Mean Induction Actuator Cylinder Method

In QBlade, the aerodynamic forces acting on a Vertical Axis Wind Turbine (VAWT) rotor can be modeled using the Mean Induction Actuator Cylinder (AC) method. This theory interlinks the conservation of momentum with the blade element theory, adapted specifically for the rotating kinematics of a vertical axis turbine. A fundamental premise of this specific implementation is the assumption of a spatially constant induction factor across the entire rotor swept area. By utilizing this uniform flow assumption, the AC method provides a highly robust and computationally efficient representation of the steady aerodynamic loads and overall rotor performance parameters, particularly avoiding the numerical divergence issues often found in local streamtube-based models at high tip speed ratios.

Mean Induction Actuator Cylinder Theory

The Actuator Cylinder method (see Madsen1) treats the VAWT rotor not as a series of independent streamtubes, but as a continuous, permeable cylindrical actuator surface.

In QBlade’s implementation, the forces generated by the blade elements are computed based on the local relative flow and then integrated across the entire azimuthal domain to calculate a global Thrust Coefficient \(C_{T}\). Instead of relying on a local iterative momentum balance, the Actuator Cylinder method maps this integrated global \(C_T\) directly to a single, spatially constant mean global induction factor \(a_0\). This mapping is achieved using a polynomial correlation derived from the work of Madsen and Cheng (see Cheng et al.2):

\[\begin{align} a_0 = k_3 C_T^3 + k_2 C_T^2 + k_1 C_T + k_0. \end{align}\]

This approach is inherently stable because the polynomial fit automatically captures the turbulent wake physics present at high thrust states, preventing the numerical breakdown associated with the standard momentum equations.

It is crucial to emphasize that, unlike the Double Multiple Streamtube (DMS) method which calculates distinct induction values for every individual streamtube, this algorithm assumes the wake induction does not vary with azimuthal position or blade height. Once the constant mean induction \(a_0\) is computed, it is applied uniformly across the entire rotor domain to determine the relative inflow velocities, \(W\), at each azimuthal blade position:

\[\begin{align} W = V \sqrt{(X - \sin\theta)^2 + F^2 \cos^2\theta \cos^2\delta}, \end{align}\]

where \(V = (1-a_0)V_\infty\) is the uniform induced velocity, \(X = \frac{r \omega}{V}\) is the local tip speed ratio, \(\theta\) is the azimuthal angle, \(\delta\) is the local blade inclination angle, and \(F\) is the tip loss factor.

Mean AC Corrections

To improve the accuracy of the VAWT simulations and account for three-dimensional effects, two main correction methods are implemented into the AC algorithm:

• Tip Loss Correction: A Willmer modification of the Prandtl tip loss method is applied to account for the finite span of the blades and the resulting tip vortices (see Paraschivoiu3). The tip loss factor \(F\) directly scales the geometric inflow angle calculation.

Mean AC Load Integration & Rotor Performance

The AC algorithm evaluates the normal and tangential blade forces at discrete azimuthal steps over a full revolution. These sectional loads are then integrated along the blade height (accounting for the local blade inclination angle \(\delta\) and chord variations) and averaged over the \(360^{\circ}\) azimuthal domain.

The total mean torque \(Q\), power \(P\), and thrust \(T\) are subsequently normalized using the freestream dynamic pressure and the rotor swept area to yield the dimensionless turbine performance coefficients:

\[\begin{align} C_P = \frac{P}{\frac{1}{2} \rho V_\infty^3 A_{ref}}, \quad C_T = \frac{T}{\frac{1}{2} \rho V_\infty^2 A_{ref}}, \quad C_Q = \frac{Q}{\frac{1}{2} \rho V_\infty^2 A_{ref} R_{max}}. \end{align}\]

1

Helge Aagaard Madsen. The actuator cylinder: a flow model for vertical axis wind turbines. Technical Report, Aalborg University, 1982.

2

Zhengshun Cheng, Helge Aagaard Madsen, Zhen Gao, and Torgeir Moan. A fully coupled method for numerical modeling and dynamic analysis of floating vertical axis wind turbines. Renewable Energy, 107:604–619, 2017. doi:10.1016/j.renene.2017.02.028.

3

Ion Paraschivoiu. Wind turbine design: with emphasis on Darrieus concept. Polytechnic International Press, 2002. ISBN 978-2553009310.