Tangential Force and Suction Force

Tangential force Earlier, in Sec. 1-3-2, the wing-fixed components of the aero­dynamic force, normal force, and tangential force were introduced together with the flow-field-fixed components of lift and drag. For small angles of attack, the normal force is almost equal to the lift, whereas the tangential force deviates considerably from the drag, even for small angles of attack. The tangential force T is taken as positive in the direction from the wing leading edge to the trailing edge. When limiting the angle of attack to small or moderately large values (see Fig. 1-7д), the tangential force coefficient cT = TfAq becomes, with T — —X,

cT = cD – cLa (3-141)

where єis the coefficient of total drag as composed from profile drag and induced drag from Eq. (3-125). By introducing cD from Eq. (3-125b) and a = (daldcL)cL into Eq. (3-141), the tangential force coefficient for elliptic circulation distribution becomes

(3’142)

With dcLjda from Eq. (3-8Ob) for the simple lifting-line theory and from Eq. (3-98) for the extended lifting-line theory, Eq. (3-142) yields

cl

CT = cDp–r – (3-143fl)

-f-~1 2 і

ct = cdp—————————— ——Сд (3-143b)

where к = uA/c’Loa. In Fig. 3-58, the difference (cDp — cT)lc2L from Eq, (3-143) is shown against the aspect ratio. Accordingly, for large aspect ratios this difference, and thus the tangential force, are independent of the aspect ratio.

Figure’3-59 illustrates the dependence of the tangential force coefficient on the lift coefficient for wings of various aspect ratios A. The profile drag coefficient had been taken to be cDp « 0.05. It is remarkable that the coefficient of the tangential force assumes negative values when the lift coefficient cL> 0.5. In this case the

Figure 3-58 Tangential force coefficient cj vs. aspect ratio a, (1) Based on the simple lifting-line theory, Eq. (3-143a). (2) Based on the extended lifting-line theory, Eq, (3-143b). Сдр = coefficient of profile drag; = 2ж.

Figure 3-59 Lift coefficient vs. tangential force coefficient Cj for wings of various aspect ratios a, from Eq. (3-1436). For comparison, the drag polars C£>(c^) are also shown.

tangential component of the resultant of the aerodynamic forces is directed upstream along the wing chord. The drag polar curves cD(cL) are also included in Fig. 3-59.

Suction force The discussions about the drag of wings of infinite span of Sec. 24-2 have shown that the flow around the leading edge of an inclined profile produces a suction force in an inviscid fluid (Fig. 2-12a). This is the result of the strong underpressure in the vicinity of the leading edge. Now, the suction force on wings of finite span will be examined. The suction force is a part of the induced drag [Eq. (3-124)], with the total drag being split into profile drag and induced drag. Equation (3-125) is therefore the expression for the drag coefficient with suction force. It has been pointed out in Sec. 24-2 that no suction force exists for very sharp leading edges. In this case, the flow around the leading edge causes local separation, eliminating the strong underpressure that results in a suction force. Rather, the resultant force of the pressure distribution over sharp-edged noses acts normal to the wing surface and, therefore, has the component La in the incident flow direction. Thus the drag coefficients with and without suction force are, respectively,

CD — c Dp + cDi (3-144<z)

cD ~ cDp (3-1446)

The difference of the drag coefficients of Eqs. (3-144<z) and (3-1446) yields the suction force coefficient cs — S/Aqm :

CS ~ cLa~cDi [18]

Comparison with Eq. (3-142) yields

cS = cDp~cT (3-146)

Consequently, the quantity of Fig. 3-58 is a direct measure of the suction force. In particular, cs = CjJ2tt for wings of very large span. This result is in agreement with Eq. (2-77), remembering that the lift coefficient for smooth leading-edge flow cLS is zero for symmetric profiles.

In conclusion, a few experimental results on wings of small aspect ratio according to Hansen [44] will be presented, confirming the above considerations. Figure 3-60 shows polar curves for a slender circular disk. To show the effect of the suction force, the leading edge of the disk was formed in several ways, as can be seen from Fig. 3-60. The suction force increases with the leading-edge nose radius. The theoretical curves for the drag coefficient with and without suction force from Eqs. (3-144л). and (3-144Z?) are added in this figure. The tests show the expected result, namely, that the measured drag coincides with the theoretical curve, including suction forces, when the leading edge is well rounded. When the leading edge is very sharp, however, the measured drag lies close to the theoretical curve without suction force. All measurements with differently formed leading edges lie between the two theoretical curves.