On (non-)conservative body forces, vorticity generation and energy conversion in ideal flows

Abstract

Usually, the load on lifting bodies in incompressible, inviscid flow is determined by integration of the pressure on the body surface, once the flow is solved. Prandtl (Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch- Physikalische Klasse, vol. 1918, pp. 451-477) proposed an opposite method in which the body force field is the source term in the equation of motion to solve the flow. This force field method has not been used intensively but has regained importance in modern wind energy research. However, an analysis of which type of body force field generates vorticity and converts energy, and which body force field does not, is lacking. Prandtl's method is adopted here, but with the addition that the force field is allowed to be conservative or non-conservative. The relation between conservative/non-conservative body forces, vorticity generation, potential and kinetic energy and Helmholtz's vorticity theorems are derived. Similarly, the load on lifting bodies in two and three dimensions is classified as (non-)conservative, with some examples. To show that the force field method is consistent with the method where the load is output of an analysis, the expression for the Kutta-Joukowsky load and the relation between bound and trailing vorticity of a wing have been rederived using the force field method. The analysis confirms that this relation between bound and trailing vorticity is not governed by Helmholtz's theorems, as is often assumed, but by the non-conservative force field generating vorticity.