Chapter 3. Drag Force and Its Coefficient

i

Chapter 3

Drag Force and its Coefficient

Table of Content

Chapter 3 ........................................................................................................................... 90

Drag Force and its Coefficient .......................................................................................... 90

3.1. Introduction ............................................................................................................ 90

3.2. Drag Classification................................................................................................. 91

3.3. Drag Polar .............................................................................................................. 95

3.4. Calculation of CDo .................................................................................................. 99

3.4.1. Fuselage ........................................................................................................ 100

3.4.2. Wing, Horizontal Tail, and Vertical Tail ...................................................... 103

3.4.3. High Lift Devices .......................................................................................... 105

3.4.4 Landing Gear ................................................................................................. 107

3.4.5. Strut ............................................................................................................... 108

3.4.6. Nacelle .......................................................................................................... 109

3.4.7. Cooling Drag ................................................................................................. 109

3.4.8. Trim Drag...................................................................................................... 111

3.4.9. CDo of Other Parts ......................................................................................... 111

3.4.10. Overall CDo .................................................................................................. 117

3.5. Wave Drag ........................................................................................................... 118

3.5.1. Exact Method ................................................................................................ 119

3.5.2. Approximate Approach ................................................................................. 122

3.6. CDo at various Configurations .............................................................................. 123

3.6.1. Clean Configuration ...................................................................................... 124

3.6.2. Take-Off Configuration ................................................................................ 124

3.6.3. Landing Configuration .................................................................................. 125

3.6.4. The Effect of Speed and Altitude on CDo ...................................................... 126

Problems ......................................................................................................................... 130

References ....................................................................................................................... 134

List of Tables

Table 3. 1. Typical values of CDo and e for several aircraft ............................................. 99

Table 3. 2. The values of A and B for different types of flaps ......................................... 106

Table 3. 3. Correction factor (Kc) for equation 3.34 ...................................................... 117

Table 3. 4. CDo of major components of Gates Learjet 25 .............................................. 118

Table 3. 5. CDo of several aircraft ................................................................................... 124

ii

List of Figure

Figure 3. 1 Examples of shapes having drag values nearly independent of Reynolds

number. ............................................................................................................................. 92

Figure 3. 2. Drag classification ........................................................................................ 93

Figure 3. 3. Variations of Do and Di versus velocity ......................................................... 94

Figure 3. 4. Variations of Drag versus velocity ................................................................ 95

Figure 3. 5. Typical variations of CD versus CL ................................................................ 96

Figure 3. 6. The glider Schleicher ASW22 with wing span of 25 meter and CDo of 0.016 97

Figure 3. 7. Agricultural aircraft Dromader, PZL M-18 with Cdo of 0.058 .................... 98

Figure 3. 8. Wing leading edge sweep angle .................................................................... 98

Figure 3. 9. Major components of Boeing 737 contributing to CDo ................................ 100

Figure 3. 10. Wing gross area and wing net area ........................................................... 102

Figure 3. 11. Mean Aerodynamic Chord (MAC) ............................................................ 104

Figure 3. 12. variations of lift coefficient versus drag coefficient for NACA 631-412 . 106

Figure 3. 13. Wing section at the flap location (Plain flap) ........................................... 107

Figure 3. 14. Landing gear and its fairing ...................................................................... 108

Figure 3. 15. Wing and landing gear struts in Cessna 172............................................. 109

Figure 3. 16. The use of cowl flaps to control engine cooling air. ................................. 110

Figure 3. 17. Wing-fuselage interference drag ............................................................... 112

Figure 3. 18. The UHF radio antenna of U-2S Dragon Lady......................................... 113

Figure 3. 19. Boeing Vertol CH-113 Labrador ............................................................... 113

Figure 3. 20. Example of a pitot tube mounted under the wing of a Cessna 172 ........... 114

Figure 3. 21. Drag rise due to compressibility for swept wing-body combinations ....... 115

Figure 3. 22. Incremental drag coefficient due to compressibility (Ref. 6). ................... 116

Figure 3. 23. Drag coefficient versus Mach number (Ref. 6) ......................................... 117

Figure 3. 24. F-35 Joint Strike Fighter, a new generation of fighters ............................ 119

Figure 3. 25. Geometry for drag wave ............................................................................ 120

Figure 3. 26. Geometry for Example 3.1 ........................................................................ 121

Figure 3. 27. Wave drag for several aircraft (Ref. 9) ..................................................... 123

Figure 3. 28. Drag for various Mach number (Ref 6) ..................................................... 126

Figure 3. 29. The variation of drag force without considering the compressibility effects

......................................................................................................................................... 127

Figure 3. 30. The variation of drag force with considering the compressibility effects (Ref

6) ..................................................................................................................................... 127

Figure 3. 31. Top-view of Boeing 757 transport aircraft ................................................ 130

Figure 3. 32. Three-view of amphibian airplane Lake LA-250....................................... 132

Figure 3.33. F-16A three-view ........................................................................................ 133

90

Chapter 3

Drag Force and its Coefficient

3.1. Introduction

In chapter 2, major forces affecting aircraft motion are discussed. One group of those

forces is aerodynamic forces that split into two forces: Lift force or lift, and Drag force or

drag. A pre-requisite to aircraft performance analysis is the ability to calculate the aircraft

drag at various flight conditions. One of the jobs of a performance engineer is to

determine drag force produced by an aircraft at different altitudes, speeds and

configurations. This is not an easy task, since, this force is a function of several

parameters including aircraft configuration. As it was discussed, the drag is a function of

aircraft speed, air density, and its configuration. Each aircraft is designed with a unique

configuration, thus, aircraft performance must take into account this configuration. The

configuration effect of aircraft drag is calculated through the drag coefficient (CD), plus

an area that relates to the aircraft.

An aircraft is a complicated three-dimensional vehicle, but for simplicity in calculation,

we assume that the drag is a function a two-dimensional area and we call it reference

area. This area could be any area including tail area, wing area and fuselage cross

sectional area, or fuselage side area, or fuselage surface area, or even aircraft top-view

area. No matter what the area is, the drag must be the same. This unique drag comes from

the fact that the drag coefficient is a function of this area. Therefore if we choose a small

reference area, the drag coefficient will be large, but if we choose a large reference area,

the drag coefficient will be small. In the air vehicle with a small wing area (e.g. high-

speed missile), the fuselage cross sectional are is considered as the reference area.

In aircraft with large wing, the top-view of wing (in fact gross wing) is the reference area.

The measurements of these areas are easy and they include the most important

aerodynamic part of aircraft. This simplified reference area is compensated with the

complicated drag coefficient:

DSCVD 2

2

1 (3.1)

The drag coefficient is non-dimensional parameter, but it takes into account every

aerodynamic configuration of the aircraft including, wing, tail, fuselage and landing gear.

This coefficient has two main parts (as will be explained in the next section). The first

part is referred to as lift-related drag coefficient or induced drag coefficient (CDi) and the

second part is called zero-lift drag coefficient (CDo). The calculation of the first one is an

easy job, but it takes a long time to determine the second part. In large transport aircraft,

this task is done by a group of up to twenty engineers for a time period of up to six

months. For this reason, the large portion of this chapter is devoted to the calculation of

91

CDo. This calculation not only is time consuming but also is very sensitive, since it

influences every aspect of aircraft performance. Drag is the enemy of flight and its cost.

One of the primary functions of aerodynamicists is to reduce this coefficient. Aircraft

designers are also very sensitive about this coefficient, because any change in the external

configuration of aircraft will change this coefficient and finally aircraft direct operating

cost. As a performance engineer, you must be able to estimate the CDo of any aircraft just

by looking at its three-view with an accuracy of about 30%. An you spend more time for

calculation, this estimation will be more accurate, but never exact unless you use an

aircraft model in wind tunnel or flight test measurements with real aircraft. The method

presented in this chapter is 95% accurate for subsonic aircraft and 85% for supersonic

aircraft.

3.2. Drag Classification

Drag force is the summation of all forces that resist against aircraft motion. The

estimation of the drag of a complete airplane is a difficult and challenging task, even for

the simplest configurations. We will consider the separate sources of drag that contribute

to the total drag of an aircraft. The real shape of drag force as a function of speed is

parabola. It means that there are some parameters that will decrease drag as the velocity

increases and there are some other parameters that will increase drag as the velocity

increases. This observation shows us a good direction for drag classification. Although

the drag and the drag coefficient can be expressed in a number of ways, for reasons of

simplicity and clarity, the parabolic drag polar will be used in all main analyses. Different

references and books use different terminology, so it may confuse students and engineers.

In this section, a list of definitions of various types of drag is presented, and then a

classification of all of these drag forces is described.

Induced Drag: The drag that results from the generation of a trailing vortex system

downstream of a lifting surface of finite aspect ratio.

