7 - Perturbation Methods

Summary

For the small-disturbance solution techniques that are treated in this book, approximations to the exact mathematical problem formulation are made to facilitate the determination of a solution. Since for incompressible and irrotational flow the governing partial differential equation is linear, the approximations are made to the body boundary condition. For example, for the three-dimensional wing in Chapter 4, only terms linear in thickness, camber, and angle of attack are kept and the boundary condition is transferred to the x–y plane. The solution technique is therefore a “first-order” thin wing theory.

The small-disturbance methods developed here can be thought of as providing the first term in a perturbation series expansion of the solution to the exact mathematical problem and terms that were neglected in determining the first term will come into play in the solution for the following terms. In this book we will follow the lead of Van Dyke and use the thin-airfoil problem as the vehicle for the presentation of the ideas and some of the details of perturbation methods and their applicability to aerodynamics. First, the thin-airfoil solution will be introduced as the first term in a small-disturbance expansion and the mathematical problem for the next term will be derived. An example of a second-order solution will be presented and the failure of the expansion in the leading-edge region will be noted. A local solution applicable in the leading-edge region will be obtained and the method of matched asymptotic expansions will be used to provide a solution valid for the complete airfoil.