Generalised actuator disk theory: wake development with turbulent entrainment

TL;DR

This work extends classical actuator disk theory by embedding wake development through turbulent entrainment within a hybrid control-volume framework, enabling predictions of $U(x)$, $P(x)$, and wake width $\sigma(x)$ across both upwind and downwind regions. The model couples mass and momentum balances with a pressure Poisson-based closure, and introduces a physically based entrainment velocity $U_e$ that blends wake-shear and ambient-turbulence effects, yielding a generalized Bernoulli equation and an iteratively solvable $C_T$–$a$ relation. Key findings show entrainment accelerates wake recovery, alters far-wake growth (linear with ambient turbulence, $x^{1/3}$ in the wake-shear case), and can slightly raise the maximum power coefficient beyond the Betz limit, especially for higher entrainment levels. Together, these insights provide a more realistic and self-consistent framework for predicting rotor performance and wake evolution, with potential extensions to more complex wake profiles and non-linear pressure effects.

Abstract

Classical actuator disk theory, developed more than a century ago, provides an idealised description of turbine rotor performance. It treats a rotor as an infinitesimally-thin permeable disk and applies the governing flow equations over a streamtube encompassing the disk. A well-known limitation of the theory is its assumption of ideal flow downstream of the disk, which restricts its applicability to short downwind distances before turbulence and mixing processes governing the wake evolution take hold. As a result, the classical theory also leads to unphysical predictions for highly-loaded rotors. Turbulent axisymmetric wakes, by contrast, represent an extensively-studied canonical free shear flow with much of the progress and its applications to wind turbines limited to the far-wake dynamics. In this work, we introduce a generalised actuator disk theory based on a hybrid stream-tube and wake control volume, that seamlessly integrates classical actuator disk analysis with wake turbulence modelling at arbitrary distances from the rotor. The resulting model, while still idealised, can be used to predict variations in velocity, pressure, and cross-sectional flow area as function of position, both upstream and downstream of the rotor disk. Furthermore, by accounting for turbulent entrainment in the wake development, it provides more realistic predictions of thrust and power coefficients for highly-loaded disks.

Figures (10)

• Figure 1: Schematic of the control volume (CV) used in the new proposed actuator disk theory. The upstream part of the CV is a streamtube, while the downstream part follows the wake borders and therefore there is flow entrainment from the lateral area. The outlet is at an arbitrary downstream location with a diameter of $\sigma(x)$, pressure $P=P(x)$ and velocity $U=U(x)$. The disk is located at $x=0$.
• Figure 2: (a) Pressure force exerted on an infinitesimal lateral area. (b) Schematic of a spherical CV with an infinitely large radius surrounding the disk. For this CV, unlike the one shown in figure \ref{['fig:schematic']} that is used in our present model formulation, lateral pressure forces are internal forces and do not appear in the momentum equation.
• Figure 3: Comparison of the predictions of equation \ref{['eq:PPE_solution_semiinf_f=g=0']} with LES data under laminar inflow conditions.
• Figure 4: Variations of flow properties for different entrainment coefficients $E_1$ ($E_2 = 3E_1$, $I = 0.05$) at two axial induction factors: (a) $a = 0.25$ and (b) $a = 0.45$.
• Figure 5: Variations of entrainment velocity with downwind distance, where $U_e$ is the total entrainment velocity (equation \ref{['eq:entrainment_velocity']}), $U_e^w$ is the wake-shear driven entrainment velocity (equation \ref{['eq:entrainment_wake_shear']}), and $U_e^{b}$ is the background-turbulence driven entrainment velocity (equation \ref{['eq:entrainment-atmospheric']}).