Parasite Drag: The total drag of an airplane minus the induced drag. Thus, it is the drag

not directly associated with the production of lift. The parasite drag is composed of many

drag components, the definitions of which follow.

Skin Friction Drag: The drag on a body resulting from viscous shearing stresses over its

wetted surface. Frequently, the drag of a very streamlined shape such as a thin, flat plate

is expressed in terms of a skin friction drag. This drag is a function of Reynolds number.

There are mainly two cases where the flow in the boundary layer is entirely laminar or

entirely turbulent over the plate. The Reynolds number is based on the total length of the

plate in the direction of the velocity. In a usual application, the boundary layer is

normally laminar near the leading edge of the plate undergoing transition to a turbulent

layer at some distance back along the surface.

A laminar boundary layer begins to develop at the leading edge and grows in thickness

downstream. At some distance from the leading edge the laminar boundary becomes

unstable and is unable to suppress disturbances imposed on it by surface roughness or

fluctuations in the free stream. In a short distance the boundary layer undergoes transition

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to a turbulent boundary layer. The layer suddenly increases in thickness and is

characterized by a mean velocity profile on which a random fluctuating velocity

component is superimposed. The distance, x, from the leading edge of the plate to the

transition point can be calculated from the transition Reynolds number

Form Drag (sometimes called Pressure Drag): The drag on a body resulting from the

integrated effect of the static pressure acting normal to its surface resolved in the drag

direction. Unlike the skin friction drag that results from viscous shearing forces tangential

to a body’s surface, form drag results from the distribution of pressure normal to the

body’s surface. In an extreme case of a flat plate normal to the flow, the drag is totally the

result of an imbalance in the normal pressure distribution. There is no skin friction drag

present in this case. As with skin friction drag, form drag is generally dependent on

Reynolds number. Form drag is based on the projected frontal area. As a body begins to

move through a fluid, the vorticity in the boundary layer is shed symmetrically from the

upper and lower surfaces to form two vortices of opposite rotation. A number of shapes

having drag values nearly independent of Reynolds number are illustrated in figure 3.1

Figure 3. 1 Examples of shapes having drag values nearly independent of Reynolds number.

Interference Drag: The increment in drag resulting from bringing two bodies in

proximity to each other. For example, the total drag of a wing-fuselage combination will

usually be greater than the sum of the wing drag and fuselage drag independent of each

other.

Trim Drag: The increment in drag resulting from the aerodynamic forces required to

trim the airplane about its center of gravity. Usually this takes the form of added induced

and form drag on the horizontal tail.

Profile Drag: Usually taken to mean the total of the skin friction drag and form drag for

a two-dimensional airfoil section.

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Cooling Drag: The drag resulting from the momentum lost by the air that passes through

the power plant installation for purposes of cooling the engine, oil, and accessories.

Base Drag: The specific contribution to the pressure drag attributed to the blunt after-end

of a body.

Wave Drag: Limited to supersonic flow, this drag is a pressure drag resulting from non-

canceling static pressure components to either side of a shock wave acting on the surface

of the body from which the wave is emanating.

Figure 3. 2. Drag classification

The material to follow will consider these various types of drag in detail and will present

methods of reasonably estimating their magnitudes. Figure 3.2 illustrates the drag

classification.

For most conventional aircraft, we divide drag into two main parts; lift related drag, and

non lift-related drag. The first part is called induced drag (Di), because this drag is

induced by lift. The second part is referred to as zero-lift drag (Do), since it does not have

any influence from lift. Figure 3.3 shows the behavior of these two drags as function of

velocity.

io DDD (3.2)

a. Induced drag: The induced drag is the drag directly associated with the

production of lift. This results from the dependency of the induced drag on the

angle of attack. As the angle of attack of the aircraft (i.e. lift coefficient) varies,

this type of drag is changed. The induced drag is itself consists of two parts. The

first part originates from vortices around wing, tail, fuselage and other

Drag

Zero-lift Drag Induced Drag

Skin friction

Drag

Form

Drag

Miscellaneous

Drag

Wave

Drag

Vortex

Drag

Wave Drag

Landing

gear

Fuselage Tail Wing Strut Nacelle

Interference

Drag

Trim

Drag

Cooling

Drag

CL

dependent

Volume

dependent

Compressibility

Drag

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components. The second part is because of air compressibility effect. In low

subsonic flight, it is negligible, but is high subsonic and transonic flight, must be

taken into account. In supersonic flight, wave drag (Dw) is added to the induced

drag. The reason is to account for the existence of shock wave. The wing is the

major responsible aircraft component for lift production. Thus, about 80% of the

induced drag comes from wing; about 10% comes from tail; and the rest originate

from other components.

iDi SCVD 2

2

1 (3.3)

In this equation, the coefficient CDi is called induced drag coefficient. The method

to calculate this coefficient will be introduced in the next section.

Figure 3. 3. Variations of Do and Di versus velocity

b. Zero-lift drag: The Zero-lift drag includes all types of drag that does not depend

on production of lift. Every aerodynamic component of aircraft (i.e. the

components that are in direct contact with flow) generates zero-lift drag. Typical

components are wing, horizontal tail, vertical tail, fuselage, landing gear, antenna,

engine nacelle, and strut.

oDo SCVD 2

2

1 (3.4)

In this equation, the coefficient CDo is called zero-lift drag coefficient. The

method to calculate this coefficient will be introduced in section 3.4.

From the equations 3.3 and 3.4 we can conclude that drag coefficient has two

components:

95

iDoDD CCC (3.5)

The calculation of CDi is not a big deal and will be explained in the next section; but the

calculation of CDo is very challenging and difficult. Major portion of this chapter is

devoted about how to calculate CDo. In fact, the main idea behind this chapter is about

estimation of CDo.

3.3. Drag Polar

As figure 3.3 and equation 3.5 show, drag is composed of two terms, one proportional to

V2 and the other inversely proportional to V

2. The first term, called zero-lift drag

represents the aerodynamic cleanness with respect to frictional characteristics, and shape

and protuberances such as cockpit, antennae, or external fuel tanks. It increases with the

aircraft velocity and is the main factor in determining the aircraft maximum speed. The

second term represents induced drag (drag due to lift). Its contribution is highest at low

velocities, and it decreases with increasing flight velocities. If we combine these two

curves (Di and Do), we will have a parabolic curve such as shown in figure 3.3. The

parabolic drag model is not exact; but accurate enough for the purpose of performance

calculation.

Figure 3. 4. Variations of Drag versus velocity

Although the drag and the drag coefficient can be expressed in a number of ways, for

reasons of simplicity and clarity, the parabolic drag polar will be used in all main

analyses. This is true only for subsonic flight. For the existing supersonic aircraft, the

drag cannot be adequately described by such a simplified expression. Exact calculations

must be carried out using extended equations or tabular data. However, the inclusion of

more precise expressions for drag at this stage will not greatly enhance basic

understanding of performance, and thus, will be included only in some calculated

96

examples and exercises. Note that the curve begins from stall speed as an aircraft is not

able to maintain level flight at any speed lower than stall speed.

The same conclusion is true for the variation of drag coefficient (CD) versus lift

coefficient (CL) as shown in figure 3.4. We can mathematically describe such curve with

a second order parabolic curve:

2bxay (3.6)

where y may be replaced with CD and x may be replaced with CL. Therefore, drag

coefficient versus lift coefficient is modeled with the following parabolic model:

2

LD bCaC (3.7)

Figure 3. 5. Typical variations of CD versus CL

Now we need to find “a” and “b” in this equation. In a symmetrical parabolic curve, the

parameter “a” is the minimum value for parameter “y”. In a parabolic curve of CD versus

CL, the parameter “a” must be the minimum amount of drag coefficient (CDmin). We refer

this minimum value of drag coefficient as CDo as it means the value of CD when the lift is

zero. The corresponding value for “b” in equation 3.7 must be found through experiment.

Aerodynamicists have found this parameter with the symbol of “K”, and refer to it as

induced drag correction factor. The induced drag correction factor is inverse proportional

to the wing aspect ratio (AR) and wing Oswald efficiency factor (e). The exact

relationship is as follows:

97

eARK

1 (3.8)

The wing aspect ratio is the ratio between the wing span (b) and its mean aerodynamic

chord (MAC or C). It can be found in practice from the wing area (S) as follows:

S

bAR

2

(3.9)

The wing Oswald efficiency factor represents the efficiency of a wing in producing lift

and is a function of wing aspect ratio and its leading edge sweep angle (LE) (See figure

3.8).

1.3cos045.0161.415.068.0 LEARe (3.10a)

64.0045.0178.1 68.0 ARe (3.10b)

Equation 3.10a is for wings with leading edge sweep angles of more than 30 degrees and

equation 3.10a is for rectangular wings (without sweep). Table 3.1 shows wing Oswald

efficiency factor for several aircraft types. The value of e is decreased in high angle of

attack (i.e. low speed) up to about 30%.

