The Edge Singularity of an Actuator Disc With a Constant Normal Load

Abstract

All rotor and propeller design methods using momentum theory are based on the concept of the actuator disc, formulated by Froude. In this concept, the rotor load is represented by a uniform pressure jump. This pressure jump generates infinite pressure gradients at the edge of the disc, leading to a velocity singularity. The subject of this paper is the characterization of this velocity singularity assuming inviscid flow. The edge singularity is also the singular leading edge of the vortex sheet emanating from the edge. The singularity is determined as a simple bound vortex of order O(1), carrying an edge force Fedge = −ρ Vedge × Γ. The order of Fedge equals the order of Vedge. This order is determined by a radial momentum analysis. The classical momentum theory for actuators with a constant, normal load Δp appears to be inconsistent: the axial balance provides a value for the velocity at the actuator, with which the radial balance cannot be satisfied. The only way to obtain consistency is to allow the radial component of Fedge to enter the radial balance. The analysis does not resolve on the axial component of Fedge. A quantitative analysis by a full flow field calculation has to assess the value of Fedge for the various actuator disc flow states. Two other solutions for the edge singularity have been published. It is shown that both solutions do not comply with the governing boundary conditions.

Impulsively started planar actuator surfaces in high-Reynolds-number steady flow

“…By τ ∼ 10 the sections of the vortex sheets not entrained in the spirals have overtaken the dipole system and meet each other on or near the centreline. After τ = 10 the vortex sheets are in close proximity and instabilities, originating in the tip spiral (see Moore 1974;Pullin 1978;van Kuik 2003van Kuik , 2004a, grow and spread along the sheets. Referring to figure 4 the opposing vortex sheets approaching each other is a similar scenario to the meeting of boundary layers behind a cylinder in impulsively started flow; small, unstable waves form in the vortex sheets, causing them to break down into coherent vortex pairs.…”

“…We define the origin at the edge of the actuator surface, as in figure 2, and analyse the infinitesimal area A = π 2 . Stokes' theorem can be applied to a closed contour, C, enclosing the edge area A: Johnson, A. Wojcik, K. R. Drake and I. Eames This derivation is the same as in van Kuik (1991Kuik ( , 2003 except that the surface behaviour in (3.1) and (3.2) is coupled to the time-dependent flow field it induces. Equation (3.2) states that fluid particles passing through the surface edge will gain circulation and this is illustrated in figure 2.…”

A comparison of nonlinear actuator disk methods for the performance analysis of ducted marine propellers

“…Since its first appearance in the momentum theory of Rankine 1 and Froude, 2 several research works dealing with the actuator disk method have been carried out in the marine/aeronautical propellers and in the wind/tidal turbines technological fields (see, for instance, Glauert,3 Lanzafame and Messina, 4 Sørensen, 5 Hansen et al, 6 Oosterveld,7 but this list should not be regarded as exhaustive). Also, really recent developments of the theory are available in the literature like those due to van Kuik et al, 8 van Kuik, 9,10 and Sørensen and Mikkelsen. 11 In 1962, Wu 12 derived the governing equation of the inviscid through-flow field around an open actuator disk in terms of the Stokes stream function and laid the theoretical foundations of several successive works.…”

“…Since s () is known, the wake can be meshed through a n zs  n rs grid, where n zs and n rs are the number of axial and radial stations, respectively. The Stokes stream function at all mesh points can be now obtained by numerically integrating equation (17), so that the tangential vorticity at the grid points can be evaluated by means of equation (10). Once !…”

Moving actuator surfaces: a new concept for wind turbine aerodynamic analysis

“…Especially in operation where the blade is partly, or completely stalled, predictive models meet major difficulties to ensure accuracy: reasons for this poor performance can be attributed to the model itself, or to the inputs provided to the model, like the airfoil aerodynamics characteristics. Amongst the sources of discrepancies, the way models handle blade tip vortices is recognized as a key issue by many workers in the field [2,3,4]. Theoretically, only vortex [18] or full CFD [19] analyses naturally model the tip vortices and its interaction with the flow.…”