chord = 1.05 m (3.46 ft).

• Empennage

V‐tail: S_V = 4.4 m² (49.34 ft²), Span = 3.6 m (11.82 ft). MAC = (8.35 ft), t/c = 4%.

H‐tail: S_H = 11.063 m² (65.3 ft²), Span = 11.85 m (32.3 ft). MAC = (9.73 ft), t/c = 4%.

Nacelle/pylon – engine buried in fuselage – no nacelle and pylon

Aircraft cruise performance where the basic drag polar has to be computed (Figure 11.22).

Drag estimated at cruise altitude = 36 152 ft and Mach number = 0.6 (have compressibility drag).

Ambient pressure = 391.68 lb ft⁻², Re/ft = 1.381 × 10⁶.

Design C_L = 0.365, Design Mach number = 0.896 (M_crit is at 0.9), Maximum Mach number = 2.0.

11.22.2 Computation of Wetted Areas, Re and Basic C_F

The aircraft is first dissected into isolated components to obtain Re, the wetted area and the basic 2D flat plate C_F of each component as listed next. Note that there is no correction factor for C_F at Mach 0.6 (no compressibility drag). The C_F compressibility correction factor (compute this from Figure 11.24b) at Mach 0.9 and Mach 2.0 will be applied later.

• Fuselage:

Fuselage wetted area = A_wf = 1474 ft²

Fuselage Re = 69 × 1.381 × 10⁶ = 9.53 × 10⁷ (length trimmed to what is pertinent for Re)

Use Figure 11.24, bottom graph to obtain basic C_Ff = 0.0021

• Wing

Wing wetted area = A_ww = 1144.08 ft²

Wing Re = 15.19 × 1.381 × 10⁶ = 2.1 × 10⁷

Use Figure 11.24, bottom graph to obtain basic C_Fw = 0.00 257

• Empennage (same procedure as for wing)

V‐tail wetted area = A_wVT = 235.33 ft²

V‐tail Re = 8.35 × 1.381 × 10⁶ = 1.2 × 10⁷

Use Figure 11.24, bottom graph to obtain the basic C_{F_V‐tail} = 0.00 277

H‐tail wetted area = A_wHT = 3811.72 ft².

H‐tail Re = 9.73 × 1.381 × 10⁶ = 1.344 × 10⁷.

Use Figure 11.24, bottom graph to obtain basic C_{F_H‐tail} = 0.002 705.

11.22.3 Computation of 3D and Other Effects to Estimate Component C_Dpmin

A component‐by‐component example is given next.

• Fuselage:

As before, at Mach 0.6, the basic C_Ff = 0.0021

3D effects (Eqs. 11.10–11.12):

Wrapping: ΔC_Ff = C_Ff × 0.025 × (length/diameter) × Re^−0.2

• = 0.025 × 0.0021 × (9.66) × (9.53 × 10⁷)^‐0.2
• = 0.000 507 × 0.0254 = 0.0 000 129 (0.6% of basic C_Ff)

Supervelocity: ΔC_Ff = C_Ff × (diameter/length)^1.5 = 0.0021 × (1/9.66)^1.5

• = 0.0021 × 0.033 = 0.0 000 693 (3.3% of basic C_Ff)

Pressure: ΔC_Ff = C_Ff × 7 × (diameter/length)³ = 0.0021 × 7 × (0.1035)³

• = 0.0147 × 0.00 111 = 0.0 000 163 (0.8% of basic C_Ff)

Other effects on fuselage (this time intake has to be included – see Section ):

It is to be noted here that [3] suggests to apply a factor of 1.284 to include most of the other effects except for intake. Therefore, unlike the civil aircraft example, it is simplified to the following only:

Intake (little spillage – rest taken in 3D effects): 2%

Total ΔC_Ff increment is given in Table 11.21.

Therefore, in terms of equivalent flat plate area, f, it becomes = C_Ff × A_wF.

Add canopy drag, C_Dπ = 0.08 (approximated from Figure 11.4).

Therefore c_anopy = 0.08 × 4.5 = 0.4 ft². f_fus = 3.3 + 0.36 = 3.66 ft²

• Wing:

From before, at Mach 0.06, the basic C_F = 0.00 259.