Figure 3. 6. The glider Schleicher ASW22 with wing span of 25 meter and CDo of 0.016

Employing the induced drag correction factor, now we have a mathematical model for

variation of drag coefficient versus lift coefficient.

2

LDD KCCCo (3.11)

This equation is sometimes referred to as aircraft “drag polar”. The main challenge in

this equation is the calculation of zero-lift drag coefficient. Table 3.1 shows typical

values of CDo of several aircraft. Gliders are the most aerodynamic efficient aircraft (with

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CDo as low as 0.01) and gliders are the least aerodynamic efficient aircraft (with CDo as

high as 0.08) .The lift coefficient is readily found from equation 2.3. Compare the glider

Schleicher ASW22 that has a CDo of 0.016 (Figure 3.6) with the agricultural aircraft

Dromader, PZL M-18 that has a CDo of 0.058 (figure 3.7).

Figure 3. 7. Agricultural aircraft Dromader, PZL M-18 with Cdo of 0.058

Figure 3. 8. Wing leading edge sweep angle

Comparison between equations 3.7 and 3.11 yields the following relationship.

2

LD KCCi (3.12)

So induced drag factor is proportional with the square of lift coefficient. Figure 3.4 shows

the effect of lift coefficient (induced drag) on drag coefficient.

99

No. Aircraft type CDo e

1 Subsonic jet 0.014-0.02 0.75-0.85

2 Large turboprop 0.018-0.024 0.8-0.85

3 Twin-engine piton prop 0.022-0.028 0.75-0.8

4 Small GA with retractable landing gear 0.02-0.03 0.75-0.8

5 Small GA with fixed landing gear1 0.025-0.04 0.65-0.8

6 Agricultural aircraft with crop duster 0.07-0.08 0.65-0.7

7 Agricultural aircraft without crop duster 0.06-0.065 0.65-0.75

8 glider 0.01-0.015 0.8-0.9

9 Supersonic jet 0.02-0.04 0.6-0.8

Table 3. 1. Typical values of CDo and e for several aircraft

3.4. Calculation of CDo

The equation 3.11 implies that the calculation of aerodynamic force of drag is dependent

on zero-lift drag coefficient (CDo). Since the performance analysis is based on aircraft

drag, the accuracy of aircraft performance analysis is higly rely on the estimation of CDo.

This section is devoted to the calculation of zero-lift drag coefficient and is the most

important section of this chapter.

As figure 3.9 illustrates, external aerodynamic components of an aircraft are all

contributing in aircraft drag. Although only wing; and with some extent, tail; has

aerodynamical function (i.e. to produce lift), but every component that has contact with

air flow, is doing some types of aerodynamic function (i.e. producing drag). Thus in order

to calculate zero-lift drag coefficient of an aircraft, we must include every contributing

item. The CDo of an aircraft is simply the summation of CDo of all contributing

components.

...HLDoSoNoLGovtohtowofoo DDDDDDDDD CCCCCCCCC (3.13)

Every component has a positive contribution, and no component has negative component.

In majority of conventional aircraft, wing and fuselage are each contributing about 30%-

40% (totally 60%-80%). All other components are contributing about 20%-40% to CDo of

aircraft. In some aircraft (e.g. hang glider), there is no fuselage, so it does not have any

contribution in CDo; instead the human pilot plays the role of fuselage.

In equation 3.13, CDof, CDow, CDoht, CDovt, CDoLG, CDoN, CDoS, CDoHLD, are respectively

representing fuselage, wing, horizontal tail, vertical tail, landing gear, nacelle, strut, high

lift device (such as flap) contributions in aircraft CDo. The three dots at the end illustrates

that there are other components that are not shown here. They include non-significant

components such as antenna, pitot tube, stall horn, wire, and wiper. In each subsection of

1 This also refers to a small GA with retractable landing gear during take-off

100

this section, a technique is introduced to calculate the contribution of each component in

CDo of aircraft. The major reference for these techniques and equations is Reference 1.

Figure 3. 9. Major components of Boeing 737 contributing to CDo

3.4.1. Fuselage

The zero-lift drag coefficient of fuselage is given by the following equation:

S

SffCC

f

fo

wet

MLDfD (3.14)

where, Cf is skin friction coefficient and is non-dimensional number. It is determined

based on the Prandtl relationship as follows:

58.2

10 Relog

455.0fC (Turbulent flow) (3.15a)

Re

327.1fC (Laminar flow) (3.15b)

Equation 3.15a is for purely turbulent flow and equation 3.15b is for purely laminar flow.

Most aircraft are frequently experiencing a combination of laminar and turbulent flow

wing

Horizontal

tail

Landing Gear

Nacelle

fuselage wing

Horizontal

tail

Vertical tail Vertical tail

101

over fuselage and other component. There are aerodynamic references2 that recommend

formula to evaluate the ratio between laminar and turbulent flow over any aerodynamic

component. The transition point from laminar to turbulent flow may be evaluated by

these references. For simplicity they are not reproduced here. Instead, you are

recommended to assume that the flow is completely turbulent. This provides a better

result; since over-estimation of drag is much better than its under-estimation. As a rule of

thumb, in low subsonic flight, the flow is mostly laminar, but in high subsonic and

transonic speed, it becomes mostly turbulent. Supersonic and hypersonic flight

experiences a complete turbulent flow over every component of aircraft.

The parameter Re is called Reynolds number and has a non-dimensional value. It is

defined as

VLRe (3.16)

Where is the air density, V is aircraft true airspeed, is air viscosity, and L is the

length of the component in the direction of flight. For the fuselage, L it the fuselage

length.

The second parameter in equation 3.14 (fLD) is a function of length to diameter ratio. It is

defined as

D

L

DLf LD 0025.0

601

3 (3.17)

where L is the fuselage length and D is its maximum diameter. If the cross section of the

fuselage is not a circle, you must find its equivalent diameter. The parameter is a

function of flight regime and Mach number (M) and is defined as follows:

21 M (0.0< M < 0.9) (3.18)

The third parameter in equation 3.14 (fM) is a function of Mach number (M). It is defined

as

45.108.01 MfM (3.19)

The last two parameters in equation 3.14 are Swetf and S, where are respectively the

wetted area of the fuselage and the wing reference area.

A comment is in order regarding the reference area S in equation 3.14. This is nothing

other than just a reference area, suitably chosen for the definition of the force and

moment coefficients. Wetted area is the actual surface area of the material making up the

2 Such as Ref. 10

102

skin of the airplane - it is the total surface area that is in actual contact with, i.e., wetted

by, the fluid in which the body is immersed. Indeed, the wetted surface area is the surface

on which the pressure and shear stress distributions are acting; hence it is a meaningful

geometric quantity when one is discussing aerodynamic force. However, the wetted

surface area is not easily calculated, especially for complex body shapes.

Figure 3. 10. Wing gross area and wing net area

In contrast, it is much easier to calculate the planform area of a wing (See figure 3.10),

that is, the projected area that we see when we look down on the wing, including fuselage

in between two parts of the wing. For this reason, for wings as well as entire airplanes,

the wing planform area is usually used as S in the definitions of CL, CD, and Cm.

Similarly, if we are considering the lift and drag of a cone, or some other slender, missile

like body, then the reference area S is frequently taken as the base area of the fuselage.

The point here is that S is simply a reference area that can be arbitrarily specified. This is

done primarily for convenience.

Whether we take for S the planfonn area, base area, or any other area germane to a given

body shape, it is still a measure of the relative size of different bodies which are

geometrically similar. And what is important in the definition of CL, CD, and Cm is to

divide out the effect of size via the definitions. The moral to this story is as follows:

Whenever you take data for CL, CD, or Cm from the technical literature, make certain that

you know what geometric reference area was used for S in the definitions and then use

that same defined area when making calculations involving those coefficients.

