Lift-Dependent Drag Items

The total drag coefficient includes the parasite drag and other components:
C_D = C_{D_p} + C_{D_vortex} + C_{D_{lift-dependent viscous}} + C_{D_{compressibility}}

This is sometimes written:
C_D = C_{D_p} + C_{D_i}+ C_{D_c}

The second term, is often called the induced drag, but it includes more than just the invicid drag associated with induced velocities from the wake. For purposes of this analysis, the "induced" drag is customarily divided into viscous and inviscid parts. The inviscid (vortex) drag includes a zero-lift term due to twist, and lift-dependent parts that depend on the twist and planform. The remaining portion of the "induced" drag, the so-called viscous part, is chiefly due to the increase of skin friction and pressure drag with changes in angle of attack. Such increases come about because of the increased velocities on the upper surface of the wing leading to higher shear stresses and more severe adverse gradients with corresponding increase in pressure drag. As in the case of parasite drag, the "induced" drag also includes several miscellaneous effects not accounted for in a simple theoretical study. Additional empirically-estimated terms arise from fuselage vortex drag, nacelle-pylon interference, changes in trim drag with angle of attack, and a change in drag due to engine power effects (either inlet or exhaust).

Inviscid Part

For twisted wings, the inviscid drag may be written:

The last term is present at zero-lift and is the zero-lift drag due to twist. The first term is the vortex drag associated with the untwisted wing.

The factor, s, accounts for the added lift-dependent drag caused by the modification of the span loading due to the addition of the fuselage. Its value is presented in the figure below for various ratios of the fuselage width (or diameter) to wing span. The values of this factor are obtained from a solution for the minimum induced drag of a lifting line in combination with a circular fuselage of infinite length and at zero angle of attack. A simple explanation of this effect is available for interested readers using this link. Although the analysis was made only for a mid-wing location, the results are used in this method for all wing locations. These results are probably slightly conservative for application to low-wing designs. However, the use of these results for wing installations with large root incidence angles does not fully predict the detrimental effect of the fuselage on the wing span-load distribution. It has been shown that with large wing incidences there is a much greater deficiency in the lift "carry-over" on the fuselage. The value of 's' is usually between .965 and .985. For initial studies assuming a value of .975 will lead to no more than a 1% error in lift dependent drag, but if the chart below is available, use that.

Apart from this factor, the expression for inviscid inviscid drag of the wing alone shows how planform and twist affect the drag. Simple finite wing theory shows that if the distribution of lift over the wing is elliptical, the inviscid drag is minimized with a given span, lift, and flight condition. We can make the span loading nearly elliptical with suitable choices of wing planform and twist and so should be able to approach the ideal minimum value quite closely. We can use the expression above, in fact, to solve for the twist angle that produces the minimum C_{D_i}for a given planform. Generally, twists that are somewhat greater than that required for minimum induced drag are used. This is often done to improve handling or reduce induced drag at low speeds. Thus, the total inviscid drag is somewhat greater than the ideal minimum: C_L² / p AR.

For most transport-like configurations taper ratios are chosen in the 0.20 to 0.35 region where the value of u is close to 0.99. (The values of u, v, and w depend only on the planform.) The lift-dependent twist term can actually contribute a negative drag increment. If the taper and sweep are higher than ideal, for example, the wing can be "washed-out" (negative twist) to bring the loading closer to elliptical.

Rather than evaluate the u, v, and w terms in the expression above, designers generally now rely on computations of a specific wing planform and twist distribution to estimate the vortex drag. If a wing-body analysis code is available, the lift carry-over can be estimated well and there is no need for th fuselage s-term either. In many cases, though, initial wing design studies will be performed without the fuselage and the fuselage correction factor, s, appled to these results.

Trim Drag

When the tail of an airplane carries some load, several drag components are increased: the tail itself has vortex drag and lift-dependent viscous drag, but the lift of the wing must be changed to obtain a specified airplane C_L:

C_{L_Airplane} = C_{L_Airplane} + C_{L_tail} (S_tail / S_wing)

The increase in wing C_L means that the wing vortex and lift-dependent viscous drag increases. In addition, wing compressibility drag is affected.

To compute this, we first must calculate the lift carried by the tail. For most transport aircraft without active controls this is about 5% of the airplane lift, but in the wrong (downward) direction. We could then compute the vortex drag of the combined wing/tail system and then add in viscous and compressibility increments. The difficulty with this is that unless we know the airplane center of gravity (CG) location, we cannot compute the tail load and in the early stages of the analysis, we do not know the airplane CG location. Sometimes we make rough estimates of the CG. When this is not possible, we can rely on more detailed computations done on other aircraft which show trim drag of about 1% to 2% of airplane drag. Airplane designs can easily be created with very high trim drag values, though. We will discuss this in connection with tail design in subsequent chapters.

Viscous Part

Over most of the flight regime of interest, the viscous part of the "induced" drag may be approximated by a parabolic variation with C_L. Thus we write:
C_{D_{i_viscous}} = K C_{D_p} C_L²

Ideally, this drag contribution should be estimated for the individual airplane components, with factors such as the influence of wing leading edge geometry, wing camber, wing thickness ratio, wing sweep, pylon interference, fuselage upsweep, tail induced drag, power effects, etc. taken into account. Since the information required to do this usually does not exist in preliminary design, it is assumed that a new airplane will be similar enough to previous airplanes that the viscous part of the lift-dependent drag can be represented by the equation above, with the K factor determined from previous flight test data. The wing contribution, including the effect of sweep, is included separately from the contributions of the other components. The form of the expression for lift-dependent viscous drag may be derived by combining simple sweep theory with the equation for airfoil supervelocities due to circulation.

The value of the factor K has been determined from flight test data for the DC-8-62 and 63 and for the DC-9-10, -20, and -30 airplanes to be approximately 0.38.

When each of these effects is added together, the total drag is seen to vary quadratically with C_L. In fact, apart from the lift dependent twist term, the drag polar is a parabola and would form a straight line when plotted vs. C_L² . Since the lift-dependent twist term is usually very small, we expect that the C_D vs. C_L² will be nearly straight. This is often the case. The drag polar can thus be approximated, over most of the range of interest by the two-parameter expression:

'e' is a parameter which expresses the total variation of drag with lift. It is sometimes called the span efficiency factor or Oswald efficiency factor after Dr. W.B. Oswald who first used it. It would be 1.0 for an elliptically-loaded wing with no lift-dependent viscous drag, but for practical aircraft 'e' varies from about 0.75 to 0.90.

We can predict the value of 'e' by computing the inviscid drag from a lifting surface method and adding the lift-dependent viscous drag:
C_D = C_{D_p} + C_{D_{i_inviscid}} + K C_{D_p} C_L² = C_{D_p} + C_L² / ( p AR e_inviscid ) + K C_{D_p} C_L²

So if C_D = C_{D_p} + C_L² / ( p AR e)

then:

The figure below shows a typical variation in e with aspect ratio, sweep, and C_{D_p} . The chart was constructed by assuming u = 0.99 and s = 0.975, and it works quite well, although the calculation should be done in detail for a specific airplane. C_{D_p} for jet transports typically varies from about .0140 for aircraft with small ratios of body wetted area to wing wetted area (707 or DC-8) to .0210 for short range aircraft with a relatively large fuselage. The wide-body tri-jets lie in the middle of this range. Note that this plot shows typical values, the actual value of 'e' for a particular airplane should be computed as described above.

Aircraft with wing-mounted propellers have a further reduction in 'e' due to the downwash behind inclined propellers. The exact effect is difficult to calculate but a reduction of about 4% is reasonable.