A variational theory of lift

Abstract

In this paper we revive a special, less-common, variational principle in analytical mechanics (Hertz’ principle of least curvature) to develop a novel variational analogue of Euler's equations for the dynamics of an ideal fluid. The new variational formulation is fundamentally different from those formulations based on Hamilton's principle of least action. Using this new variational formulation, we generalize the century-old problem of the flow over a two-dimensional body; we developed a variational closure condition that is, unlike the Kutta condition, derived from first principles. The developed variational principle reduces to the classical Kutta–Zhukovsky condition in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the Kutta condition being a manifestation of viscous effects. Rather, we found that it represents conservation of momentum. Moreover, the developed variational principle provides, for the first time, a theoretical model for lift over smooth shapes without sharp edges where the Kutta condition is not applicable. We discuss how this fundamental divergence from current theory can explain discrepancies in computational studies and experiments with superfluids.

Can lift be generated in a steady inviscid flow?

“…Gonzalez and Taha [2] proposed a variational theory of lift by applying Hertz' principle of least curvature that is rarely used in physics to inviscid flow to determine the airfoil circulation as an alternative to the Kutta condition. It is claimed that this theory challenges the accepted wisdom about the Kutta condition being a manifestation of viscous effects.…”

“…Their main argument is that the lift can be determined in inviscid flow and the role of the fluid viscosity is not critical in lift generation. Here, it is necessary to examine the physical meanings of the variational principle proposed by Gonzalez and Taha [2].…”

“…Applying Hertz' principle of least curvature in analytical mechanics to steady inviscid flow, Gonzalez and Taha [2] proposed the functional where a is the acceleration of fluid, Ω is a fluid domain, and x = (x 1 , x 2 ) are the 2D Cartesian coordinates. In Eq.…”

“…A complete and focused clarification on this question is lacking. Due to the recent publication by Gonzalez and Taha [2] who challenged the conventional understanding of the role of the fluid viscosity in lift generation, this intriguing question attracts a considerable renewed attention, which should be examined carefully in some technical aspects related to lift generation.…”

Low Reynolds Number Wing Design for Unmanned Aerial Vehicle: A Case Study

“…Lift force for example has got many theories of lift for over a hundred years now based on Newton's 3 rd law, Bernoulli's principle, Kutta-Zhukovsky condition for sharp edge airfoils (Hoffman and Johnson, 2008;Babinsky, 2003;Coutinho, 2014;McLean, 2012). Moreover, the amazing work of the team at the University of California with a new theory of lift "a variational theory of lift" seems to solve a great part of the puzzle and will create new chapters of aerodynamic books (Gonzalez and Taha, 2022). With all these variables, the design of the UAV's wing can take place within the same Reynolds number range and similar mission of an existing aircraft design method based on aircraft design references.…”