Lifting-line theory for a wing in non-uniform flow

Abstract

Prandtl's theory of the lifting line gave the answer to most of the questions in the aerodynamic design of airplane wings. Thus the three-dimensional wing theory became a standard tool of airplane designers. One restriction involved in the conventional wing theory is the uniformity of the undisturbed flow in which the wing is placed. Now there are many important cases which do not satisfy this condition. For instance, in the case of a wing spanning an open jet wind tunnel, the velocity of the air stream has a maximum at the center of the jet and drops to zero outside of the jet. Another example is the problem of the influence of the propeller slip-stream on the characteristics of the wing. Here the higher velocity of the propeller slip-stream makes the application of the Prandtl wing theory difficult. Such cases led several authors to investigate the problem of a wing in non-uniform flow. Some investigators found a satisfactory solution of the problem for the case of “stepwise” velocity distribution. In this case the flow in regions of uniform velocity can be determined by using Prandtl's concepts with additional continuity conditions at the boundaries between such regions. On the other hand, the problem of a continuously varying velocity field seems to need an appropriate treatment. K. Bausch [1] has tried to modify the Prandtl theory for the case of small inhomogeneity in the air stream; however, besides the restriction of slight deviation from uniform flow, his method encounters a further difficulty in estimating the error introduced by the approximations. The seriousness of this difficulty becomes evident when one tries to compare the results of Bausch with that of F. Vandrey [2] . Vandrey considers the problem with variable velocity as the limiting case of a wing in a stepwise velocity field, and his result seems to differ from that of Bausch. Recently R. P. Isaacs [3] has investigated the same problem, but the authors have not yet had the opportunity to study his work.