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9.5 Aircraft Drag Formulation

263

Figure 9.2. Aircraft drag polar

Figure 9.15), because at a high CL (i.e., low speed, high angle of attack), there could be additional drag due to separation; at a very low CL (i.e., high speed), there could be additional drag due to local shocks. Both effects are nonlinear in nature. Most of the errors in estimating drag result from computing CDp, three-dimensional effects, interference effects, excrescence effects of the parasite drag, and nonlinear range of aircraft drag. Designers should keep CDw = 0 at LRC and aim to minimize to CDp 0 (perceived as the design point).

An aircraft on a long-range mission typically has a weight change of more than 25% from the initial to the final cruise condition. As the aircraft becomes lighter, its induced drag decreases. Therefore, it is more economical to cruise at a higher altitude to take advantage of having less drag. In practical terms, this is achieved in the step-climb technique, or a gradual climb over the cruise range.

9.5 Aircraft Drag Formulation

A theoretical overview of drag is provided in this section to show that aircraft geometry is not amenable to the equation for an explicit solution. Even so, CFD is yet to achieve an acceptable result for the full aircraft.

Recall the expression in Equation 9.2 for the total aircraft drag, CD, as:

CD = CDparasite + CDi + CDw = CDpmin + CDp + CDi + CDw

where CDparasite = CDfriction + CDpressure = CDpmin + CDp

At LRC, when CDw 0, the total aircraft drag coefficient is given by:

CD = CDpmin + CDp + CDi

(9.3)

The general theory of drag on a 2D body (Figure 9.3a) provides the closedform Equation 9.4. A 2D body has infinite span. In the diagram, airflow is along the x direction and wake depth is shown in the y direction. The wake is formed due to viscous effects immediately behind the body, where integration occurs. The subscript denotes the free-stream condition. Consider an arbitrary CV large enough

264

Aircraft Drag

(a) 2D body and wake in CV

(b) 3D aircraft in CV

Figure 9.3. CV approach to formulate aircraft drag

in the y direction where static pressure is equal to free-stream static pressure (i.e., p = p). Wake behind a body is due to the viscous effect in which there is a loss of velocity (i.e., momentum) and pressure shown in the figure. Measurement and computation across the wake are performed close to the body; otherwise, the downstream viscous effect dissipates the wake profile. Integration over the y direction on both sides up to the free-stream value gives:

D = Dpress + Dskin = ( pp)dy +

ρu(Uu)dy

−∞

−∞

 

 

=

[( pp) + ρu(Uu)] dy

(9.4)

−∞

An aircraft is a 3D object (Figure 9.3b) with the additional effect of a finite wing span that produces induced drag. In that case, the previous equation can be written as:

b/2

 

 

 

 

D = Dskin + Dpress + Di =

[( pp) + ρu(Uu)]dxdy

(9.5)

−∞ −b/2

where b is the span of the wing in the x direction (i.e., the axis system has changed). The finite-wing effects on the pressure and velocity distributions result in induced drag Di embedded in the expression on the right-hand side of Equation 9.5. Because the aircraft cruise condition (i.e., LRC) is chosen to operate below Mcrit, the wave drag, Dw , is absent; otherwise, it must be added to the expression. Therefore, Equation 9.5 can be equated with the aircraft drag expression as given in Equation 9.3. Finally, Equation 9.5 can be expressed in nondimensional form, by dividing

1/2ρU2SW. Therefore,

CD =

=

1

SW

1

SW

 

 

b/2

 

 

1

ρ

U2

+ ρU

 

U

 

 

 

 

 

( p p)

 

2ρu

1

 

u

 

dxdy

−∞ −b/2

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b/2

 

Cp + ρU1

Udxdy

(9.6)

 

 

2ρu

u

 

−∞ −b/2

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