Wing gross area

Wing net area

103

3.4.2. Wing, Horizontal Tail, and Vertical Tail

Since wing, horizontal tail and vertical tail are three lifting surface, we treat them in a

similar manner. The zero-lift drag coefficients of wing (woDC ), horizontal tail (

htoDC ), and

vertical tail (vtoDC ), are respectively given by the following equations:

4.0

004.0

min

ww

wwwo

dwet

MtcfD

C

S

SffCC (3.20)

4.0

004.0

min

htht

hththto

dwet

MtcfD

C

S

SffCC (3.21)

4.0

004.0

min

vtvt

vtvtvto

dwet

MtcfD

C

S

SffCC (3.22)

In these equations, Cfw, Cfht, Cfvt are similar to what we defined for fuselage in formula

3.15. The only difference is that the equivalent value of L in Reynolds number (equation

3.23) for wing, horizontal tail, and vertical tail are their mean aerodynamic chord (MAC

or C ). In another word, the definition of Reynolds number for a lifting surface is

CVRe (3.23)

where the mean aerodynamic chord is calculated by

11

3

2rCC (3.24)

where Cr is root chord (See figure 3.11), and id taper ratio (the ratio between tip chord

to root chord). The parameter ftc is a function of thickness ratio and is given by

4

maxmax

1007.21

c

t

c

tf tc (3.25)

where max

c

tis the maximum thickness (t) to chord (C) ratio of a wing, or a tail. The

maximum thickness to chord ratio for wing is about 12% to18%, and for the tail is about

9% to 12%. The parameter Swet in equation 3.12 is the wetted area and is difficult to

calculate, because of the curvature of the wing or tail. Since the wing and tail are not too

thick, it may be assumed that is about twice of the net area (see figure 3.10). The

parameter Cdmin in formula 3.20, 3.21, and 3.22 is the minimum drag coefficient of the

cross section (airfoil) of the wing or tail. It can be readily read extracted from a Cd-Cl

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curve of the airfoil. One example is illustrated in figure 3.12 for a NACA airfoil of 631-

412. Reference 3 is a rich collection of information for a variety of NACA airfoils.

Figure 3. 11. Mean Aerodynamic Chord (MAC)

Example 3.1: Consider a cargo aircraft with the following features:

m = 380,000 kg S = 567 m2, MAC = 9.3 m, (t/c)max = 18%, Cdmin = 0.0052

This aircraft is flying at sea level with the speed of 400 knot. If the aircraft CDo is 2.3

times the wing CDo (i.e. CDow), what is the aircraft CDo?

Solution:

8

51031.1131000380

107894.1

3.95144.0400225.1Re

VL (3.16)

Root Chord

Tip Chord

Mean Aerodynamic Chord

Centerline Chord

105

605.0340

5144.0400

a

VM (1.20)

We assume that the boundary layer is turbulent, so

00212.0

1031.1log

455.0

Relog

455.058.28

10

58.2

10

fC (3.15a)

9614.0606.008.0108.0145.145.1 MfM (3.19)

742.118.010018.07.211007.214

4

maxmax

c

t

c

tf tc (3.25)

0102.0004.0

0052.0

576

57629614.0742.100212.0

004.0

4.0

4.0

min

wo

ww

wwwo

D

dwet

MtcfD

C

C

S

SffCC

(3.20)

Therefore the aircraft zero-lift drag coefficient is

023.00102.03.23.2 woo DD CC

3.4.3. High Lift Devices

High lift devices are parts of wing to increases lift when deflected. They are usually

employed during take-off and landing. Two main groups of High lift devices are leading

edge HLD (flap) and trailing edge High lift devices (slat). There are many types of wing

trailing edge flaps such as split flap, plain flap, single-slotted flap, fowler flap, double-

slotted flap, and triple-slotted flap. They are deflected down to increase the camber of the

wing, so CLmax will be increased. The most effective method used on all large transport

aircraft is the leading edge slat. A variant on the leading edge slat is a variable camber

slotted Kruger flap used on the Boeing 747. The main effect of wing trailing edge flap is

to increase the effective angle of attach of the wing without actually pitching the airplane.

The application of high lift devices has a few negative side-effects including an increase

in aircraft drag (as will be measured in CDo).

106

Figure 3. 12. variations of lift coefficient versus drag coefficient for NACA 631-412

3.4.3.1. Trailing Edge High Lift Device

The increase in CDo due to application of trailing edge high lift device (flap) is given by

the following formula:

Bf

f

D AC

CC

flapo

(3.26)

No Flap type A B

1 split flap 0.0014 1.5

2 plain flap 0.0016 1.5

3 single-slotted flap 0.00018 2

4 double-slotted flap 0.0011 1

5 Fowler 0.00015 1.5

Table 3. 2. The values of A and B for different types of flaps

107

where Cf/C is the ratio between average extended flap chord to average wing chord (see

figure 3.13) at the flap location and is usually about 0.2. Do not confuse this Cf with skin

friction coefficient. The parameters “A” and “B” are given in the table 3.2 based on the

type of flap. The f is the flap deflection in degrees (usually less than 50 degrees).

Figure 3. 13. Wing section at the flap location (Plain flap)

3.4.3.1. Leading Edge High Lift Device

The increase in CDo due to application of leading edge high lift device is given by the

following formula:

woslo Dsl

D CC

CC

(3.27)

where Csl/C is the ratio between average extended slat chord to average wing chord. The

CDow is the wing aero-lift drag coefficient without extending high lift devices (including

slat).

3.4.4 Landing Gear

The landing gear (or undercarriage) is the structure (usually wheels) that supports an

aircraft and allows it to move across the surface of the ground when it is not flying.

Landing gear usually includes wheels equipped with shock absorbers for solid ground,

but some aircraft are equipped with skis for snow or floats for water, and/or skids. To

decrease drag in flight, some landing gears are retracted into the wings and/or fuselage

with wheels flush against the surface or concealed behind doors; this is called retractable

gear.

Cf

C

108

When landing gear is fixed (not retraced), it produces an extra drag for the aircraft. It is

sometimes responsible for increase in drag as high as 50%. The increase in CDo due to

application of landing gear is given by the following formula:

n

i

DDS

SCC i

o

1

lg

lglg

(3.28)

where Slg is the frontal area of each wheel, and S is the wing reference area. The

parameter CDlg is the drag coefficient of each wheel; that is 0.3, when it has fairing; and

is 0.15 when it does not have fairing (see figure 3.14). The frontal area of each wheel is

simply the diameter (dg) times its width (wg).

gg wdS lg (3.29)

Figure 3. 14. Landing gear and its fairing

As it is observed in equation 3.28, every wheel that is experiencing air flow must be

accounted for drag. For this reason, index “i” is used. The drag of the strut of landing

gear is introduced in the next section. In some aircraft, a fairing is used to decrease the

drag of a non-retracted gear (see figure 3.14b). The parameter “n” is the number of

wheels in an aircraft.

3.4.5. Strut

Landing gear is often attached to the aircraft structure through strut. These struts are

producing an extra drag for aircraft. In some aircraft (such as hang gliders), their section

is a symmetrical airfoil in order to reduce the aircraft drag. In some old aircraft, wings are

attached through few struts to support wing structure (see figure 3.15). Modern aircraft

dg

tg

a. Without fairing b. With fairing

109

use advanced material for structure that are stronger and there are no need for any strut to

support their wings.

Figure 3. 15. Wing and landing gear struts in Cessna 172

The increase in CDo due to application of strut is given by the following formula:

n

i

sDD

S

SCC

iosso

1

(3.30)

where Ss is the frontal area of each strut (its diameter times its length), and S is the wing

reference. The parameter CDos is the drag coefficient of each strut; that is 0.1, when it has

airfoil section; and is 1 when it does not have airfoil section. The parameter “n” is the

number of struts in an aircraft. It is observed that using an airfoil section for a strut

decreases its drag at about 10 times. However its manufacturing cost is increased.

3.4.6. Nacelle

If the engine is not buried inside fuselage, it must be in contact with air. In order to

reduce the engine drag, in most aircraft engine is located inside an aerodynamic cover

called nacelle. It can be considered that the nacelle is similar to the fuselage, except it

fineness ratio is higher. Thus, the nacelle zero-lift drag coefficient (CDon) will be

determined in the same way as fuselage. In the case that the nacelle fineness ratio (i.e.

nacelle length to nacelle diameter ratio) is below 2, assume it as 2. This parameter is used

in the equation 3.17.

3.4.7. Cooling Drag

Cylinder heads, oil coolers, and other heat exchangers require a flow of air through them

for purposes of cooling. Usually, the source of this cooling air is the free stream possibly

augmented to some extent by a propeller slipstream or bleed from the compressor section

of a turbojet (See figure 3.16). As the air flows through the baffling air experiences a loss

strut

110

in total pressure extracting energy from the flow same time however heat is added to the

flow. If the rate at which the heat is being added to the flow is less than the rate at which

energy is being extracted from flow the energy and momentum flux in the exiting flow

after it has expanded the free stream ambient pressure will be less than that of the

entering flow The result is a drag force known as cooling drag.

Figure 3. 16. The use of cowl flaps to control engine cooling air.

It is a matter of bookkeeping as to whether to penalize the airframe or the engine for this

drag. Some manufacturers prefer to estimate the net power lost to the flow and subtract

this from the engine power. Thus, no drag increment is added to the airplane. Typically,

for a piston engine, the engine power is reduced by as much as approximately 6% to

account for the cooling losses. Because of the complexity of the internal flow through a

typical engine installation, current methods for estimating cooling losses are

semiempirical in nature.

