Topology of Lift and Drag

In deRham's theory, a closed 1-form, A, consists of two parts: a perfect differential and a harmonic part. In three dimensions, such covariant vector fields consist of a pure gradient (with zero curl, but perhaps finite divergence), and a non-gradient harmonic part (that has zero divergence and zero curl). Such flows have zero vorticity. The harmonic part cannot exist on a topological domain which is simply connected.

The two dimensional theory of Lift (such as that championed by Joukowski) considers the topologically non-simply connected domain exterior to an airfoil of arbitrary shape. The Lift force is orthogonal to the velocity field (similar to a Lorentz or so called Magnus force) such that the power (force times velocity) is zero. Joukowski showed that the Lift is due to the line integral of the velocity along any closed path in the exterior domain that encloses the wing. The value of such an integral is defined as the net circulation, and as Joukowski proved, depends entirely on the harmonic component of the vector field, and is
independent of the airfoil shape.

If the harmonic component of the 1-form vanishes so does the Lift. In two dimensions, the shape of the airfoil does nothing except to insure that a translation from an initial condition at rest to a constant translational velocty will induce a harmonic component of the flow vector field. Any mechanism that will produce the same harmonic component will produce the same Lift.

In two dimensions such flows have zero vorticity and zero drag. In three dimensions, a wing of finite aspect ratio will induce a flow component which has finite curl, or vorticity, which leads to the generation of tip vortices and induced drag. High performance sail planes have very high aspect ratios just to minimize this topological effect.

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