Unsteady two‐dimensional potential‐flow model for thin variable geometry airfoils

Abstract

ABSTRCTIn the present work, analytical expressions for distributed and integral unsteady two-dimensional forces on a variable geometry airfoil undergoing arbitrary motion are derived under the assumption of incompressible, irrotational, inviscid fl ow. The airfoil is represented by its camber line as in classic thin-airfoil theory, and the defl ection of the airfoil is given by superposition of chord-wise defl ection mode shapes. It is shown from the expressions for the forces that the infl uence from the shed vorticity in the wake is described by the same time lag for all chord-wise positions on the airfoil. This time-lag term can be approximated using an indicial function approach, making the practical calculation of the aerodynamic response numerically very effi cient by use of Duhamel superposition. Furthermore, the indicial function expressions for the time-lag terms are formulated in their equivalent state-space form, allowing for use of the present theory in problems employing the eigenvalue approach, such as stability analysis.The analytical expressions for the integral forces can be reduced to Munk's steady and Theodorsen's unsteady results for thin airfoils, and numerical evaluation shows excellent agreement with other unsteady two-dimensional thin-airfoil results.Apart from the obvious applications within active load control/reduction, the present theory can be used for various applications which up to the formulation of the present theory have been possible only using much more computational costly methods. An example that highlights this feature is the propulsive performance of a soft heaving propulsor presented in this paper.

Indicial lift response function: an empirical relation for finite‐thickness airfoils, and effects on aeroelastic simulations

“…They show that the wake memory effects do not depend on how the change in quasi-steady loading is generated. The same behavior is reported in the work by Gaunaa [20], where the aerodynamic forces due to arbitrary motion and deformation of an airfoil are derived under thin airfoil assumptions. Gaunaa shows that the quasi-steady loading of the airfoil can be represented by an equivalent three-quarters chord downwash w 3/4 ; the equivalent downwash w 3/4 encompasses, in a single term, all the sources of quasi-steady loading, as, for instance, the airfoil linear motion, the angle of attack and its angular rate, the camber-line deformation and its time derivatives.…”

Iterative learning control applied to a non-linear vortex panel model for improved aerodynamic load performance of wind turbines with smart rotors

“…In summary, the model given above is a relatively simple (compared to a full Navier-Stokes simulation) but realistic model of the flow and actuation which can be used to investigate control schemes aimed at damping fluctuations in the lift using trailing-edge devices for load control. The model used here is an alternative to the thin-airfoil-type models used in other studies, e.g., by Gaunaa (2010). It is simpler to implement the thin-airfoil type models, with less ad hoc modelling, and is based on a direct solution of the Euler equations, taking explicit account of the thickness of the airfoil.…”

Effect of Smart Rotor Control Using a Deformable Trailing Edge Flap on Load Reduction under Normal and Extreme Turbulence

“…The unsteady aerodynamic behaviors of other sections, e.g., inboard section, will vary little across all simulations and not exert much effect on the comparisons in control performance due to non-existence of DTEF. On the other hand, the unsteady aerodynamics of flap action can be much more correctly computed using unsteady models, e.g., newly developed models by Gaunaa and Andersen et al [28,29], who provide very promising dynamic stall and dynamic inflow models to predict unsteady aerodynamic forces and moments on an airfoil section undergoing arbitrary motion like the present case with deformable trailing edge flap. Integration of these models into Aerodyn code will undoubtedly further improve aerodynamic environment of smart rotor system and will be conducted in our future work.…”