Fuselage Upsweep Drag
The drag due to the upward curvature of the aft fuselage is the sum of a fuselage pressure drag increment due to the upsweep and a drag increment due to a loss of lift. Because of the loss of lift, the airplane must fly at a higher wing lift coefficient in order to maintain the required net airplane CL. This causes an increase in lift-dependent drag. The total upsweep drag may be written:
The change in drag with CL (i.e. dCD / dCL ) varies with both airplane lift coefficient and Mach number (by virtue of its dependence on the wing compressibility characteristics.) For a first approximation, a single value may be used; 0.04 is typical. The geometric parameter used to correlate upsweep drag with fuselage shape is the vertical displacement of the fuselage centerline in the tail cone above the fuselage reference plane. The vertical position of the center of cross-sectional area is measured, not at the end of the fuselage, but at a point that is located 75% of the total upsweep length. The parameter is thus (h/l).75 lt. This is to minimize the effect of modifications at the very aft end of the fuselage that do not produce much change in the effective upsweep.
The total upsweep drag increment (including each of the two terms discussed previously) increases with the parameter, (h/l).75 lt, according to the following expression, derived from wind tunnel data:
CDpupsweep = 0.075 (h/l).75 lt
The subscript p denotes the fact that this CD is nondimensionalized by fuselage maximum cross-sectional area, rather than reference wing area. To obtain the increment in CD based on wing area, remember to multiply by the ratio of fuselage cross section area to wing area. Typical values of CDpupsweep are around 0.006. This translates into about 0.0007 based on wing area for a DC-9.
Two points are of interest with regard to aft-fuselage upsweep:
1. Tests of fuselage shapes in the absence of the wing yield results that greatly overestimate the magnitude of the upsweep drag.
2. Wind tunnel test results have indicated that the loss of lift due to upsweep is significantly greater than just the download on the aft fuselage, which suggests that there is a flow change over the wing and forward fuselage due to the aft-fuselage upsweep. Also, the net change in pitching moment due to upsweep is an increased nose-down moment instead of a nose-up moment that might be expected. As a result, the loss in lift does not complement the download on the tail that is required to trim the airplane. In fact, the effect of upsweep is to slightly increase the airplane trim drag.