The previously described method applies to two-dimensional airfoils, but can be used effectively in estimating the drag rise Mach number of wings when the effects of sweep and other 3-D effects are considered.
In Figure 7 the mean thickness ratio t/c is the average t/c of the exposed wing weighted for wing area affected just as the mean aerodynamic chord, MAC , is the average chord of the wing weighted for wing area affected. The mean thickness ratio of a trapezoidal wing with a linear thickness distribution is given by:
t/cavg = (troot + ttip) / (Croot + Ctip)
This equation for t/cavg is based on a linear thickness (not linear t/c) distribution. This results from straight line fairing on constant % chord lines between airfoils defined at root and tip. The same equation is valid on a portion of wing correspondingly defined when the wing has more than two defining airfoils. The entire wing t/cavg can then be determined by averaging the t/cavg of these portions, weighting each t/cavg by the area affected. Note that Croot and Ctip are the root and tip chords while troot and ttip are the root and tip thicknesses. b is the wing span and y is the distance from the centerline along the span.
Almost all high speed subsonic and supersonic aircraft have sweptback wings. The amount of sweep is measured by the angle between a lateral axis perpendicular to the airplane centerline and a constant percentage chord line along the semi-span of the wing. The latter is usually taken as the quarter chord line both because subsonic lift due to angle of attack acts at the quarter chord and because the crest is usually close to the quarter chord.
Figure 8. Velocity Components Affecting a Sweptback Wing
Sweep increases Mcc and MDiv. The component of the freestream velocity parallel to the wing, V||, as shown in figure 8 does not encounter the airfoil curvatures that produce increased local velocities, reduced pressures, and therefore lift. Only the component perpendicular to the swept span, Vn , is effective. Thus on a wing with sweep angle, L:
V0eff = V0 cos L
M0eff = M0 cos L
q0eff = q0 cos2 L
The meaningful crest critical Mach number, Mcc, is the freestream Mach number at which the component of the local Mach number at the crest, perpendicular to the isobars, first reaches 1.0. These isobars or lines of constant pressure coincide closely with constant percent chord lines on a well-designed wing.
Since q0effective is reduced, the CL based on this q and the Cp at the crest, also based on qoeffective will increase, and Mcc and MDiv will be reduced. Furthermore, the sweep effect discussion so far has assumed that the thickness ratio is defined perpendicular to the quarter chord line. Usual industry practice is to define thickness ratio parallel to the freestream. This corresponds to sweeping the wing by shearing in planes parallel to the freestream rather than by rotating the wing about a pivot on the wing centerline. When the wing is swept with constant freestream thickness ratio, the thickness ratio perpendicular to the quarter chord line increases. The physical thickness is constant but the chord decreases. The result is a further decrease in sweep effectiveness below the pure cosine variation. Thus, there are several opposing effects, but the favorable one is dominant.
In addition to increasing Mcc, sweepback slightly increases the speed increment between the occurance of Mach 1.0 flow at the crest and the start of the abrupt increase in drag at MDiv. Using a definition for MDiv as the Mach number at which the slope of the CD vs. M0 curve is 0.05 (i.e. dCD/dM = 0.05), the following empirical expression closely approximates MDiv:
MDiv = Mcc [ 1.02 +.08 ( 1 - Cos L ) ]
The above analysis is based on two-dimensional sweep theory and applies exactly only to a wing of infinite span. It also applies well to most wings of aspect ratio greater than four except near the root and tip of the wing where significant interference effects occur.
The effect of the swept wing is to curve the streamline flow over the wing as shown in Figure 9. The curvature is due to the deceleration and acceleration of the flow in the plane perpendicular to the quarter chord line.
Figure 9. Stagnation Streamline with Sweep
Near the wing tip the flow around the tip from the lower to upper surface obviously alters the effect of sweep. The effect is to unsweep the spanwise constant pressure lines known as isobars. To compensate, the wing tip may be given additional structural sweep, Figure 10.
Figure 10. Highly Swept Wing Tip
It is at the wing root that the straight fuselage sides more seriously degrade,the sweep effect by interfering with curved flow of figure 9. Airfoils are often modified near the root to change the basic pressure distribution to compensate for the distortions to the swept wing flow. Since the fuselage effect is to increase the effective airfoil camber, the modification is to reduce the root airfoil camber and in some cases to use negative camber. The influence of the fuselage then changes the altered root airfoil pressures back to the desired positive camber pressure distribution existing farther out along the wing span.
This same swept wing root compensation can be achieved by adjusting the fuselage shape to match the natural swept wing streamlines. This introduces serious manufacturing and passenger cabin arrangement problems so that the airfoil approach is used for transports. Use of large fillets or even fuselage shape variations is appropriate for fighters. The designing of a fuselage with variable diameter for transonic drag reasons is sometimes called 'coke-bottling'. At M= 1.0 and above, there is a definite procedure for this minimization of shock wave drag. It is called the "area rule" and aims at arranging the airplane components and the fuselage cross-sectional variation so that the total aircraft cross-sectional area, in a plane perpendicular to the line of flight, has a smooth and prescribed variation in the longitudinal (flight) direction. This is discussed further in the section on supersonic drag.
Figure 11. 'Coke-Bottled' Fuselage
The estimates provided by Figure 7 and the equation for MDiv assume that the wing root intersection has been designed to compensate for the 'unsweeping' effect of the fuselage either with airfoil or fuselage fairing treatment. If this is not done, MDiv will be reduced or there will be a substantial drag rise at Mach numbers lower than MDiv. For all aircraft there is some small increase in drag coefficient due to compressibility at Mach numbers below MDiv as illustrated in Figure 4.