The calculation of the cooling drag is highly configuration-dependent so that it is

unfeasible to present here a general method which will apply to most engine installations.

Instead, suffice it to say that one should consider cooling drag in predicting the

performance of an airplane and this is best done in consultation with the engine

manufacturer. Large manufacturers of turbine engines will provide computer programs

for estimating installation losses for their engines. Engine cooling drag coefficient (CDoen)

is given by the following relationship:

VS

PTKC e

Deno

281051.4 (3.31)

where P is the engine power (hp), T is the air temperature (K), is the relative density of

the air, V is the aircraft velocity (m/sec), and S is the wing reference area (m2). The

111

parameter Ke is a coefficient that depends on the type of engine. It varies between 1 and

3.

3.4.8. Trim Drag

Basically, trim drag is not any different from the types of drag already discussed. It arises

mainly as the result of having to produce a horizontal tail load in order to balance the

airplane around its pitching axis. Any drag increment that can be attributed to a finite lift

on the horizontal tail contributes to the trim drag. Such increments mainly represent

changes in the induced drag of the tail.

The trim drag is usually small, amounting to only 1 or 2% of the total drag of an airplane

for the cruise condition. Reference 5, for example, lists the trim drag for the Leajet Model

25 as being only 1.5% of the total drag for the cruise condition. To examine this further,

we begin with the sum of the lifts developed by the wing and tail that must equal the

aircraft’s weight. One can easily derive the horizontal tail lift coefficient (CLt) as

t

LLLS

SCCC

wt (3.32)

where CL is aircraft lift coefficient, CLw is wing lift coefficient, and St is horizontal tail

area. Then trim drag will be

2

2 1

t

LL

tt

LtDDS

SCC

AReCKCC

wttitrimo (3.33)

where et is horizontal tail span efficiency factor, and ARt is the horizontal tail aspect

ratio.

3.4.9. CDo of Other Parts

So far, we introduced several techniques to calculate CDo of aircraft major components.

There are other factors and parts that are producing drag and contribute in total CDo of

aircraft. These parts are introduced in this section.

1. Interference

When two shapes intersect or are placed in proximity, their pressure distributions and

boundary layers can interact with each other, resulting in a net drag of the combination

that is higher than the sum of the separate drags. This increment in the drag is known as

interference drag. Except for specific cases where data are available, interference drag is

difficult to estimate accurately. Two examples are: 1. placing an engine nacelle in

proximity to a rear pylon on a tandem helicopter (like a CH-47), and 2. the interference

drag between the rotor hub and pylon for a helicopter.

Figure 3.17 shows a wing abutting the side of a fuselage. At the fuselage-wing juncture a

drag increment results as the boundary layers from the two airplane components interact

and thicken locally at the junction. This type of drag penalty will become more severe if

112

surfaces meet at an angle other than 90o. In particular, acute angles between intersecting

surfaces should be avoided. Reference 4, for example, shows that the interference drag of

a 45% thick strut abutting a plane wall doubles as the angle decreases from 90o to

approximately 60°. If acute angles cannot be avoided, filleting should be used at the

juncture.

Figure 3. 17. Wing-fuselage interference drag

In the case of a high-wing configuration, interference drag results principally from the

interaction of the fuselage boundary layer with that from the wing’s lower surface. This

latter layer is relatively thin at positive angles of attack. On the other hand, it is the

boundary layer on the upper surface of a low wing that interferes with the fuselage

boundary layer. This upper surface layer is appreciably thicker than the lower surface

layer. Thus, the wing-fuselage interference drag for a low- wing configuration is usually

greater than for a high-wing configuration.

The available data on wing-fuselage interference drag are sparse. Reference 4 presents a

limited amount but, there is no correlation with wing position or lift coefficient. Based on

this reference, an approximate drag increment caused by wing-fuselage interference is

estimated to equal 4% of the wing’s profile drag for a typical aspect ratio and wing

thickness.

Although data such as those in Reference 4 may be helpful in estimating interference

drag, an accurate estimate of this quantity is nearly impossible. For example, a wing

protruding from a fuselage just forward of the station where the fuselage begins to taper

may trigger separation over the rear portion of the fuselage. Sometimes interference drag

can be favorable as, for example, when one body operates in the wake of another. Race

113

car drivers frequently use this to their advantage in the practice of “drafting.” Reference 2

offers more details about interference drag.

2. Antenna

The antenna of most aircraft is located outside of the aircraft. They are in contact with the

air, hence they produce drag. There are mainly two types of antenna: a. rod or wire, and

b. blade. Figure 3.18 shows a rod antenna of U-2 aircraft. Figure 3.19 illustrates two

blade antenna of Boeing Vertol CH-113/113A.

Figure 3. 18. The UHF radio antenna of U-2S Dragon Lady

The large blade antenna of U-2S is the UHF radio antenna. The thin whip as an ADF

antenna. The whip was originally straight. The bend in the upper portion of the whip

antenna was introduced to provide clearance for the Senior Span/Spur dorsal pod. Most

U-2 aircraft seem to now use the bent whip, even if they are not capable of carrying the

Senior Span/Spur dorsal pod.

Figure 3. 19. Boeing Vertol CH-113 Labrador

Look on the bottom from the starboard side of Boeing Vertol CH-113/113A. Note the

two blade antenna, as well as two more whip antenna. These represent the right-most

antenna in the remaining two pair of the six, described in the previous view. Also seen, is

the towel-rack 'loran' antenna. These are only fitted to the former-Voyager airframes. A

carry-over from it's Army days. The item hanging down, in the distance, is the sectioned

flat plate that covers the ramp hinge, when the ramp is closed.

114

To calculate CDo of antenna, rod antenna is treated as a strut, and blade antenna is

considered as a small wing. The very equations that are introduced for strut (Section

3.4.5) and wing (Section 3.4.2) are employed to determine the CDo of the antenna.

3. Pitot Tube

A pitot tube is a pressure measuring instrument used to measure air flow velocity, and

more specifically, used to determine the airspeed (and sometimes altitude) of an aircraft

(See figure 3.20). The basic pitot tube simply consists of a tube pointing directly into the

air flow. As this tube contains air, a pressure can be measured as the moving air is

brought to rest. This pressure is the stagnation pressure of the air, also known as the total

pressure, or sometimes (particularly in aviation circles) the pitot pressure.

Since the pitot tube has contact with air, it has a contribution to aircraft CDo. If the aircraft

is in subsonic regime, the horizontal part of the pitot tube may be assumed as a little

fuselage and its vertical part as a strut. For supersonic flow, section 3.5 introduces a

technique to account for pitot tube zero-lift drag coefficient (CDopt).

4. Surface Roughness

The outer surface of aircraft structure (skin) has considerable role in aircraft drag. For

this reason, the aircraft skin is often painted. This not only protects the skin from

atmospheric hazards (e.g rusting), but also reduces its drag. As the surface of the skin is

more polished, the aircraft drag will be reduced. The reader is referred to specific

aerodynamic references to gain more information about the effect of surface roughness

on the aircraft drag.

Figure 3. 20. Example of a pitot tube mounted under the wing of a Cessna 172

5. Leakage

There are gaps between control surfaces (such as elevator, aileron and rudder), flaps, and

spoilers and the lifting surfaces (such as wing and tails). The air is flowing through these

tiny gaps and thus producing extra drag. This is called leakage drag. The reader is

115

referred to specific aerodynamic references to gain more information about the influence

of these gaps on the aircraft drag.

6. Rivet and Screw

The outside of the aircraft structure is covered with a skin. The skin (if it is from metallic

materials3) is attached to the primary components of an aircraft structure (such as spar

and frame) via part such as rivet or screw. In the case of the screw, the top part of the

screw could be often hidden inside skin. But the heads of most types of rivets are out of

skin, therefore they produce extra drag. Figure 3.15d illustrates both rivet and screw on

wing of Cessna-172.

7. Compressibility

Compressibility is a property of the fluid. Liquids have very low values of

compressibility whereas gases have high compressibility. Obviously, in real life every

flow of every fluid is compressible to some greater or lesser extent; hence, a truly

constant density (incompressible) flow is a myth. However, for almost all liquid flows as

well as for the flows of some gases under certain conditions, the density changes are so

small that the assumption of constant density can be made with reasonable accuracy.

Figure 3. 21. Drag rise due to compressibility for swept wing-body combinations

So far we considered incompressible flow where the density is assumed to be constant

throughout. Compressible flow is a flow in which the density is not constant. A few

important examples are the internal flows through rocket and gas turbine engines, high-

3 The Composite structures have the advantage that they do not require any rivet or screw.

116

speed subsonic, transonic, supersonic, and hypersonic wind tunnels, the external flow

over modern airplanes designed to cruise faster than 0.3 of the speed of sound, and the

flow inside the common internal combustion reciprocating engine. For most practical

problems, if the density changes by 5 percent or more, the flow is considered to be

compressible.

