The researchers’ work, published as a preprint under review in the journal Wind Energy Science, focuses on what aerodynamicists call attached-flow unsteady aerodynamics. Attached flow means the air moves smoothly over the blade surface without separating into the chaotic, energy-dissipating conditions known as stall. On the outboard region of a rotating blade, the section that produces most of the power and carries most of the aerodynamic load, the flow is usually attached even though the angle of attack and the relative velocity are constantly changing. Getting the unsteady forces right in this regime is therefore not a niche concern; it underpins essentially every engineering model of dynamic stall, since stall models are built on top of the attached-flow baseline.

The theoretical machinery for unsteady airfoil behavior dates back nearly a century to classical unsteady thin-airfoil theory, a framework that describes how lift and moment on a thin wing section respond to changes in motion and in the incoming flow. The problem, according to the Danish team, is not the theory itself but its implementation. Although the underlying mathematics is classical, existing descriptions in the literature do not, in the authors’ assessment, provide a complete and internally consistent route for putting the attached-flow contributions into the aerodynamic solvers that industry and researchers actually use. Different codes have adopted different subsets of the required effects, and the consequences of those omissions have been difficult to quantify.

Li and colleagues set out to close that gap by assembling all of the attached-flow ingredients into a single, consistent framework, expressed in a form that is ready for implementation. Their formulation covers several distinct elements that a complete engineering model must combine: the lookup of two-dimensional airfoil polar data, which relates lift, drag, and moment coefficients to the angle of attack; the memory effect of the shed wake, in which vorticity shed into the wake at earlier times continues to influence the loads at the current moment; the non-circulatory, or apparent-mass, loads that arise from the acceleration of air near the airfoil; and consistent definitions of the magnitude and direction of the resulting force. The framework is also designed to connect naturally with models of separated-flow dynamics, so that dynamic stall can be layered on top of the attached-flow foundation.

A key methodological choice in the new work is the way the shed wake memory is formulated. Rather than using the angle of attack as the aerodynamic state variable that carries the history of the flow, the researchers use the downwash velocity, the vertical velocity imparted to the flow at the airfoil. This change matters because the downwash is the quantity that more directly reflects the physical state of the circulatory system, and formulating the memory in terms of it clarifies the implications of the associated modeling choices. Starting from the dimensional loads of classical unsteady thin-airfoil theory, the authors derive both the circulatory contributions, which come from bound circulation and the shed vortex wake, and the non-circulatory contributions in coefficient form, making them directly usable in engineering codes.

The formulation is written specifically for generalized lifting-line methods, a family of approaches that includes the blade-element momentum method used in most industrial design tools, the lifting-line method, and the actuator-line method often coupled with computational fluid dynamics in research simulations. Because the two-dimensional theory itself is classical and well established, the authors direct their verification effort at the rotor level, where the implementation choices actually matter. They test the framework on two rotor cases designed to probe different aspects of the physics.

The first test case is a coned straight blade, which serves as a cross-method benchmark. By comparing simulations that include all of the required contributions with simulations that deliberately omit three of them, the researchers quantify the errors in rotor-integrated thrust and power that result from each omission. The results are striking for anyone who assumed the neglected terms were negligible. Among the three, the contribution associated with the projection of the lift direction has the largest influence on predicted power. Omitting the mid-chord heaving acceleration contribution produces a different kind of error: it eliminates the cancellation that should occur in the non-circulatory normal force, and that loss of cancellation shows up as an error in the predicted thrust. In other words, terms that might look like small refinements on paper can measurably change the numbers that engineers use to size components and estimate energy production.

The second test case involves a vertical-axis wind turbine operating from zero onset flow, a configuration chosen because it provides an elegant analytical check. In the ideal thin-airfoil limit, the theory demands that all of the circulatory and non-circulatory contributions cancel exactly, yielding zero total rotor torque. The researchers show that their implementation reproduces this cancellation, which verifies that the various pieces of the framework have been assembled consistently. A model that left out any of the required terms would fail this test, producing a spurious torque from a rotor that should, in this idealized limit, produce none. The authors note that the complete model avoids unphysical predictions of this kind, whereas incomplete implementations do not.

