Relationship of Classical Airfoil Theory to Gauss’s Principle and Variational Optimization

This article extends an earlier study to further explore the relationship of Gauss’s principle of least constraint and variational optimization to inviscid, incompressible fluid mechanics and classical airfoil theory. Development of the special and general equivalence theorems that relate flowfield velocity solutions to Euler’s equation is discussed, and important differences between them are delineated. The Lagrange multiplier theorem is introduced to define rigorously the constraint and impressed pressure components that are key for applying Gauss’s principle to fluid mechanics. Closed-form results from thin airfoil theory are used to demonstrate the role of the impressed and constraint pressure components for various airfoils. Earlier arguments disproving a new variational theory of lift derived from Gauss’s principle are extended.