Flow velocities higher than 0.3 of the speed of sound are associated with relatively large

pressure changes, accompanied by correspondingly large changes in density. The aircraft

drag at high subsonic speed is about twice as of that at Mach 0.3. Compressibility effects

on airplane aerodynamics have been important since the advent of high-performance

aircraft in the 1940s.

Now, let’s look at a wing with a free stream. The lift is created by the occurrence of

velocities higher than free stream on the upper surface of the wing and lower than free

stream on much of the lower surface. As the flight speed of an airplane approaches the

speed of sound (i.e., M > 0.65), the higher local velocities on the upper surface of the

wing may reach and even substantially exceed M = 1.0. The existence of supersonic local

velocities on the wing is associated with an increase of drag due to a reduction in total

pressure through shock waves and due to thickening and even separation of the boundary

layer due to the local but severe adverse pressure gradients caused by the shock waves.

Figure 3. 22. Incremental drag coefficient due to compressibility (Ref. 6).

The drag increase is generally not large, however, until the local speed of sound occurs at

or behind the crest of the airfoil, or the crestline, which is the locus of airfoil crests along

the wing span. The crest is the point on the airfoil upper surface to which the free stream

is tangent. The occurrence of substantial supersonic local velocities well ahead of the

crest does not lead to significant drag increase provided that the velocities decrease below

sonic forward of the crest. The incremental drag coefficient due to compressibility is

117

designated CDc. Figure 3.22 is an empirical average of existing transport aircraft data.

For this reason, the parameter in equation 3.18 and 3.24 is employed.

Figure 3. 23. Drag coefficient versus Mach number (Ref. 6)

A complete set of drag curves for a large transport jet is given in Figure 3.23 in a format

using Mach number as the abscissa. Figure 3.21 show the effect of compressibility on

three configurations.

3.4.10. Overall CDo

The overall CDo is determined as the sum of CDo of all aircraft component and factors.

The calculation of CDo contribution due to factors introduced in section 3.4.9 is

complicated. These factors sometimes are responsible for an increase in CDo up to about

50%. Thus we will resort to correction factor and estimation. The overall CDo of an

aircraft is given by

...lg

ftonoosovtohtofowoo DDDDDDDDcD CCCCCCCCKC (3.34)

No. Aircraft type Kc

1 Passenger 1.1

2 Agriculture 1.5

3 Cargo 1.2

4 Single engine piston 1.3

5 General Aviation 1.2

6 Fighter 1.1

Table 3. 3. Correction factor (Kc) for equation 3.34

118

where Kc is a correction factor and depends on several factors such as the type, year of

fabrication and configuration of the aircraft. Table 3.3 yields the Kc for several types of

aircraft.

Each component has a contribution to overall CDo of an aircraft. Their contributions vary

from aircraft to aircraft and from configuration to configuration (e.g. cruise to take-off).

Table 3.4 illustrates contributions of all major components of Gates Learjet 25. Note that

the row 8 of this table shows the contributions of other components as about 20%.

No. Component CDo of component Percent from total CDo

(%)

1 Wing 0.0053 23.4

2 Fuselage 0.0063 27.8

3 Wing tip tank 0.0021 9.3

4 Nacelle 0.0012 5.3

5 Engine strut 0.0003 1.3

6 Horizontal tail 0.0016 7.1

7 Vertical tail 0.0011 4.8

8 Other components 0.0046 20.4

9 Total CDo 0.0226 100

Table 3. 4. CDo of major components of Gates Learjet 25

3.5. Wave Drag

In supersonic speed, a new type of drag is produced and it is referred to as “shock wave

drag” or simply “wave drag”. When a supersonic flow experiences an obstacle, a shock

wave is formed. A shock wave is a thin sheet of fluid across which abrupt changes occur

in P, , V, and M. In general, air flowing through a shock wave experiences a jump

toward higher density, higher pressure, and lower Mach number. The effective Mach

number approaching the shock wave is the Mach number of the component of velocity

normal to the shock wave. This component Mach number must be greater than 1.0 for a

shock to exist.

Whenever the local Mach number becomes greater than 1.0 on the surface of a wing or

body in a subsonic freestream, the flow must be decelerated to a subsonic speed before

reaching the trailing edge. If the surface could be shaped so that the surface Mach number

is reduced to 1.0 and then decelerated subsonically to reach the trailing edge at the

surrounding freestream pressure, there would be no shock wave and no shock drag. This

ideal is theoretically attainable only at one unique Mach number and angle of attack. In

general, a shock wave is always required to bring supersonic flow back to M < 1.0. A

major goal of transonic airfoil design is to reduce the local supersonic Mach number to as

close to M = 1.0 as possible before the shock wave.

The freestream Mach number at which the local Mach number on the airfoil first reaches

1.0 is known as the critical Mach number, Mcr. The freestream Mach number at which M

119

= 1.0 at or behind the airfoil crest is called the crest critical Mach number. The locus of

the airfoil crests from the root to the tip of the wing is the crestline. Empirically, it is

found for all airfoils except the supercritical airfoil, that at 2% to 4% higher Mach

number than that at which M = 1.0 at the crest the drag rises abruptly. The Mach number

at which this abrupt drag rise starts is called the drag divergence Mach number, MDIV.

The main function of a swept wing is to reduce wave drag at transonic and supersonic

speeds.

Thus in supersonic speed, the drag coefficient is expressed by

wio DDDD CCCC (3.35)

where CDw is called wave drag coefficient. The precise calculation of CDw is time

consuming, but to give the reader a guidance we present two techniques, one easy and

approximate; and one long and accurate. For configurations more complicated than

bodies of revolution, the drag may be computed with a panel method or “Computational

Fluid Dynamic” techniques. Figure 3.24 illustrates F-35 Joint Strike Fighter, a new

generation of fighters.

Figure 3. 24. F-35 Joint Strike Fighter, a new generation of fighters

4

3.5.1. Exact Method

In this approach, an aircraft is divided into several parts such that each part must have a

corner angle and experiences a separate shock wave. Then CDw for each part is calculated

separately and then all CDw are added together. For each part, the wave drag coefficient is

given by

APM

D

AV

DC ww

Dw

22

2

1

2

1

(3.36)

4 It is claimed that F-35 will be the last manned fighters. In the future, the fighters will be unmanned.

120

where subscript infinity ( ) means that the parameter are considered in the infinity

distance from the surface. This does not really mean infinity, but it simply means at a

distance out of the effect of shock and the surface (i.e. free-stream). The parameter A is

the surface of a body at which the pressure is acting.

Dw is the wave drag and is equal to the axial component of the pressure force. The reason

is that, in supersonic speed, the induced drag may be ignored, since it has negligible

contribution, compared to the wave drag. The force due to flow pressure is given by

cos2 APDw (3.37)

where P2 represents the pressure behind a shock wave, and is the corner angle (see

figure 3.25).

M1

M2P2

Dw

P1

M1

M2P2

Dw

P1

Figure 3. 25. Geometry for drag wave

To determine the pressure behind the shock (P2), the governing equations for oblique or

normal shock or their published tables must be used. A simple technique is to use the

linearized supersonic flow theory5 where states that Cp is directly proportional to the

local surface inclination with respect to the free stream. This theory holds for any slender

two-dimensional shape. The pressure coefficient is assumed to be the linear function of

the corner angle () as

1

2

2

MC p

(3.38)

where is in radian. Then the pressure behind the shock is determined from

pp CPMCVPP 22

22

1

2

1 (3.39)

5 Ref. 7, page 272

121

Example 3.2: Consider a 15° half-angle wedge at zero angle of attack in a Mach 3 flow

of air. Calculate the wave drag coefficient. Assume that the pressure exerted over the

base of the wedge, the base pressure (P1), is equal to the free-stream pressure of 1 atm.

Solution: The physical picture is sketched in Fig. 3.26. The wave drag is the net force in

the x direction; P2 is exerted perpendicular to the top and bottom faces, and P1 is exerted

over the base. The chord length of the wedge is c. Consider a unit span of the wedge, i.e.,

a length of unity perpendicular to the xy plane. A is the planform area (the projected area

seen by viewing the wedge from the top, thus A = c x 1. The drag per unit span, denoted

by Dw, is

15tan215sin

15cos

12)sin(sin 1

2 cPcP

FAPAPD backdownup

By definition, the wave drag coefficient is

12

2

11

2

12

cPM

D

APM

DC ww

Dw

(3.36)

Figure 3. 26. Geometry for Example 3.1

Thus

15tan

4

1

2

1

12

PM

PPC

wD

The pressure behind shock (P2) is

122

pCPMPP 2

22

1 (3.39)

where

185.013

180152

1

2

22

MC p (3.38)

So

atmP 166.21185.0134.12

1 2

2

Finally the CDw is

1.0)15tan(

134.1

1166.2415tan

42

1

2

1

12

PM

PPC

wD

Note that this drag coefficient is based on the wedge area. If it was part of an aircraft, it

must be calculated based on the aircraft wing area.