Why should anyone outside the wind energy community care about coefficient bookkeeping in engineering models? Because the stakes are enormous. Wind turbine blades are among the largest flexible structures ever built by humans, and their design margins depend on load predictions generated by aeroelastic codes that run millions of simulated seconds of operation. If a simulation systematically mispredicts power or thrust because an unsteady aerodynamic term was dropped decades ago and never revisited, the consequences ripple through certification, blade design, and the economics of offshore wind. The Danish team’s results show that omitting some effects can noticeably change predicted power and loads, which means the industry’s standard simulations may carry errors that are neither random nor negligible.

The study also carries a broader lesson about how engineering science accumulates. Classical theories do not come with implementation manuals, and each generation of engineers translates them into code under practical constraints, sometimes losing pieces of the physics along the way. By deriving every attached-flow contribution in coefficient form, clarifying the role of the downwash-based memory formulation, and demonstrating the errors that follow from common omissions, Li, Gaunaa, and Pirrung have provided what they describe as an implementation-ready method that combines the required aerodynamic effects in one consistent framework. Tested across rotor cases spanning conventional horizontal-axis and vertical-axis machines, the framework supports more reliable aerodynamic and aeroelastic simulations of wind turbines. As rotors grow larger and more flexible, and as the industry pushes turbines into more turbulent offshore environments, the unsteady forces that this work puts on a firmer footing will only become more important to capturing how these giants of the energy transition actually behave in the sky.

Subject of Research: Unsteady attached-flow airfoil aerodynamics for wind turbine simulation models

Article Title: Unsteady airfoil aerodynamics in attached flow: From unsteady thin-airfoil theory to wind turbine implementation

Article References: Li, A., Gaunaa, M., & Pirrung, G. R. (2026). Unsteady airfoil aerodynamics in attached flow: From unsteady thin-airfoil theory to wind turbine implementation. https://doi.org/10.5194/wes-2026-168

Image Credits: AI Generated

DOI: 10.5194/wes-2026-168

Keywords: wind energy, unsteady aerodynamics, thin-airfoil theory, blade-element momentum, shed wake memory, dynamic stall, aeroelasticity, lifting-line methods, vertical-axis wind turbine, rotor loads, Technical University of Denmark, Wind Energy Science

The research, published in the journal Results in Engineering, was carried out by Maziyar Moghtadaeifar, Ramin Farzadi, and Majid Bazargan, who built their design around a radical idea that has been gathering momentum in wind engineering: the J-shaped airfoil. A conventional blade cross-section, such as the standard NACA0021 profile used here as a baseline, is a closed teardrop shape. A J-shaped airfoil takes that profile and simply removes part of the pressure side, the face of the blade that faces away from the turbine’s rotation axis, leaving a hollow, scoop-like section bounded by a wall just one millimeter thick. The opening begins at the point of maximum thickness and runs all the way to the trailing edge, transforming the blade into something between a wing and a curved trough.

That hollow matters. When wind sweeps across a J-shaped blade, air circulates inside the concave cavity, and the resulting changes in pressure distribution alter how lift and drag act on the rotor. Earlier experimental and numerical work had shown that such profiles can improve self-starting, because the open cavity behaves somewhat like a drag device at low speeds, but often at a cost to peak efficiency. The innovation in the new study is to combine this profile with a helical configuration: three blades, each twisted sixty degrees around the rotation axis, mounted on a rotor with a radius of 0.99 meters and a height of 1.15 meters. Helical blades smooth the torque output because some section of some blade is always optimally positioned relative to the wind, reducing the violent fluctuations that plague straight-bladed designs and making the rotor less dependent on the angle at which it happens to have stopped.

To test the concept, the team ran three-dimensional unsteady Reynolds-averaged Navier-Stokes simulations using the shear-stress transport k-omega turbulence model, a combination that previous validation studies have identified as among the most accurate for capturing the complex, separated flows around vertical-axis rotors, including the transient phenomenon of dynamic stall. The computational domain stretched thirty-two rotor radii downstream and twelve radii in the other directions, with a blockage ratio of just 1.61 percent, and a sliding-mesh cylinder carried the spinning blades. The researchers verified their setup against published experimental data for two different variable-pitch turbines, then ran careful mesh and time-step independence studies, refining the grid until the cycle-averaged torque coefficient changed by less than one percent and settling on a time step of half a degree of rotation per step.

