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Optimal smoothing length scale for actuator line models of wind turbine blades

The actuator line model (ALM) is a commonly used method to represent lifting surfaces such as wind turbine blades within Large-Eddy Simulations (LES). In the ALM the lift and drag forces are replaced by an imposed body force which is typically smoothed over several grid points using a Gaussian kernel with some prescribed smoothing width $ε$. To date, the choice of $ε$ has most often been based on numerical considerations related to the grid spacing used in LES. However, especially for finely resolved LES with grid spacings on the order of or smaller than the chord-length of the blade, the best choice of $ε$ is not known. In this work, a theoretical approach is followed to determine the most suitable value of $ε$. Firstly, we develop an analytical solution to the linearized flow response to a Gaussian lift and drag force and use the results to establish a relationship between the local and far-field velocity required to specify lift and drag forces. Then, focusing first on the lift force, we find $ε$ and the force center location that minimize the square difference between the velocity fields induced by the Gaussian force and 2D potential flow over Joukowski airfoils. We find that the optimal smoothing width $ε^{\rm opt}$ is on the order of 14-25\% of the chord length of the blade, and the center of force is located at about 13-26\% downstream of the leading edge of the blade, for the cases considered. These optimal values do not depend on angle of attack and depend only weakly on the type of lifting surface. To represent the drag force, the optimal width of the circular Gaussian drag force field is shown to be equal to the momentum thickness of the wake.