Analysis and evaluation of the momentum theory errors as applied to propellers R. Bontempo and M. Manna * Dipartimento di Ingegneria Industriale, Universit` a degli Studi di Napoli Federico II, Via Claudio 21, 80125 Naples, Italy Abstract The paper offers an analytical formulation of the two errors embodied in the momentum theory. The first one originates from to the use of the differential form of the axial momentum equation and the second one from the linearisation of the tangential velocity terms. Both errors are evaluated comparing the axial velocity at the disk as predicted by the momentum theories with that one obtained thorough a semi-analytical actuator disk method based on the exact solution of the flow. Several cases characterised by different values of the thrust and advance coefficient are analysed, and the range of validity of the momentum theories is discussed in depth. Keywords: axial momentum theory; generalised momentum theory; actuator disk. 1 Introduction Nowadays, several numerical methods are available for the performance analysis of propellers. Among others, lifting line [1–4], lifting surface [5–10], panel methods [11– 15] and, obviously, CFD based methods [16–27] are the most popular ones. Despite this great variety of models, the so-called Blade Element/Momentum theory (BEM), which originates from the pioneering works of Rankine [28], Froude [29], Froude [30] and Drzewiecki [31], still constitutes the most employed analysis method for open rotors. A few recent works dealing with the BEM the- ory are provided in the bibliography [32–43], but the list should not be regarded as exhaustive. The great success of this method is mainly due to its formidable simplic- ity, robustness and low computational cost. Moreover, as shown by Gur and Rosen [44], more advanced meth- ods do not show a significant accuracy enhancement in the performance prediction when flight conditions near to the design point are analysed. Actually, the BEM relies on the combined use of two different theories: the momentum and the blade element theory. The first of these two theories, i.e. the momen- tum theory, returns the flow field induced by an actuator disk modelling the propeller. From the flow field, the angle of attack can be evaluated along the rotor span, so that the thrust and torque distributions can finally be obtained through the Blade Element Theory (BET). Two different variants of the momentum theory exist. In the first one, the so-called Axial Momentum Theory (AMT), the effects of the wake rotation are completely neglected, while in the second variant, the Generalised Momentum Theory (GMT), these effects are taken into account. However, both in the AMT and GMT, the axial veloc- * Corresponding author: e-mail address: mar- cello.manna@unina.it; tel: +39-081-7683287 ity at the disk, which has to be employed to obtain the angle of attack along the blade and ultimately the pro- peller performance characteristics, is evaluated by means of simplifying assumptions or even through wrong equa- tions. For example, as stressed by Glauert [45], the axial momentum equation is wrongly applied in its differential form (see also Rosen and Gur [46]). Goorjian [47] proved, through an analytical procedure, that this equation is wrong since it leads to a contradiction when combined with the other equations of the GMT. Moreover, the equations appearing in the GMT are customarily simplified by neglecting the second order terms involving the tangential velocity. An analytical evaluation of this error is due to Phillips [48] who anal- ysed the very simplified case of a propeller with uniform axial and angular velocity at the disk and at downstream infinity. He found that, for typical values of the thrust and advance coefficient, an error of order 5% is normally introduced by neglecting the nonlinear swirl terms. Summarising two kinds of error are typically intro- duced in the momentum theory. The first one is due to the use of the differential form of the axial momentum equation and the second one to the linearisation of the swirl velocity terms. In this paper, both these errors are accurately evaluated by comparing the momentum theo- ries results with those of the semi-analytical (SA) method of Conway [49]. This latter method relies on the exact solution of the flow through an arbitrarily loaded actua- tor disk. However, since the exact solution is provided in an implicit formulation, a semi-analytical and iterative procedure has been implemented to evaluate it [49, 50]. Similarly to [51, 52], a thorough analysis of the errors re- lated to this semi-analytical procedure has been carried out (this analysis is not reported in this paper for the sake of brevity). By doing so, a high accuracy of the es- timate of the momentum theories errors can be ensured. 1 AIAA Journal, Vol. 54, No. 12 (2016), pp. 3840-3848. http://dx.doi.org/10.2514/1.J055131