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Turbine theory: Tower disturbance |
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Turbine theory: Tower disturbance |
The presence of a tower can optionally be accounted for by choosing which tower influence and tower shadow models modify the undisturbed wind field.
The effect of the tower on the flow, at the aerodynamic centre of each blade mid-segment, is determined before the aerodynamic loading is calculated. To calculate this disturbed wind velocity, the point of closest approach on the tower is first identified. This is the point on the tower axis which is, instantaneously, closest to the blade segment's aerodynamic centre. At this point, the undisturbed relative wind velocity, normal to the tower, $\vec{u\urm{n}}$, is first determined, and then the disturbance to the flow, $\Delta \vec{v}$, at the aerodynamic centre is calculated according to the choice of tower influence and shadow models. These models are framed using the non-dimensional components $x$ and $y$, measured from the point of closest approach, on the tower, to the aerodynamic centre, normalised by the tower radius at the closest approach. Here, $x$ is in the direction of $\vec{u\urm{n}}$ and $y$ is perpendicular to this and the tower axis.
If none is selected for both the tower influence and tower shadow, then there is no effect: the disturbed wind field is identical to the undisturbed, and we have simply \begin{equation} \Delta \vec{v} = 0 \end{equation} Otherwise, the total flow disturbance is the sum of that associated with the tower influence, $\Delta \vec{v\urm{i}}$, and the tower shadow, $\Delta \vec{v\urm{s}}$, \begin{equation} \Delta \vec{v} = \Delta \vec{v\urm{i}} + \Delta \vec{v\urm{s}} \end{equation}
If the closest approach is beyond either end of the tower, e.g. the aerodynamic centre is above the tower, the disturbance to the flow is linearly scaled to zero over one tower radius.
The disturbed wind velocity, $\vec{v}$, is found by adding the total flow disturbance as calculated above, $\Delta \vec{v}$, to the undisturbed relative velocity at the aerodynamic centre. This is reoriented to be with respect to the nominal rotor plane frame before being used in the aerodynamic calculation.
Based on potential theory, these models modify the wind field both upwind and downwind of the tower. They are commonly used to capture the change in the wind field experienced by an upwind rotor, due to the so-called tower dam effect. Two tower influence models, adapted from AeroDyn, are available: potential, which employs the classical analytic solution to potential flow around a cylinder; and Bak, a correction to the potential solution, parametrised by the tower drag coefficient.
The disturbance to the flow, due to the potential influence model is given by \begin{equation} \begin{aligned} \Delta v\urm{i,x} &= - |\vec{u\urm{n}}| \frac{x^2 - y^2}{d^2} \\ \Delta v\urm{i,y} &= - |\vec{u\urm{n}}| \frac{2 x y}{d^2} \end{aligned} \end{equation} where \begin{equation} d = x^2 + y^2 \end{equation} $\Delta v\urm{i,x}$ and $\Delta v\urm{i,y}$ are the disturbance to the flow in the $x$ and $y$ directions respectively.
The disturbance to the flow due to the Bak influence model is given by \begin{equation} \begin{aligned} \Delta v\urm{i,x} &= - |\vec{u\urm{n}}| \frac{x\urm{s}^2 - y^2}{d\urm{s}^2} + |\vec{u\urm{n}}| \frac{C\urm{d} x\urm{s}}{ 2 \pi d\urm{s} } \\ \Delta v\urm{i,y} &= - |\vec{u\urm{n}}| \frac{2 x\urm{s} y}{d\urm{s}^2} + |\vec{u\urm{n}}| \frac{C\urm{d} y}{ 2 \pi d\urm{s} } \end{aligned} \end{equation} where $x\urm{s} = x + 0.1$, $C\urm{d}$ is the tower drag coefficient at the closest approach and \begin{equation} d\urm{s} = x\urm{s}^2 + y^2 \end{equation}
These models are empirical approximations of the velocity deficit in the wake behind the tower and so only modify the wind field downwind of the tower. They are therefore appropriate when modelling a downwind rotor. Two shadow models, again adapted from AeroDyn, are available: Powles and Eames. Both shadow models are parametrised by the tower drag coefficient.
The disturbance to the flow due to the Powles shadow model is given by \begin{equation} \Delta v\urm{s,x} = \begin{cases} - |\vec{u\urm{n}}| \max\left\{0.5, \frac{C\urm{d}}{d^{1/4}} \cos^2 \left( \frac{\pi y}{2 d^{1/4}} \right) \right\} & \text{if $x \gt 0$} \\ 0 & \text{otherwise} \end{cases} \end{equation} where \begin{equation} d = x^2 + y^2 \end{equation} and $C\urm{d}$ is the tower drag coefficient at the closest approach.
The disturbance to the flow due to the Eames shadow model is given by \begin{equation} \Delta v\urm{s,x} = \begin{cases} - |\vec{u\urm{n}}| \max\left\{0.5, \frac{C\urm{d}}{y\urm{w} \sqrt{2 \pi}} \exp{\left( -\frac{y^2}{2 y\urm{w}^2} \right)} \right\} & \text{if $x \gt 0$} \\ 0 & \text{otherwise} \end{cases} \end{equation} where \begin{equation} y\urm{w} = I\urm{t} x \end{equation} $I\urm{t}$ is the tower turbulence intensity at the closest approach.