On the applicability of the actuator line method for unsteady aerodynamics

A linear theory for unsteady aerodynamic effects of the actuator line method (ALM) was developed. This theory is validated using two-dimensional ALM simulations, where we compute the unsteady lift generated by the plunging and pitching motion of a thin airfoil in uniform flow, comparing the results with Theodorsen’s theory. This comparison elucidates the underlying characteristics and limitations of ALM when applied to unsteady aerodynamics. Numerical simulations were conducted across a range of chord lengths and oscillation frequencies. Comparison of ALM results with theoretical predictions shows consistent accuracy, with all Gaussian parameter choices yielding accurate results at low reduced frequencies. Furthermore, the study indicates that selecting a width parameter ratio of $\varepsilon/c$ (the Gaussian width parameter over the chord length) between 0.33 and 0.4 in ALM yields the closest alignment with analytical results across a broader frequency range. Additionally, a proper definition of angle of attack for a pitching airfoil is shown to be important for accurate computations. These findings offer valuable guidance for the application of ALM in unsteady aerodynamics and aeroelasticity.

💡 Research Summary

The paper presents a comprehensive investigation of the Actuator Line Method (ALM) for unsteady aerodynamic applications, developing a linear theory that links ALM‑generated forces to the classical Theodorsen solution for a thin airfoil undergoing harmonic plunging and pitching motions. Starting from Theodorsen’s exact potential‑flow formulation, the authors derive an expression for the unsteady lift coefficient that isolates the circulatory component, represented by the complex Theodorsen function C(k). They then model the ALM body force as a lift line smeared with a three‑dimensional Gaussian kernel of width ε, and insert this distributed force into the inviscid vorticity equation. By solving the linearized vorticity equations for a periodic force, they obtain analytical expressions for the induced vorticity fields ω_x and ω_z, which depend on the reduced frequency based on the smearing length, k_ε = Ω ε²/(2U∞).

Using Helmholtz’s theorem, the induced velocity at the actuator line is expressed as an integral over the vorticity field, leading to a complex transfer function κ(k_ε) that relates the non‑dimensional induced velocity to the non‑dimensional circulation. κ(k_ε) is shown to be a function solely of k_ε and captures the phase lag and amplitude attenuation introduced by the Gaussian smearing. The final lift coefficient expression becomes C_l = C_l,QS + a₀ κ(k_ε) c Γ*, where C_l,QS is the quasi‑steady lift term, a₀ is the static lift‑curve slope, c is the chord, and Γ* is the non‑dimensional circulation. This formulation explicitly demonstrates how the choice of ε influences the unsteady response.

Numerical validation is performed with the open‑source Dedalus code, solving the incompressible Navier‑Stokes equations in two dimensions for a thin airfoil at Re = 10⁵. The authors vary the chord length to achieve ε/c ratios of 0.2, 0.33, 0.4, and 0.5, while keeping ε = 1 and the grid spacing Δy = Δz = 4 (ensuring ε/Δ ≈ 4). Harmonic motions are imposed with h(t) = h₀ sin Ωt (h₀ = 0) and α_g(t) = α₀ sin Ωt (α₀ = 1°), covering reduced frequencies k from 0.05 to 1.0.

Results show that at low reduced frequencies (k < 0.2) all ε/c values reproduce Theodorsen’s amplitude and phase within a few percent, indicating that the Gaussian smearing does not significantly affect the quasi‑steady regime. As the frequency increases, only the intermediate smearing ratios ε/c ≈ 0.33–0.4 maintain errors below 5 % in both magnitude and phase. Smaller ε/c (0.2) leads to under‑prediction of lift and excessive phase lag, while larger ε/c (0.5) causes over‑prediction and reduced phase lag. The study attributes these trends to the balance between vorticity concentration (small ε) and excessive diffusion (large ε).

A second key finding concerns the definition of the effective angle of attack for pitching motions. The standard ALM practice of α = α_g + arctan(u_y/u_z) neglects the contribution of pitch rate, resulting in noticeable discrepancies at higher frequencies. By augmenting the angle of attack with a pitch‑rate term, α_m = α_g + arctan(u_y/u_z) + b