On the generation of vorticity by force fields in rotor- and actuator flows

Abstract

In most rotor design methods, the blade load is found by a blade element analysis in an iterative procedure with flow solvers like actuator disc and -line analyses as well as momentum balances. For the flow solvers the force field is the input. In most other aerodynamic analyses the force field is the output result instead of input. This is done by applying boundary conditions at the lifting surface with which the flow is solved and the pressure at the surface, so the load, is determined (only inviscid flows are considered here). Both approaches are consistent, but appear to differ with respect to the generation of vorticity. In the lifting surface approach, usually Helmoltz's laws are used to show that bound and free vorticity is conserved instead of being generated, while in the force field approach vorticity is generated instead of conserved. It is shown that both methods are consistent since sometimes Helmholtz's laws are incorrectly referred to. These laws have been derived in absence of non-conservative forces, while the surface pressure distribution is shown to be such a force field. Besides this, the question is discussed how a force field creates vorticity in an inviscid flow, since some papers consider viscosity to be necessary to generate vorticity. A literature study contradicts this, showing that in inviscid flows vorticity is generated by tangential pressure gradients or, equivalently, a non-uniform force field. This makes the Euler equation including the force field term well suited to express the generation of vorticity in characteristics of the force field. A comparison of the convection of vorticity in the wake of a disc, rotor blade and wing shows several differences. The azimuthal vorticity in the disc wake does not depend on vorticity conservation laws, in contrast to the axial and radial components. For a rotor and wing all components are governed by vorticity conservation.