3D effects (Eqs. 11.15–11.17):

Supervelocity: ΔC_Fw = C_Fw × 1.4 × (aerofoil thickness/chord ratio)

• = 0.00 257 × 1.4 × 0.05 = 0.00 018 (7% of basic C_Fw)

Pressure: ΔC_Fw = C_Fw × 60 × (aerofoil thickness/chord ratio)⁴×

• = 0.00 257 × 60 × (0.05)⁴ × (6/3.73)^0.125
• = 0.1542 × 0.00 000 625 × 1.06 = 0.00 000 102 (0.04% of basic C_Fw)

Interference: ΔC_Fw for thin high wing take 3% of C_Fw.

Other effects:

Excrescence (non‐manufacturing, e.g. control surface gaps etc.):

Table 11.22 summaries Vigilante wing ΔC_Fw correction (3D and other shape effects).

Therefore, in terms of equivalent flat plate area, f, it becomes = C_Fw × A_ww

• Empennage

Since it follows the same procedure as in the case of wing, this is not repeated. The same percentage increment as the case of wing is taken as a classroom exercise. It may be emphasised here that in industry, one must compute systematically as shown in the case of wing.

V‐tail: Wetted area, A_wVT = 235.33 ft²

• Basic C_{F_V‐tail} = 0.00 277, f_VT = 1.12 × 0.00 277 × 235.33 = 0.73 ft²

H‐tail: Wetted area, A_wHT = 388.72 ft²

• Basic C_{F_H‐tail} = 0.002 705, f_HT = 1.12 × 0.002 705 × 388.72 = 1.18 ft²

11.22.4 Summary of Parasite Drag (ISA Day, 36 182 ft Altitude and Mach 0.9)

Wing reference area S_w = 700 ft², C_Dpmin = f/S_w.

A Vigilante parasite drag summary is given in Table 11.23.

As was indicated in Section 11.16 [3], offers a correlated factor of 1.284 to include all so‐called other effects. Therefore, the final flat plate equivalent drag f_aircraft = 1.284 × 8.87 = 11.39 ft². 28.4% = 11.62 ft² to include military aircraft excrescence.

It gives C_Dpmin at Mach 0.6 = 11.39/700 = 0.01 627 ([3] gives 0.1645, close enough).

This is the C_Dpmin at flight Mach number before compressibility effects start to show up, that is, it is seen as C_Dpmin at incompressible flow. At higher speed there is C_F shift to a lower value. The C_Dpmin estimation needs to be repeated with lower C_F at 0.9 Mach and at 2.0 Mach. To avoid repetition to account for compressibility, a factor of 0.97 (ratio of values at Mach 0.9 and Mach 0 in Figure 11.24, bottom graph – a reduction of 3%) is taken at 0.9 Mach. A factor of 0.8 (a reduction of 20%) is taken at 2.0 Mach as shown in Table 11.24. At compressible flow, add the wave drag. At supersonic speed it is contributed to by shock waves.

To stay in line with the methodology presented here, the following values of ΔC_Dp have been extracted from [3].

11.22.5 ΔC_Dp Estimation

The data for ΔC_Dp given in Table 11.24 is extracted from Ref. [3] and is approximate.

11.22.6 Induced Drag

Formula used C_Di = C_L²/(3.14 × 3.73) = C_L²/11.71.

The ΔC_Di values used in Vigilante are given in Table 11.25.

11.22.7 Supersonic Drag Estimation

Supersonic flight would have bow shock‐wave that is a form of compressibility drag, which is evaluated at zero C_L. Drag increases with change of angle of attack. The difficulty arises in understanding the physics involved with increase in C_L. Clearly, the increase, though lift dependent, has little to do with viscosity unless shock interacts with boundary layer to increase pressure drag. Since the very obligation of design is to avoid such interaction up to a certain C_L, this book treats compressibility drag at supersonic speed as being composed of compressibility drag at zero C_L (that is C_{D_shock}) plus compressibility drag at higher C_L (that is ΔC_Dw).

