Potential flow tower shadow model

In Ashes, to simulate the tower shadow for an upwind, the flow around a tower can be modeled according to the potential flow theory description of a flow around a cylinder (see Wikipedia).
We now show the equations that describe the local flow velocity around the tower in the following situation: a cylinder of radius 
$$a$$
 is placed in a uniform incoming wind of velocity 
$$V_0$$
.
$$O$$
is the center of the cross section of the cylinder and we calculate the modified wind velocity at point 
$$P$$
. When computing the modified wind velocity in Ashes, only the projection of the wind in the cross-sectional plane of the cylinder, noted 
$$V_{0,h}$$
, is considered.
Note:
  • If the wind has a component parallel to the axis of the tower, that component is ignored. This is the case for example for the vertical component of turbulent wind when computing the tower shadow effect for vertical towers.
  • The cross-sectional plane is the one perpendicular to the axis of the tower part casting the shadow, so parts that are not vertical are handled as well (see Which parts of the structure cast a shadow below).
  • To calculate the tower shadow at a given blade station, the local incoming wind velocity at the location of the blade station, is used (rather than the mean wind speed or the wind speed at hub high)

The location of point 
$$P$$
 is described by the polar coordinates 
$$r$$
 and
$$\theta$$
 in a polar coordinate system which origin is 
$$O$$
 and which direction is given by 
$$V_{0,h}$$
. 
$$e_r$$
 and 
$$e_\theta$$
 are the unit vectors of this coordinate system. The figure below illustrates this set-up. In this figure, 
$$\theta$$
 is positive.



A tower shadow is not necessarily produced by a tubular tower. When using truss towers, a tower shadow can be calculated based on a user-defined radius (see shadow radius parameter in the Aerodynamics tab).


The resulting wind velocity (i.e. the velocity of the wind affected by the tower) is noted 
$$V$$
 in the following figure, and is decomposed in the polar coordinate system into 
$$V_{r}$$
 and 
$$V_{\theta}$$
.



The components of 
$$V$$
 are then computed according to potential flow theory as
$$V_r = \left(1-\left(\frac{a}{r}\right)^2\right)|V_{0,h}|\cos{\theta}$$
 for the component along 
$$e_r$$
$$V_{\theta}= -\left(1+\left(\frac{a}{r}\right)^2\right)|V_{0,h}|\sin{\theta}$$
 for the component along 
$$e_\theta$$

where 
$$|V_{0,h}|$$
 is the magnitude of 
$$V_{0,h}$$
.

1 Which parts of the structure cast a shadow

The shadow is not cast by the support section as a whole, but by each straight run of elements with a circular cross section it is made of. A run is a chain of beam elements that lie on the same line: an ordinary tubular tower is a single run, while a multi-rotor support section made of a mast and a horizontal arm is two.

Every rotor is shadowed by all the runs of the support section its RNA is connected to, not only by the one it is mounted on. On a multi-rotor turbine, the blades of the rotor on top of the mast sweep past the arm sticking out at hub height, and the blades of the rotor at the end of the arm sweep past the mast; both effects are included.

Each run is treated like a tower of its own: the equations above are applied in the plane perpendicular to its axis, so runs that are inclined or horizontal are handled just like a vertical mast. The radius 
$$a$$
 is the radius of the run's cross section at the height of the blade station (the tapered value for a tower defined From parameters). Just beyond the upper end of a run the radius is smoothed to zero over a short distance, and past either end the run casts no shadow at all.

Where several runs can shadow the same blade station, each one contributes its own disturbance of the flow and the contributions are added.

Note: adding the contributions of several runs is a first-order superposition. It is accurate when the runs are far apart relative to their diameters, which is the case for a mast and an arm, and it slightly overestimates the combined deficit for runs that are close together, since the way each one alters the flow around the other is not modelled.

Note: the potential flow solution is not defined inside the cylinder or on its axis, which a blade station can reach when a rotor sweeps around the arm it is mounted on. A station inside a run is treated as lying on its surface (
$$a/r$$
 capped at 1), and a station exactly on the axis is left undisturbed by that run.



Note:  we have shown how to evaluate the wind behind the tower. For upwind turbines (which is the most common case in the industry), it is necessary to evaluate the wind in front of the tower. Obviously, the equations stay the same, but the − sign in the definition of 
$$V_\theta$$
can be confusing. This is due to the fact that in this case, θ will be larger than 90 degrees. The figure below illustrates the coordinates of point P when calculating the wind velocity in front of the tower.


If this note makes things even more confusing, just ignore it. It will make sense if you try to write down the equations and start doubting about the definition of 
$$\theta$$
 or the polar coordinate system.