# b) Simulation using Virtual Disks

The context

In this part, the idea is to simulate an existing wind farm and evaluate its performances. The farm under study is the following : It is composed by 5 wind turbines Repower MM70 aligned and separated from a distance of $2.85 D_{blades}$.

Here, the wind is represented by the vectors of the top right of the figure.

The geometrical characteristics of the MM70 are gathered in the figure : In StarCCM+: first, the domain is created, then a first simulation is launch. To finish the results are exploited.

How does this simulation work ?

The wind turbines are represented by actuator disks (in blue here): The "virtual disk" is a model in StarCCM+ which simulate the presence of a wind turbine by modifying the Navier-Stokes equations adding a forcing term: in the cells composing the virtual disk, two forces are added to the equation solved by the software : a normal force, and a tangential one.

StarCCM+ takes as inlet :  and geometrical consideration of the turbine : the position, the orientation...

With that, the software adds two forces to the momentum equation for every cell whose center is inside the virtual disk :

• a Thrust force $$F^{\perp}_{cell}=T \frac{V_{cell}}{\sum V_{cell}}$$

where T is the real thrust force : $$T=\frac{1}{2} \rho_0 U_0^2 C_t \pi (R_0^2-R_1^2)$$

V the volume of the cell, $U_0$ the velocity impacting the turbine, $C_t$ the thrust coefficient (given by the curve), $\rho_0$ the density of the air and $R_0$ $R_1$ the geometry of the turbine.

• a Torque $$F^{//}_{cell}=Q \frac{r_{cell}^2 V_{cell}}{\sum r_{cell}^2 V_{cell}}$$

where Q is the real torque of a wind turbine : $$Q=\frac{P}{\Omega}$$

V the volume of the cell, r its distance from the center P the power produced (given by the power curve) and $\Omega$ the rotation rate.