The role of conservative forces in rotor aerodynamics

Abstract

AbstractThe theory to predict the performance and loads on rotors (propellers, screws, windmills) has a history of more than a century. Apart from modern computational fluid dynamics and vortex panel models taking the true blade geometry into account, most other models proceed from an infinitely thin actuator disc or line. These models assume an externally defined force field distributed at the disc or line, representing the loads on the real rotor. Given this force field, the flow is solved by momentum balances or by the equations of motion. The use of external force fields was discussed in textbooks of the first decades of the 20th century, but has received little attention since then. This paper investigates the higher-order effect of adding thickness to the actuator disc or changing the actuator line to a blade with cross-sectional dimensions. For the generation of a Rankine vortex by a force field acting on an actuator disc with thickness, an exact solution has been found in which not only the thrust and torque determine the flow, but also a radial force. This force is conservative, in contrast to the other force components. For rotor blades, a conservative normal and radial force acting on the chordwise bound vorticity is present. This explains the experimentally observed inboard motion of the tip vortex of model wind turbine rotors before the wake induction field drives it outboard. Simulations by computational fluid mechanics and a vortex panel code reproduce the inboard motion, but an actuator line analysis, in which the chordwise vorticity is absent, does not. The conservative load is only $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1\mbox{--}2\, \%$ of the thrust on the entire blade but ${\approx }10\, \%$ of the thrust at the tip ($r/R>0.9$). Conservative forces at the disc and rotor blade vanish for vanishing disc thickness or blade cross-section, so play no role in any of the infinitely thin actuator disc or line methods. However, if higher-order effects of non-zero dimensions are to be modelled, the conservative force field has to be included.