To compute compressibility drag at zero C_L, the following empirical procedure is adopted, taken from [3]. Compressibility drag of an object depends on its thickness parameter, in case of the fuselage it is the fineness ratio and in case of wing it is the thickness to chord ratio. The fuselage (including the empennage) and wing compressibility drags are separately computed and then added along with the interference effects. Extensive use of graphs (Figures 11.25–11.31) is required for the empirical methodology. Compressibility drag values for both at Mach 0.9 and Mach 2.0 are estimated.

Drag estimation at Mach 0.9 follows the same method as worked out in the civil aircraft example and is tabulated in Section 11.19.

Fuselage compressibility drag (includes empennage contribution) at Mach 2.0:

Thickness parameter is fuselage fineness ratio.

• Step 1. Plot fuselage cross‐section along fuselage length as shown in Figure 11.22 and obtain the maximum cross‐section, S_π = 45.25 ft² and the fuselage base S_b = 12 ft². Find the ratio
  • Find (1 + Sb/Sπ) = 1 + 12/45.25 = 1.27.
  • Find Sπ/Sw = 45.25/700 = 0.065.
• Step 2. Get the fuselage fineness ratio l/d = 73.3/7.788 = 9.66 (d is minus intake width).
  • Obtain (l/d)² = (9.66)² = 93.3.
• Step 3. Use Figure 11.26 to get C_Dπ(l/d)² = 18.25 at M_∞ = 2.0 for (1 + S_b/S_π) = 1.27
  • This gives C_Dπ = 18.25/93.3 = 0.1956.
  • Convert this to the fuselage contribution of compressibility drag expressed in terms of wing reference area, C_Dwf = C_Dπ × (S_π/S_w) = 0.1956 × 0.065 = 0.01 271.

Wing compressibility drag at Mach 2.0:

• Step 1. Obtain design C_{L_DES} from Figure 11.27 for supersonic aerofoil for
  • AR × (t/c)⅓ = 3.73 × (0.05)⅓ = 1.374. It gives CL_DES = 0.352.
  • Test data of CL_DES from [3] gives 0.365, which is close enough. Test data is used.
• Step 2. Obtain from Figure 11.28 the two‐dimensional design Mach number, M_{DES_2D} = 0.784.
  • Using Figure 11.29 obtain ΔMAR = 0.038 for 1/AR = 0.268.
  • Using Figure 11.29 obtain ΔMDΛ¼ = 0.067 for Λ¼ = 37.5°.
• Step 3. Make the correction to get design Mach as M_DES = M_{DES_2D} + ΔM_AR + ΔM_DΛ¼.
  

• Then ΔM = M_∞ − M_DES = 2.0 − –0.889 = 1.111.
• Step 4. Compute (t/c)^5/3 × [1 + (h/c)/10)] = (0.05)^5/3 × [1 + (0)/10)] = 0.00679.
• Step 5. Compute ARtan Λ_LE = 3.73 × tan43 = 3.73 × 0.9325 = 3.48.
• Step 6. Use Figure 11.30 and the corresponding value of AR_{tan ΛLE} = 3.48 in Step 5.
  

• Compute ΔC_{DC_WING} = 0.675 × 0.00 679 = 0.00458

Finally, interference drag at supersonic speed has to be added to fuselage and wing compressibility drag.

The procedure for estimating wing‐fuselage interference drag is given next.

• Step 1. Compute (fuselage diameter at maximum area/wing span) = 7.785/53.14 = 0.1465
• Step 2. With a taper ratio, λ = 0.19, compute (1 − λ)cosΛ_¼ = (1 − 0.19) cos37.5 = 0.643.
• Step 3. Using Figure 11.31 obtain ΔC_{D_INT} × [(1 − λ)cosΛ_¼] = 0.00 048
  • Compute ΔC_{D_INT} = 0.00 048/0.643 = 0.00 075.

Summary of compressibility drag of a Vigilante at zero lift

Compressibility drag at Mach 0.6 (3% reduction on account C_F change) and Mach 0.9 (20% reduction on account C_F change) has been done as in the case of a civil aircraft and is given in Table 11.26 along with drag at Mach 2.0.

11.22.8 Total Aircraft Drag

Total Vigilante drag at the three Mach numbers is tabulated in Table 11.27. Figure 11.8 gives the Vigilante drag polar at the three aircraft speeds. Figures 11.24–11.31 are re‐plotted in this chapter from [3].