The first headline result concerns how the J-shaped profile itself performs. Across low and medium tip speed ratios, the ratio of blade speed to wind speed that governs rotor behavior, the J-shaped helical turbine outperformed a geometrically identical machine with conventional NACA0021 blades. At a tip speed ratio of 1.11, for example, the J-shaped rotor produced a torque coefficient 26.59 percent higher than the full-profile machine, with the advantage concentrated in the upwind passage between azimuth angles of thirty and one hundred twenty degrees, where dynamic stall dominates. The cavity appears to reshape the internal flow structure in a way that enlarges the pressure difference between the two sides of the blade, and the first torque peak shifts to a later azimuth angle and grows taller. Averaged across all the operating conditions examined, the profile swap alone was worth roughly a ten percent improvement in torque.

But the J-shape carried a hidden liability that emerged at high speed. At a tip speed ratio of 1.77, the open-tipped J-shaped turbine’s performance collapsed, falling more than fifty percent below the conventional rotor. The culprit was the blade tip. Because the J-profile is hollow, vortices do not merely shed from the tip the way they do on a solid wing; they can leak out through the open cavity itself, spilling high-energy flow and destroying the pressure difference the blade depends on. Tip vortices, the swirling structures that form wherever lift-generating surfaces end abruptly, are a familiar efficiency tax on all rotors, but the open J-section makes the leakage far more damaging precisely when the turbine is spinning fast enough to generate its best power.

The team’s fix was disarmingly simple: cap the tips. They added thin NACA0021-shaped plates, one millimeter thick, to both ends of each blade, sealing the hollow section like lids on a tube. The effect was enormous. At the problematic tip speed ratio of 1.77, the caps boosted the torque coefficient by roughly 122 percent compared with the uncapped J-shaped turbine, lifting it about eight percent above even the conventional full-profile machine. Averaged over all five tip speed ratios studied, from 0.44 to 1.77, the caps raised the J-shaped turbine’s torque coefficient by about 46 percent, and the capped J-shaped design beat the conventional helical turbine by about 48 percent overall. In power-coefficient terms, obtained by multiplying torque by tip speed ratio, the capped machine dominated across the operating range.

By slicing each blade into twenty sections along its height and tracking the torque of each, the researchers traced exactly where the gains came from. At a mid-range tip speed ratio of 1.33, the upwind half of the rotor improved by just over twenty percent with caps installed, while the downwind half gained a more modest eight percent, with the largest benefits near the tips themselves. Flow visualizations revealed the mechanism: without caps, the vortex forming inside the blade’s concave cavity escapes outward over the open edge; with caps in place, the vortex is redirected toward the trailing edge, and pressure builds inside the cavity, strengthening the force that drives rotation. The caps essentially convert a leaking bucket into a sealed one, letting the J-profile’s pressure advantage accumulate instead of draining away through the blade ends.

Perhaps most important for the urban applications that motivate this whole line of research, the capped design starts better. The team held the rotor stationary in a seven-meter-per-second wind and computed the starting torque at azimuth angles from zero to 105 degrees in fifteen-degree increments. The capped J-shaped turbine beat the conventional helical rotor at every single angle, with improvements ranging from about four percent to a striking 46.6 percent at ninety degrees, and an average gain of roughly 21.5 percent across all positions. For a technology whose central weakness has always been the dead zone where lift-based rotors cannot overcome their own drag, a design that simultaneously raises efficiency, smooths torque, and strengthens self-starting is a meaningful step forward. The authors caution that they fixed the cut ratio and cap thickness rather than optimizing them, so further gains may remain on the table. But the message of the study is clear: sometimes the path to better wind energy is not a bigger machine or a fancier material, but a hollow blade with its ends sealed shut, turning the city’s messy gusts into cleaner, steadier power.

Subject of Research: Aerodynamic performance and self-starting behavior of a helical J-shaped vertical-axis wind turbine with capped blade tips

Article Title: Aerodynamic performance and self-starting characteristics of a novel helical J-shaped vertical-axis wind turbine with blade-tip caps

Article References: Moghtadaeifar, M., Farzadi, R., & Bazargan, M. (2026). Aerodynamic performance and self-starting characteristics of a novel helical J-shaped vertical-axis wind turbine with blade-tip caps. Results in Engineering, 32, Article 113308. https://doi.org/10.1016/j.rineng.2026.113308

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113308

Keywords: vertical-axis wind turbine, J-shaped airfoil, helical blades, blade-tip caps, tip vortices, self-starting torque, CFD simulation, URANS, SST k-omega, dynamic stall, urban wind energy, power coefficient