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1/2 The lift-to-drag ratios CL/CD , CL /CD , CL /CD 3/2 and CL /CD Arne Karlsson, Dept. of Aeronautical and Vehicle Engineering, KTH March 10, 2004 2/3

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1/2 The lift-to-drag ratios CL /CD , CL /CD , CL /CD and CL /CD are important performance parameters for an airplane. They all show up, among else, in the analysis of cruise performance, see for example [2] and references therein. One can also show that in gliding (unpowered) flight the lift-to-drag ratio CL /CD is equal to the ratio between the horisontal and the vertical velocity components or the ratio between the horisontal distance travelled (relative the surrounding air mass) and the altitude loss (also relative the surrounding air). In sailplane and sailflying therminology this ratio is called the glide ratio [3, p. 471]. Parts, but not all, of the derivations given below can also be found in different parts of [1, Chapt. 5]. Since the drag coefficient CD is a function of the lift coefficient CL and since the lift coefficient is a function of the angle of attack α all these lift-to-drag ratios are functions of α. It will be shown below that they have a maximum value at some angle of attack. One can also show that these ratios are functions of the airplane speed V in steady and level flight. The airplane speeds giving the maxima of the lift-to-drag ratios are also given below. In this paper we will assume the drag polar to be parabolic, i.e.

CD = CD0 + CDi = CD0 + K C 2L

(1)

where CDi = K C 2L is the lift-induced drag coefficient. With this drag polar the liftto-drag ratios above can be written in a unified form as C Lǫ CLǫ = CD CD0 + K C2L

(2)

where ǫ is either 1/2, 2/3, 1 or 3/2. The maxima for these ratios are found from d dCL



CLǫ CD ⇒



=

(CD0 + K C 2L ) ǫCLǫ−1 − CLǫ · 2KCL =0 [CD0 + K C 2L ]2

ǫC ǫ−1 L



CD0 + K 1

CL2

2 − KCL2 = 0 ǫ 

2 ǫ

1/2

2/3

1

3/2

CDi

1 CD0 3

1 CD0 2

CD0

3CD0



CL

CLǫ CD



max



CD0 3K

1/4



CD0 2K

3 CD0 2

4 CD0 3

CD 



CD0 3K

3 4CD0



CD0 2K

1/3



CD0 K

2CD0

2 3CD0

3CD0 K

4CD0

1 

2 CD0 K



3CD0 K

3/4

1 4CD0

Table 1: Induced drag coefficient CDi , lift coefficient CL and drag coefficient CD giving maxima of the lift-to-drag ratios CLǫ /CD . and these maxima are given if the induced drag coefficient and the lift coefficient takes the values CDi, (C ǫ /CD ) L

= max

2 KCL, (CLǫ /CD )

ǫ CD0 = 2−ǫ max

CL,(C ǫ /CD )

⇒

L

= max



ǫ CD0 2−ǫ K (3)

which gives the drag coefficient CD, (C ǫ /CD ) L

max

= CD0 + CDi, (C ǫ/CD ) L

= max

2 CD0 2−ǫ

(4)

The corresponding maxima in the lift-to-drag ratios are 

 C Lǫ  ǫ CD0 ǫ/2 2 − ǫ = 2−ǫ K 2CD0 CD max 

(5)

These results are summarized in table 1. Now assume linear characteristic in the CL -α curve. Then CL =

dCL α = CLα α dα

(6)

Note that this implies that the angle of attack is measured from the zero-lift line, i.e. α is the so called absolute angle of attack. Also note that the slope CLα of the CL -α curve is a constant for a given airplane fixed by its geometry if linearity is assumed in the CL -α curve. From this and the analysis above the angles of attack giving the maxima in the lift-to-drag ratios is obtained 1 α(CLǫ /CD ) = max CLα



ǫ CD0 2−ǫ K

(7)

1/2

2/3

3/2

3

THE LIFT-TO-DRAG RATIOS CL /CD , C L /CD , CL /CD AND CL /CD

and the magnitude of the maxima are obtained from eq. (5) above. For a thin airfoil in incompressible flow the theoretical value on CLα is 2π radian−1 . The value on this slope for real airfoils typically differs from this by up to ±5%. Methods to estimate the slope CLα of the CL -α curve for different wing configurations and speeds are found on pp. 81-92 in [1]. From fig. 2.36 and in the discussion of this figure in the text it is also argued that the lift is essentially the same for the whole airplane as for the wing alone. In level flight the lift L must balance the weight W of the airplane. Hence the lift coefficient is L 2W CL = 1 2 = (8) ρV S ρV 2 S 2 With this expression for the lift coefficient the lift-to-drag ratios can be written CLǫ 2W = ρV 2 S CD 

or C Lǫ = CD



2W ρS

ǫ ⎡

⎣CD0

ǫ ⎡ ⎣CD0 V 2ǫ

+K

+K





2W ρV 2 S

2W ρS

2

V

2 ⎤−1 ⎦

⎤ −1

−2(2−ǫ) ⎦

(9)

The velocities giving the maxima in the lift-to-drag ratios is obtained after combining eq. (3) and eq. (8). These velocities are V(C ǫ /CD ) L

= max

  

2 ρ

2−ǫ K ǫ CD0



W S

1/2

(10)

Note the influence from altitude (through the density ρ) and the wing loading W/S on this speed.

References [1] Anderson Jr., J.D.: Aircraft Performance and Design, McGraw-Hill, 1999. [2] Karlsson, A.: Cruise Performance, 2004. [3] Raymer, D.P.: Aircraft Design: A Conceptual Approach, 2nd Ed., AIAA Educational Series, 1992.

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