Curved-flow breakthrough sharpens the simulation of vertical-axis turbines

Vertical-axis turbines have long promised a distinctive kind of renewable energy: power harvested from wind, rivers and tidal currents regardless of which direction the flow comes from, with simple installation and the potential for denser turbine arrays than conventional horizontal-axis farms. Yet the aerodynamics of these machines are notoriously difficult to capture in computer simulations, because their blades sweep through curved streamlines rather than the straight, uniform flow that most classical airfoil theory assumes. A new peer-reviewed study published in Wind Energy Science by Grégoire Winckelmans of UCLouvain and colleagues at Université Laval presents a substantially improved set of models for these curvature effects, together with a novel actuator line method that, for the first time in this context, explicitly enforces the aerodynamic moment as well as the forces acting on the blade.

The central difficulty is geometric. When the ratio of blade chord to rotor arm length is not small, the curvature of the flow around a rotating blade changes the aerodynamic coefficients compared with those measured or computed in a uniform stream. Researchers have known since the pioneering work of Migliore and colleagues in 1980 that a symmetric airfoil in curved flow behaves, in terms of lift and moment, much like a cambered airfoil at a non-zero incidence in straight flow, an effect captured by the concepts of virtual camber and virtual incidence. Later work by Akimoto and co-authors in 2013 formalized this with a bidirectional conformal mapping, and studies by Rainbird and Bianchini confirmed the phenomenon experimentally and numerically. What has been missing, the new paper argues, is a complete and consistent treatment that works for every blade attachment point, every pitch angle, and all the way up to stall.

The team’s approach is deliberately economical. The correction models are allowed only one input: the standard aerodynamic coefficients of the airfoil in uniform flow, here obtained from wall-resolved computational fluid dynamics of a NACA0015 airfoil at a high Reynolds number of 6.0 million. Everything else is an analytical modification of those coefficients, expressed through the classic analogy with potential flow along curved streamlines. For the normal force and the aerodynamic moment, the authors extend known models so that they cover all possible attachment points of the blade to the rotor arm and remain valid up to stall, a regime where many earlier formulations broke down. Crucially, they show that the precise form of the correction matters: a seemingly natural alternative formula would predict stall angles several degrees away from those found in the reference simulations.

The most striking conceptual advance concerns the tangential force, the component along the direction of blade travel that ultimately determines how much torque, and therefore power, a vertical-axis turbine produces. The authors demonstrate that the potential flow analogy, which works well for the normal force and moment, is fundamentally flawed for this component. A straight potential flow past a cambered airfoil produces only a normal force and a moment, whereas an inviscid flow past a rotating airfoil also carries a tangential force. Their new model derives this inviscid tangential force from a simple physical requirement: because the flow is inviscid, the moment at the root of the rotating arm must be exactly zero, which forces the tangential force to compensate for the aerodynamic moment. Inviscid simulations confirmed this condition numerically, and the parasitic drag from viscosity is then added on top.

Validation came in two stages. First, the team compared their corrected coefficients against wall-resolved CFD of a NACA0015 airfoil rotating with a chord-to-radius ratio of 2/7, a deliberately severe case in which curvature effects are strong. They tested the full range of pitch angles up to the two stall angles, which are themselves shifted asymmetrically by the curvature, and for two attachment points: mid-chord and quarter-chord. The agreement was close throughout. For the unpitched mid-chord case, for example, the model predicted a normal force coefficient of about minus 0.454 against a measured value of minus 0.431, and incorporating a small correction for the deformed-airfoil moment brought the moment predictions into even tighter alignment with the reference data.

The second stage moved to a full unsteady turbine configuration: a single-blade vertical-axis turbine with the same NACA0015 profile, operating at a tip speed ratio of 3.25, the value at which power production is maximized and at which the flow does not separate dynamically at any point in the rotation cycle. To simulate this efficiently, the authors implemented their corrected models in an actuator line method, a moderate-fidelity technique that replaces the meshed blade with equivalent volumetric forces in the Navier-Stokes equations. This bypasses the need to resolve the thin boundary layers on the blade surface, cutting the mesh from roughly 328,000 cells in the wall-resolved reference to about 37,000 cells in the actuator line simulation, while still reproducing realistic loads and wake behavior.

The genuinely novel ingredient is the enforcement of the aerodynamic moment. A standard actuator line distributes forces over a Gaussian kernel centered on the blade, but it cannot impose a pure moment. The team solved this by replacing the moment with an annular force field, a swirling distribution of forces around the control point that produces the same net torque without any net force. This field is then integrated analytically over the same two-dimensional template of grid cells used for the forces, guaranteeing that the moment is fully transmitted to the flow solver regardless of the chosen kernel size. The result is an actuator line that imposes both the aerodynamic forces and the aerodynamic moment, and the authors show that the two must be treated consistently: if the moment is not enforced, the inviscid part of the tangential force must be ignored, and if the moment is enforced, that force component must be included.

The comparisons against the wall-resolved reference simulation were favorable across the entire rotation cycle. The cycle-averaged coefficients for the normal force, tangential force and moments agreed well, and the predicted power coefficient tracked the reference closely, with the corrected actuator line substantially outperforming an uncorrected version. When the actuator line was centered at the blade attachment point at mid-chord, the upstream half of the cycle was reproduced with excellent accuracy, eliminating a small shift in the peak power coefficient that appeared when the line was centered at the quarter-chord aerodynamic center instead. Small discrepancies persisted in the downstream half of the cycle, where the blade flies through its own wake and the effective angle of attack becomes harder to predict, but the overall agreement gives the authors confidence in the framework.

The implications reach well beyond a single test case. Because the corrections require nothing more than standard uniform-flow airfoil data, any actuator line user can apply them to an existing turbine model simply by knowing the chord-to-radius ratio and the blade attachment geometry. The moment-enforcement technique also opens the door to simulating cambered blades, which carry an intrinsic aerodynamic moment even in straight flow, and to multi-turbine cluster simulations that would be computationally prohibitive with wall-resolved methods. The authors note that their models are currently limited to flows without fast unsteady effects or dynamic stall, and that extending them to thicker airfoils and to the combined effects of curvature and thickness are natural next steps. For a technology whose commercial fortunes depend on cheap, trustworthy simulation, the ability to capture curved-flow aerodynamics accurately at a fraction of the computational cost may prove a decisive enabler for the next generation of vertical-axis wind, river and tidal energy systems.

Subject of Research: Flow curvature correction models and actuator line simulation of vertical-axis turbine aerodynamics

Article Title: Improved modeling of flow curvature effects and actuator line method with aerodynamic moment, with application to vertical-axis turbines

Article References: Winckelmans, G., Rochefort, P., Villeneuve, T., Trigaux, F., Duponcheel, M., & Dumas, G. (2026). Improved modeling of flow curvature effects and actuator line method with aerodynamic moment, with application to vertical-axis turbines. Wind Energy Science, 11(9), 3615-3651. https://doi.org/10.5194/wes-11-3615-2026

Image Credits: AI Generated

DOI: 10.5194/wes-11-3615-2026

Keywords: vertical-axis turbines, flow curvature, actuator line method, aerodynamic moment, NACA0015 airfoil, virtual camber, CFD, wind energy, turbine simulation, tangential force, potential flow, tip speed ratio