On the Convergence, Node Collocation, and the Disc-Edge Singularity of a Vortex-Ring/Vortex-Cylinder Free-Wake Model for the Uniformly Loaded Actuator Disc

Free-wake vortex-ring models are the simplest way of describing a uniformly loaded actuator disc consistently with the Euler equations, i.e., including the radial velocity that accompanies slipstream contraction or expansion. This paper examines the numerical behavior of this model class using an independent open-source FORTRAN 90/95 implementation and a 1:1 Python (v3.12) replica, for the propeller case (cT=1) and the Betz case (cT=−8/9). Three results are reported. First, the two convergence measures in common use—the residual of the wake (sheet) equations and the deviation of the power coefficient cP from momentum theory—are shown not to be equivalent: the latter has a discretization floor and is not monotone, so it is unsuitable as a stopping criterion, and accuracy figures obtained with it are sometimes misleading. Second, the discrete Kelvin–Helmholtz saw-tooth mode is stabilized by a damping factor proportional to z, which reaches the residual floor within a few hundred instead of 104 iterations; the remaining cP fluctuation band reflects the unresolved disc-edge region and is removed by node collocation with a spacing-proportional vortex kernel, which converges to a unique fixed point with machine-level residuals and cP−(−16/27)=+4.7×10−5. Third, the converged solutions show a bounded edge strength with fitted exponent a=0.00±0.04 in γ∝sa, differing from both the s−1/2 spiral and the constant-γ proposals in the literature; it is explained why single-valued, regularized sheet discretizations cannot decide this question.