3.5.2. Approximate Approach6

In this approach, we consider an aircraft as a whole and we will not divide it into several

parts. Wave drag coefficient consists of two components; volume-dependent wave drag

(CDwv) and lift-dependent wave drag (CDwl). The volume-dependent wave drag is a

function of aircraft volume and much greater than lift-dependent wave drag. The reason

is that in the supersonic speed, the lift coefficient (CL) is very small.

vwlww DDD CCC (3.40)

a. The lift-dependent wave drag is given by

2

22

2

1

L

MSCKC Lwl

Dlw

(3.41)

where L represents the aircraft fuselage length, S wing area, and Kwl is a parameter that is

given by

2

2

bL

SKwl

(3.42)

6 This approach is reproduced from Ref. 8.

123

where b is the wing span.

Figure 3. 27. Wave drag for several aircraft (Ref. 9)

b. The volume-dependent wave drag is written as

4

2128

SL

VKC wv

Dwv (3.43)

where V is the total aircraft volume and Kwv is a factor that is given by

L

bL

b

Kwv

21

75.01

17.1 (3.44)

where is a function of Mach number as

12 M (3.45)

In general, wave drag is considerable such that it will increase aircraft drag up to about

two to three times. Figures 3.27 shows drag coefficients of several aircraft. The drag rise

of most of them due to high Mach number is significant. Table 3.5 illustrates CDo of

several aircraft at their cruise speed.

3.6. CDo at various Configurations

Any aircraft, based on its flight condition may have various configurations. When an

aircraft retracts it landing gear, deflects its flap, rotates its control surfaces, exposes any

external component (such as gun), releases its store (missile), or opens its cargo door; it

124

has changed its configuration. In general there are three configurations that any aircraft

may adapt, they are: 1. clean configuration, 2. take-Off configuration, 3. landing

configuration.

No. Aircraft Engine No. of Engines Landing gear CDo

1 PC-9 Turboprop 1 Retractable 0.022

2 F-104 Turbojet 1 Retractable 0.016

3 Tucano Turboprop 1 Retractable 0.021

4 Boeing 747 Turbofan 4 Retractable 0.018

5 Jetstar Turbojet 4 Retractable 0.0185

6 C-5A Turbofan 4 Retractable 0.019

7 Boeing 727 Turbofan 3 Retractable 0.02

8 F-14 Turbofan 2 Retractable 0.02

9 Learjet 25 Turbofan 2 Retractable 0.022

10 C-54 Piston prop 4 Retractable 0.023

11 C-46 Turboprop 2 Retractable 0.025

12 Beech V35 Turboprop 1 Retractable 0.025

13 Cessna 310 Piston prop 2 Retractable 0.025

14 F-4C Turbojet 2 Retractable 0.03

15 Piper Pa-28 Piston prop 1 Fixed 0.047

16 Cessna 150 Piston prop 1 Fixed 0.05

Table 3. 5. CDo of several aircraft

3.6.1. Clean Configuration

The clean configuration is the configuration of an aircraft when it is in cruise condition.

In this configuration, no flap is deflected, and landing gear is retracted (if it is

retractable). Therefore the drag polar is

2Ccleanoclean LDD CKCC (3.46)

Thus clean CDo of the aircraft includes every component (such as wing, tail, and fuselage)

except flap and landing gear (if retractable). If landing gear is not retractable, the CDo

includes landing gear too. The parameter CLC is the cruise lift coefficient.

3.6.2. Take-Off Configuration

The take-off configuration is the configuration of an aircraft when it is in take-off

condition. In this configuration the aircraft has high angle of attack, flap is deflected for

take-off, and landing gear is not retracted (even if it is retractable). The take-off zero-lift

drag coefficient is given by

LGoTOoflapcleanoTOo DDDD CCCC

(3.47)

125

In take-off condition, the flaps are usually deflected down about 10-30 degrees. The take-

off CDo depends of the deflection angle of the flaps. As this angle increases, the take-off

CDo increases too. The drag polar at take-off configuration is

2TOTOoTO LDD CKCC (3.48)

where CLTO represents the lift coefficient at take-off. This coefficient does not have a

constant value during take-off, due to the accelerated nature of its motion. The CLTO at

the lift off condition (where the from wheel is just detached from the ground), may be

given by

22

9.0LO

LVS

mgC

TO (3.49)

where VLO represents the aircraft lift-off speed. The factor of 0.9 is due to the effect of

engine thrust during take-off.

3.6.3. Landing Configuration

The landing configuration is the configuration of an aircraft when it is in landing

condition. In this configuration the aircraft has higher angle of attack (even more than

take-off), flap is deflected (even more than take-off), and landing gear is not retracted

(even if it is retractable). The landing zero-lift drag coefficient is given by

LGoLoflapcleanoLo DDDD CCCC

(3.50)

In landing condition, the flaps are usually deflected down about 30-60 degrees. The

landing CDo depends of the deflection angle of the flaps. As this angle increases, the

landing CDo increases too. The landing zero-lift drag coefficient (CDoL) is often greater

than the takeoff zero-lift drag coefficient (CDoTO). The drag polar at landing configuration

is given by

2LLoL LDD CKCC (3.51)

where CLL is the lift coefficient at landing. The CLL at the landing condition is given by

22

L

LVS

mgC

L (3.52)

where VL is the aircraft landing speed. It is noticeable that the landing speed (VL) and

take-off speed (VTO) are often about 10%-30% greater than stall speed. In chapter 8, take-

off and landing performances will be discussed in detail.

126

3.6.4. The Effect of Speed and Altitude on CDo

Reynolds number is one of the influential parameters on the zero-lift drag coefficient. As

the Reynolds number increases, the boundary layer thickness decreases and thus CDo

decreases too. As equation 3.16 shows, the Reynolds number is a function of true

airspeed. Since the true airspeed is a function of altitude, it can be concluded that the

Reynolds number is also a function of altitude. Another factor affecting CDo is the

compressibility that is significant at speeds higher than Mach 0.5. The third important

factor in wave drag as introduced in section 3.5.

Considering these factors, it is concluded that the CDo is a function of mach number and

altitude:

hMfCoD , (3.53)

Adding up all important factors, we observe that at low Mach numbers, CDo is increased,

due to an increase in Reynolds number. As compressibility factor shows up in higher

subsonic speeds, the CDo increases with a higher rates. In transonic speeds, shock wave is

formed and a jump (increase) in CDo will be experienced. Therefore the CDo is directly

proportional with speed, and as speed (Mach number) is increased, the CDo is increased.

Figure 3.28 shows a typical variation of drag coefficient versus lift coefficient at various

Mach number.

Figure 3. 28. Drag for various Mach number (Ref 6)

The second factor that affects the CDo is altitude. For a specific Mach number, as the

altitude increases, the true airspeed is decreased. For instance, consider an aircraft is

flying with a speed of mach 0.5 at sea level. The true airspeed at this altitude is 170 m/sec

(0.5 x 340 = 170). If this aircraft is flying with the same Mach number at 11,000 ft

altitude, it true airspeed will be 147 m/sec (0.5 x 249 = 147). Thus, the higher altitude

means the lower Reynolds number and therefore higher CDo.

127

Figures 3.21 and 3.22 illustrate the variations of drag force for a light transport aircraft

(mass of 50,000 kg) with turbofan engine. Figure 3.29 shows this variation without

considering the compressibility effects and figure 3.30 shows this variation with

considering the compressibility effects. Both figures show these variations at various

altitudes. Comparing these two figures reveals the significance of the compressibility

effect on CDo.

Figure 3. 29. The variation of drag force without considering the compressibility effects

Figure 3. 30. The variation of drag force with considering the compressibility effects (Ref 6)

128

In conclusion, it can be assumed that at the speed of Mach numbers less than 0.7, the

variation of CDo is such that it can be considered constant. At higher Mach numbers, the

compressibility effect and wave drag must be taken into account. The second conclusion

is that, at higher altitude, total drag force is reduced. The reason is that, although at higher

altitude, the CDo is increased, but the air density is decreased. The rate of change

(decrease) in air density is faster than the rate of change (increase) of CDo. This is one of

the reasons why airlines choose to fly at higher altitude despite the need and cost of

climb.

Example 3.3:

Consider the aircraft in example 3.1 has a single-slotted flap with an average chord of

2.3. This aircraft takes off with flap angle of 20 degrees and lands with the flap angle of

35 degrees. Assume the CDo of landing gear is 0.01, K = 0.052 and take-of and landing

speed is 130 knot. Determine the aircraft CDo at take-off and landing.

