Unsteady aerodynamic loads on pitching aerofoils represented by Gaussian body force distributions

The actuator line model (ALM) is an approach commonly used to represent lifting and dragging devices like wings and blades in large-eddy simulations (LES). The crux of the ALM is the projection of the actuator point forces onto the LES grid by means of a Gaussian regularisation kernel. The minimum width of the kernel is constrained by the grid size; however, for most practical applications like LES of wind turbines, this value is an order of magnitude larger than the optimal value that maximises accuracy. This discrepancy motivated the development of corrections for the actuator line, which, however, neglect the effect of unsteady spanwise shed vorticity. In this work we develop a model for the impact of spanwise shed vorticity on the unsteady loading of an aerofoil modelled as a Gaussian body force distribution, where the model is applicable within the regime of unsteady attached flow. The model solution is derived both in the time and frequency domain and features an explicit dependence on the Gaussian kernel width. We verify the model with ALM-LES for both pitch steps and periodic pitching. The model solution is compared withTheodorsen theory and validated with both computational fluid dynamics using body fitted grids and experiment. It is concluded that the optimal kernel width for unsteady aerodynamics is approximately $40\,\%$ of the chord. The ALM is able to predict the magnitude of the unsteady loading up to a reduced frequency of $k\approx 0.2$ .