# Tower Influence

A tower shadow model, based on the work of Bak (Moriarty and Hansen1) is implemented in QBlade. This model is based on a superposition of the analytical solution for potential flow around a cylinder and a model for the downwind wake behind a cylinder, based on a tower drag coefficient.

For the potential flow around the cylinder contribution, the local velocity components $$U_{\mathrm{local}}$$ and $$V_{\mathrm{local}}$$ are affected by normalized velocity factors:

\begin{split}\begin{align} U_{\mathrm{local}} &= u\cdot U_{\infty}, \\ V_{\mathrm{local}} &= v\cdot U_{\infty}. \\ \end{align}\end{split}

In the above equations, $$U_{\infty}$$ is the free stream velocity and $$u$$ and $$v$$ are given by:

\begin{split}\begin{align} u &= 1-\frac{\left(x+0.1\right)^2 - y^2}{\left( \left(x+0.1\right)^2 + y^2\right)^2} + \frac{C_d}{2\pi} \frac{x+0.1}{\left(x+0.1\right)^2 + y^2}, \\ v &= 2\frac{\left(x+0.1\right)y}{\left( \left(x+0.1\right)^2 + y^2 \right)^2} + \frac{C_d}{2\pi} \frac{y}{\left(x+0.1\right)^2 + y^2}. \\ \end{align}\end{split}

In these equations, $$x$$ and $$y$$ are the upwind and crosswind distances normalized by the tower radius at the relevant height. $$C_d$$ is the drag coefficient of the the tower.

In addition, the tower produces a wake deficit in the downstream direction. The deficit inside the location of the wake is given by:

\begin{align} U_{\mathrm{local}} = (1-u_{\mathrm{wake}})\cdot U_{\infty}, \end{align}

where $$u_{\mathrm{wake}}$$ is given by:

\begin{align} u_{\mathrm{wake}} = \frac{C_d}{\sqrt{d}}\cos^2 \left( \frac{\pi}{2} \frac{y}{\sqrt{d}} \right) \qquad \mathrm{for} \quad |y|\leq\sqrt{d}. \end{align}

In the above equation, $$d=\sqrt{x^2+y^2}$$ is the non-dimensional radial distance from the evaluation point to the tower center. The wake width is assumed to be $$\sqrt{d}$$.

The tower shadow model only affects velocity components that are normal to the tower centerline; the z-component of the velocity, parallel to the tower centerline, remains unaffected. The tower shadow model is only used when the z-component of the evaluation point is smaller or equal to the tower height. An example of the tower shadow velocity deficit is shown in Fig. 11.