Solution:

The CDo of flap is given by

Bf

f

D AC

CC

flapo

(3.26)

From table 3.2, A = 0.00018 and B = 2, so

The CDo of flap at take-off is

0178.02000018.03.9

3.2 2

flapoDC (3.26)

The CDo of flap at landing is

0545.03500018.03.9

3.2 2

flapoDC (3.26)

The take-off CDo is

051.001.00178.0023.0 LGoTOoflapcleanoTOo DDDD CCCC (3.47)

To find aircraft CD, we need to find induced drag coefficient. The lift coefficient at take-

off is

16.2

5144.0130567225.1

38000029.029.0

22

LO

LVS

mgC

TO (3.49)

129

243.016.2052.0 22 LD KCC

i (3.12)

So CDTO is

294.0243.0051.02

TOTOoTO LDD CKCC (3.48)

For the landing:

4.2

5144.0130567225.1

3800002222

LO

LVS

mgC

L (3.52)

3.04.2052.0 22 LD KCC

i (3.12)

088.001.00545.0023.0 LGoLoflapcleanoLo DDDD CCCC (3.47)

388.03.0088.02

LLoL LDD CKCC (3.48)

Add materials about:

Supersonic flight speed measurement (pitot tube with normal shock)

130

Problems Note: In all problems, assume ISA condition, unless otherwise stated.

1. A GA aircraft is flying at 5000 ft altitude. The length of the fuselage is 7 m, wing

mean aerodynamic chord is 1.5 m, horizontal tail mean aerodynamic chord is 1.5

m, and vertical tail mean aerodynamic chord is 0.6 m. Determine Reynolds

number of fuselage, wing, horizontal tail and vertical tail.

2. The following (figure 3.31) is a top-view of Boeing 757 transport aircraft that has

a wing span of 38.05 m. Using a proper scale and using a series of measurements,

determine the wing reference (gross) area and wing net area of this aircraft.

Figure 3. 31. Top-view of Boeing 757 transport aircraft

3. Estimate the wing wetted area of Boeing 757 (problem 2). Assume the wing has a

maximum thickness of 12%.

4. The mean aerodynamic chord a trainer aircraft is 3.1 m. This trainer is cruising at

seal level with a speed of Mach 0.3. Determine skin friction coefficient of the

wing when boundary layer over the wing is: a. laminar, b. turbulent.

5. A business jet with a 31 m2 wing area and mass of 6500 kg is flying at 10,000 ft

altitude with a speed of 274 knot. If CDo = 0.026, K = 0.052, plot the followings:

131

a. drag polar

b. the variation of drag versus speed.

6. The tip chord of a wing is 6 m and its root chord is 9 m. What is the mean

aerodynamic chord?

7. The Attach aircraft Thunderbolt II (Fairchild A-10A) has the following features:

mTO = 22,221 kg, S = 47 m2, K = 0.06, CDo = 0.032, Vmax = 377 knot, VTO = 120

knot. Assume that the CDo is constant throughout all speeds.

a. Plot the Variation of Do versus speed (from take-off speed to maximum

speed)

b. Plot the Variation of Di versus speed (from take-off speed to maximum

speed)

c. Plot the Variation of total D versus speed (from take-off speed to

maximum speed)

d. What speed, the drag force is minimum at?

8. The wing of twin-turbofan airliner Boeing 777 has 31.6 degrees of leading edge

sweepback, span of 60.93 m and planform area of 427.8 m2. Determine Oswald

efficiency factor (e) and induced drag correction factor (K) of this wing.

9. Determine Oswald efficiency factor of a wing with aspect ratio of 7 and leading

edge sweep of 20 degrees.

10. A single engine aircraft has a fixed lading gear with three similar tires. Each tire

has a diameter of 25 cm and thickness of 7 cm. The landing gear does not have

fairing and wing area is 26 m2. Determine zero-lift drag coefficient of landing

gear.

11. A cargo airplane is cruising with a speed of Mach 0.47, taking off with a peed of

95 knots and landing with a speed of 88 knot. Its flap is deflected down 22

degrees during take-off and 35 degrees during landing. The aircraft has a mass of

13,150 kg, wing area of 41.2 m2, and K = 0.048. The zero lift drag coefficients of

all components are

CDow = 0.008, CDof = 0.008, CDow = 0.0064, CDoht = 0.0016, CDovt = 0.0012,

CDon = 0.002, CDoLG = 0.015, CDos = 0.004,

Determine drag force at

a. cruise

b. take-off

c. landing

12. A Sweden aircraft designer is designing a fuselage for a 36-passenger aircraft to

cruise at a Mach number of 0.55. He is thinking of two seating options: a. 12 rows

of three, and b. 18 rows of two passengers. If he selects option a, the fuselage

length would be 19.7 m with a diameter of 2.3 m. In option b the length would be

132

27.2 m with diameter of 1.55 m. What option yields the lowest fuselage zero-lift

drag coefficient?

13. The amphibian airplane Lake LA-250 (figure 3.32) has a wing with the following

features: S = 15.24 m2, b = 11.68 m, MAC =1.35 m, (t/c)max = 15%, airfoil:

NACA 4415, Cdminw = 0.0042. The aircraft has a mass of 1678 kg and is cruising

at 155 knot. Determine the wing zero-lift drag coefficient. For other information,

use the aircraft three-view that is provided.

Figure 3. 32. Three-view of amphibian airplane Lake LA-250

14. Assume that the aircraft in problem 13 has a plain flap with chord ration of 0.2.

Determine wing CDo, when the flap is deflected 30 degrees during take-off.

15. The four-seat light airplane Piper PA-34 has a horizontal tail with the following

features: Sht = 3.6 m2, b = 4.14 m, (t/c)max = 15%, Cdmint = 0.0056. The aircraft has

a mass of 2,154 kg, wing area of 19.39 m2 and is cruising at 171 knot at an

altitude of 18,500 ft. Determine the horizontal tail zero-lift drag coefficient.

16. A jet trainer has the following features:

mTO = 5630 kg, S = 25.1 m2, b = 17.4 m, e = 0.85, Vc = 270 knot, CLmax = 2.2, VTO =

1.2 Vs, VLand = 1.3 Vs, CDoclean = 0.032, CDoflapTO = 0.02, CDoflapLand = 0.035, CDoLG =

0.01.

Determine the aircraft drag at three flight conditions: a. clean, take-off, and c.

landing.

17. The wings of the Y-5B agricultural biplane are connected together thru two struts.

Each strut has a circular cross section with the length of 1.2 m and diameter of 4

cm. The total planform area of both wings is 38 m2, determine the zero-lift drag

coefficient of struts.

133

18. A single-engine small aircraft has a wing area of 25.96 m2 and a turboprop engine

that has a power of 600 hp. The aircraft is cruising at 25000 ft altitude with a

speed of 182 knot. What is its cooling drag coefficient? Assume Ke = 2.

19. The F-16 supersonic jet fighter (figure 3.33) with a mass 12,331 kg and wing area

of 27.8 m2 is cruising at an altitude of 40,000 ft at Mach number of 2.1. The wing

span is 9.45 m and length of the fuselage is 15.3 m. Determine wave drag of this

jet, if the aircraft volume is considered to be 21.3 m2.

Figure 3.33. Fighter jet F-16A three-view

20. Assume that F-16 (problem 19) is flying at a Mach number of 1.2 at the same

altitude. What is wave drag of this fighter at this flight condition? Compare the

result with the result of problem 19 and comment about your finding.

134

References

1. USAF Stability and Control Datcom, D.E. Hoak, 1978, Air Force Flight

Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio

2. B. W. McCormick, Aerodynamics, Aeronautics, and Flight Dynamics, John

Wiley, 1995

3. Theory of Wing Sections, Ira H. Abbott and A. E. von Doenhoff, 1959, Dover

4. Horner, S. R., Fluid-Dynamic Drag, Midland Park, NJ, 1965

5. Ross, Richard, and Neal, R.D. Learjet Model 25 Drag Analysis, Proceeding of the

NASA-Industry-University GA drag reduction workshop, Lawrence, KS, July 14-

16, 1975

6. R. Shevell, Fundamentals of Flight, Second Edition, 1989, Prentice Hall

7. J. Anderson, Modern Compressible Flow, Third edition, 2005, McGraw-Hill

8. Dietrich Kuchemann, Aerodynamic Design of Aircraft, Pergamon Pr, 1978

9. Aircraft Design: A Conceptual Approach, Raymer D. P., AIAA, 2006

10. J. Anderson, Fundamentals of Aerodynamics, third edition, 2005, McGraw-Hill

11. Roskam J. Airplane Design, Volume VI, 2004