#### Basic principles

In this particular implementation, the flow field is resolved in the commercial software FLUENT using a finite volume discretization of the incompressible NS equations, according to which the momentum equation is expressed as

- (12)

The velocity vector in cylindrical coordinates is given by , and the body force vector is used to model the effect of a rotor by introducing source terms in the momentum equation.

The body force vector represents a force per unit volume; the force from the rotor applied on an annular area of the differential size is therefore

- (13)

where the load is a vector of aerodynamic force per spanwise length on the rotor.

The aerodynamic forces acting on the rotor are determined from local velocities and by utilizing two-dimensional aerofoil characteristics. For every aerofoil element, the relative velocity, *V*_{rel}, and flow angle, *ϕ*, with respect to the rotor plane are determined. Note that in case the actuator disc is moving within the flow field at a given velocity, e.g. *u*, this velocity component is subtracted from the measured fluid velocity, *v*_{z}, in a cell to find the relative velocity experienced by a blade element. The lift and drag forces per spanwise length are calculated once the local angle of attack, *α*, is found taking local pitch, *γ*, into account, and lift and drag coefficients, *C*_{L} and *C*_{D} , have been determined by table lookup.

- (14)

- (15)

- (16)

- (17)

Finally, a force vector is found by projecting the lift and drag forces along the axial and tangential directions.

- (18)

A real rotor has a finite number of blades that will produce a system of tip vortices in the wake. This wake will not be the same as that behind a constantly loaded idealized actuator disc. To account for the finite number of blades, a Prandtl tip-correction factor, *F*_{c}, is implemented to correct the aerodynamic force components in (18), which then becomes

- (19)

The actuator disc appears as a one-dimensional (1D) line in an axisymmetric plane. To prevent spatial oscillations due to discontinuous variations over the actuator disc, forces are smeared axially and symmetrically away from the line using a Gaussian smoothing kernel, [15] , where

- (20)

The axial position of the actuator disc at the beginning of a given time step is indicated by *z*_{0}. In case the axial position of the actuator disc is moving within the flow field, the value of *z*_{0} will be updated at the end of a time step. The parameter *ϵ* controls the smearing and is set equal to *ϵ* = *ϵ*_{i}*Δz* with *ϵ*_{i} of the order 1 *≲ ϵ*_{i} *≲*2 so that the final distributed force as a spatial function is

- (21)

#### Moving actuator disc

Modifications to the actuator disc model have to be made to simulate a wind turbine rotor oscillating in surge motion. The approach followed in this implementation is to displace the virtual position of the actuator disc within a stationary omputational mesh. This is to ensure that realistic interaction between the actuator disc and the wake structure is modelled. [15]

Starting at an initial position, located at the origin *z* = *z*_{0} = 0, the position of the actuator disc *z*_{0} is updated at the end of every time step, as *z*_{0}(*t*) = *x*_{A} sin(*ω*_{s}*t*). The updated position is then used to determine the column index of the grid cells in which the actuator disc is located. This is indicated schematically in Figure 2, where the actuator disc is represented by the red line and the column of grid cells in which it is located is shaded in grey. In the part of the computational grid where the moving rotor can be located, indicated by dashed lines in Figure 2, equidistant cell spacing is used in both the radial and axial directions. Outside of this region, grid cells are gradually stretched in either the axial or the radial direction. Fluid velocities at the actuator disc's location need to be sampled to evaluate equations (14)-(17), which determine the volume forces that will be applied to the flow field in the next time step. Quadratic interpolation is used to determine these velocities; the velocities in the current grid cells in addition to the axial neighbouring cells directly upstream and downstream are required for this calculation. Once the loading on the actuator disc itself has been updated, the Gaussian spreading function, (21), is used to determine the corresponding volume force that should be applied to each cell in the computational grid.

The just described actuator disc model is the most physically complete and realistic model considered in this work. No additional assumptions regarding the wake geometry, definition of the axial induction factor in case of a moving rotor, its value when the rotor is highly loaded or unsteady modelling of induced velocity need to be made—all of which is necessary for the BEM models. These parameters are naturally included in the actuator disc model and result as part of the flow field solution when solving the NS equations numerically. Differences between the two engineering dynamic inflow approaches of an oscillating wind turbine rotor and the more advanced computational fluid dynamics implementation will be investigated in the next sections, where it will hopefully shed light on the validity of the engineering models for taking own motion into account.