Assumptions

• Massless kite and tether
• Straight tether
• Uniform and constant wind field (= parallel to the ground)
• Constant aerodynamic coefficients
• All forces acting on kite can be concentrated in one point $\vec{K}$
• Flight of the kite is a transition through steady states (quasi-steady theory)
• Steering of the kite is not taken into account

Tethered flight in three dimensions

Arbitrary flight state

Kite represented as point $\vec{K}$

Kite position described by
azimuth angle $\phi$
elevation angle $\beta$

Apparent wind velocity
$$\vva = \vvw - \vvk$$

The following analysis is in the
plane spanned by the radial unit
vector $\ver$ and the apparent wind
velocity $\vva$

Planar analysis

Decomposition into radial and tangential components
$$\vva = \vvar + \vvat$$

$\tau:$ local tangential plane, $\perp$ to tether

Resultant aerodynamic force
$$\vFa = \vec{L} + \vec{D}$$

Geometric similarity of force and velocity triangles
$$\frac{L}{D} = \frac{\vat}{\var} \quad \to \quad \vat = \frac{L}{D} \var$$

Radial component of apparent wind velocity
$$\var = \vwr - \vkr$$

$\vkr = f\vw$, but how to get $\vwr$

Radial wind velocity component

For this, we go back to three dimensions and
perform two subsequent orthogonal projections

1. Orthogonal projection by azimuth angle $\phi$
$\to$ multiply $\vw$ by $\cos\phi$

2. Orthogonal projection by elevation angle $\beta$
$\to$ multiply result by $\cos\beta$

Radial wind velocity component
$$\vwr = \cos\beta \cos\phi\, \vw$$

Apparent wind speed and tether force

Radial component of apparent wind velocity
$$\var = \cos\beta \cos\phi\, \vw - \vkr$$

Nondimensionsionalization and recombination
$$\left.\begin{aligned} \frac{\var}{\vw} & = \cos\beta \cos\phi - f \quad \\ \frac{\vat}{\vw} & = E\var \end{aligned} \right\} \quad \frac{\va}{\vw} = \sqrt{1 + E^2} \left( \cos\beta \cos\phi - f \right)$$

Nondimensional tether force
$$\begin{aligned} \frac{\Ft}{q S} & = \CL \sqrt{1 + \frac{1}{E^2}} \left( \frac{\va}{\vw} \right)^2 , \quad \text{with} \quad q=\frac{1}{2}\rho\vwexp{2}\\ & = \CL \sqrt{1 + \frac{1}{E^2}} \left( 1 + E^2 \right) \left( \cos\beta \cos\phi - f \right)^2 \end{aligned}$$

Mechanical power

Power harvesting factor
$$\zeta = \frac{P}{\Pw S} = \CL \sqrt{1 + \frac{1}{E^2}} \left( 1 + E^2 \right) f \left( \cos\beta \cos\phi - f \right)^2, \quad \text{with} \quad \Pw=\frac{1}{2}\rho\vwexp{3}$$

Definition of spherical coordinates

• Coordinate 1: radial distance $r$
• Coordinate 2: polar angle $\theta$
• Coordinate 3: azimuth angle $\phi$
• Other angular choices and ordering of coordinates
  are possible to define right-handed coordinate system
• This definition $(r, \theta, \phi)$ follows the ISO convention
  and is common in theoretical physics
• Unlike Cartesian coordinates $(x, y, z)$,
  spherical coordinates define a curvilinear
  coordinate system

Spherical unit vectors

• Unit vectors $\ver$, $\vetheta$ and $\vephi$ vary in space
• Transformation
  $x = r \sin\theta \cos\phi$
  $y = r \sin\theta \sin\phi$
  $z = r \cos\theta$
• $\ver = \sin\theta\cos\phi\, \vex + \sin\theta\sin\phi\, \vey + \cos\theta\, \vez$
  $\vetheta = \cos\theta\cos\phi\, \vex + \cos\theta\sin\phi\, \vey - \sin\theta\, \vez$
  $\vephi = -\sin\phi\, \vex + \cos\phi\, \vey$
  

Quasi-steady theory in spherical coordinates

This part of the lecture is covered in much detail in (Schmehl et al. 2013). For now, please refer to this book chapter.

Power plot example

Notes

• $\lambda = 0$ does not imply that the apparent wind velocity vanishes
• It does mean that the tangential component (perpendicular to the tether) of the kite velocity vanishes.