Potential flow solutions for energy extracting actuator disc flows

Abstract

The actuator disc is the oldest representation of a rotor, screw or propeller. Performance prediction is possible by applying momentum theory, giving integrated values for power and velocity. Computational fluid dynamics has provided much more flow details, but a full potential flow solution zooming in on these flow details was still absent. With the wake boundary discretized by vortex rings, flow states for energy extracting discs have been obtained for thrust coefficients up to 0.998. Boundary conditions are met with an accuracy of a few . Results from momentum theory are confirmed.Most rotor design codes use momentum theory in annulus or differential form, assuming that the axial velocity v x at the disc is uniform. However, the absolute velocity jvj is found to be uniform, and arguments for this are presented. The non-uniformity of v x is an inherent part of the flow solution caused by, in terms of momentum theory, the pressure acting at the annuli. This makes the annuli not independent from each other as assumed in current design codes. Although this was already known, it is now confirmed up to the highest thrust coefficients.Optimizing a rotor design should be carried out for the non-uniform distribution of v x . To enable this, an equation for the non-uniformity as function of thrust and radial position is presented, being a surface-fit to the calculated data. Qualitatively, the non-uniform distribution does the same as the Prandtl-Glauert-Shen tip correction applied to a uniform distribution.

Verifying the Blade Element Momentum Method in unsteady, radially varied, axisymmetric loading using a vortex ring model

“…For Case I (0), v z calculated from the FWVR model matches with that from Momentum Theory, but it is not as uniform as Momentum Theory predicts, especially at the edge of actuator disc, which was also found by Sørensen and van Kuik. 7,8,11 For Case I (a) and (b), the locally changed v z captured by the FWVR model matches generally well with that from Momentum Theory. Figure 4 shows the axial velocity v z and radial velocity v r obtained from the FWVR model at the actuator disc plane for Case I (0),(a),(b).…”

“…Madsen 9 proved that the momentum strip theory of BEM is unacceptable, especially at the tip and the root regions, using a near wake model. 10 The annuli were shown to be not independent; the pressure at the boundary of the annuli were suggested by van Kuik and Lignarolo 11 to be included in the momentum theory to remedy the effect of annuli independent assumption. Conway 12 obtained a linear solution for an actuator disc with steady heavy loads, which was extended to a semi-analytical non-linear method to consider certain distributions of radially varied loads.…”

An actuator disk method with tip‐loss correction based on local effective upstream velocities

“…This can be traced to the contribution of the pressure component acting on the lateral boundaries of the annular streamtubes to their momentum balance. 2,32 This pressure term is usually neglected when the whole disk is considered, because the control volume is entirely surrounded by a uniform pressure, but this contribution does not vanish in a discrete analysis. This increases the induction on the outboard part and decreases it on the inboard part; Equations 17 and 18 should then be modified accordingly to account for this pressure term.…”

“…Modifications have been proposed to account for the physical mechanisms causing the deviation mentioned above. In the work of van Kuik et al, 32 an annulus-correction model was derived based on a potential flow model for the wake. This correction is a function of the radial position and the global C T and is applied on the axial velocity.…”

Effects of the Approximations Embodied in the Momentum Theory as Applied to the NREL PHASE VI Wind Turbine

“…Due to the large diffusion of the BEM methods, the evaluation of the MT errors is of the utmost importance; consequently, a few works dealing with this issue have recently appeared [24,[36][37][38][39][40][41]. These studies present the comparison between the MT and AD methods based on flow equations not polluted by the typical MT approximations, e.g., CFD-AD and SA-AD.…”