## Thus the above conclusions have been confirmed

Theoretical determination of sidewash Computation of the distribution of the induced sidewash velocity for a known circulation distribution can basically be done like that of the downwash, namely, with the help of the Biot-Savart law. A few qualitative considerations may be noted first. In Fig. 7-40, a symmetric and an asymmetric circulation distribution are compared. Because the circulation distributions have been taken as constant, the symmetric distribution of Fig. 7-40a produces one horseshoe vortex, and the antimetric distribution of Fig. 740b two horseshoe vortices turning in opposite directions. It is immediately obvious that in

the middle plane, у = 0, a down wash velocity —w is obtained for the symmetric circulation distribution but a sidewash ±v for the antimetric distribution, having reversed signs on the upper and lower sides. The latter results essentially from the

counterclockwise-turning “double vortex,” shed in the middle. However, this highly idealized vortex model is insufficient to determine the induced sidewash quantitatively. The computation of the induced sidewash must be based on a variable circulation distribution I'(y), for example, like that for the sideslipping wing – fuselage system of Fig. 7-38. The sidewash velocity very close to the vortex sheet is obtained in analogy to Eq. (2-46л) as

(7-62)

where the upper sign applies above the vortex sheet and the lower sign below. The validity of this equation can also be checked by inspecting Fig. 7-38c and d. There, the slope of the circulation distribution is shown for у = 0, and the sign of the sidewash velocity v is indicated. The induced sidewash angle = v/U„ is obtained from Eq. (7-62) by introducing the dimensionless circulation distribution 7 = rjbUoo and the dimensionless coordinate in the span direction 17 —у/s as

(7-63)

By introducing the expression 7(17) = 7g(v) + Plp(v) for the circulation distribution, where 7g is the distribution in straight flight and (3y@ the additive circulation for sideslipping flight, Eq. (7-63) yields, for the efficiency factor of the vertical tail in the vortex sheet,

(7-64)

The above derivation shows that Eqs. (7-62)-(7-64) are valid for any distance behind the wing for a not-rolled-up vortex sheet.

From Eq. (7-64) it is seen that the efficiency factor changes abruptly in the vortex sheet. The quantity dy^dr] is obtained from the circulation distribution of the sideslipping wing-fuselage system. The y@ distribution for the high-wing airplane is illustrated in Fig. 7-4ІЯ. The determination of the induced sidewash outside the vortex sheet has been studied by Jacobs and Truckenbrodt [14]. By applying the Biot-Savart law, the induced sidewash angle for a given circulation distribution 7(17) is obtained from lifting-line theory as

(7-65)

with r as in Eq. (7-29). For unswept wings and a very large distance (| -> TO), Jacobs [14] gave a simple procedure for the evaluation. The solution for arbitrary wing

planforms has been studied by Gersten [10]. For large distances behind the wing it suffices to use the values for £

In conclusion, results of a few sample computations will be reported. In Fig. 7-41 the induced sidewash field is given for a high-wing system. Figure 7-4la illustrates the geometry and the additive circulation distribution 7^ due to the sideslipping. Figure 741 b represents the streamline pattern of the induced velocity field very far behind the wing, and Fig. 7-4lc gives the distribution of the sidewash factor d&vldj3 as a function of the distance from the vortex sheet for the middle plane 7? = 0. This figure demonstrates the discontinuity of the sidewash factor at the vortex sheet f = fi and the strong drop with distance from the vortex sheet. Figure 7-41d gives the distribution of the sidewash factor in the span direction for several distances from the vortex sheet.

In Fig. 7-42 for a high-wing and for a low-wing airplane the curves of constant local efficiency factor of the vertical tail Ъ§у]Ъ$ = const are shown for the transverse plane at the location of the vertical tail. The total efficiency factor of the

vertical tail is obtained from this through integration over the vertical tail height. The field of the curves 9j3y/9j3 = const is independent of the angle of attack of the airplane. There is, however, a dependence of the efficiency factor of the vertical tail on the angle of attack because, with a change of the angle of attack, the vortex sheet is displaced relative to the vertical tail (see Fig. 7-20). This influence is quite noticeable, as may be seen by comparing the cases cL — 0 and cL = 1 in Fig. 7-42. For the system of wing, fuselage, and vertical tail of Fig. l-39a, Jacobs [14] applied this method to determine the efficiency factors theoretically (Fig. 7-39c). The agreement with measurements in Fig. 7-39b is satisfactory.

The problem area of the interaction of wing, fuselage, and vertical tail at sideslipping has been investigated by Puffert [28]. The concepts established for the induced sidewash have been translated into that for the rolling wing by Michael [23] and by Bobbitt [5].