Terminology

• Velocity is a vector property.
• Speed is a scalar property, the magnitude
  of the velocity vector.
• A kite can be of fixed-wing of soft-wing type.

Mathematical notation

• Scalars are set in italic: \(a\), \(b\), \(\rho\), \(\phi\)
• In digital resources, vectors are set in bold roman: \(\vec{v}\), \(\vec{L}\), \(\vec{D}\)
• In handwritten resources, vectors are overset with an arrow: \(\overset{→}{v}\), \(\overset{→}{L}\), \(\overset{→}{D}\)
• Units are set in roman: \(\rm m\), \(\rm kg\)
• Mathematical functions are set in roman: \(\sin\), \(\cos\), \(\max\)
• Product of scalars: \(a b\)
• Scalar product of vectors: \(\vec{v}\cdot\vec{v} = v^2\)
• Cross product of vectors: \(\vec{v} = \boldsymbol\omega\times\vec{r}\)
• Cartesian unit vectors

Environmental properties

Ambient wind velocity
\[\vvw, \quad {\rm m/s}\]

Air density
\[\rho, \quad {\rm kg/m}^3\]

Dynamic wind pressure
\[q = \frac{1}{2}\rho\vw^2, \quad {\rm N/m}^2\]

Wind power density
\[\Pw = \frac{1}{2}\rho\vw^3, \quad {\rm W/m}^2\]

Gravitational acceleration
\[\vec{g}, \quad {\rm m/s}^2\]

Kite geometry

Flat wing surface area (unrolled)

Wing planform area (projected)
\[S, \quad {\rm m}^2\]

Wing span
\[b, \quad {\rm m}\]

Wing maximum chord
\[\cmax, \quad {\rm m}\]

Wing side area (projected)

Airfoil geometry

Kite mass
\[\mk, \quad {\rm kg}\]

Kite kinematics

Kite velocity
\[\vvk\]

Apparent wind velocity
\[\vva = \vvw - \vvk\]

Resultant aerodynamic force
\[\Fa = \frac{1}{2}\rho\CR S \va^2\]

Relative flow

Aerodynamic forces

Determining aerodynamic coefficients

Jelle Poland

Aerodynamic characteristics

• CFD: OpenFOAM RANS simulation
• VSM: Vortex Step Method of Cayon et al. (2023)
• WT: Wind tunnel measurements (Open Jet Facility of TU Delft)

Jelle Poland

Aerodynamic lift devices

Approach

Start from the minimal setup, the untethered wing.
Step by step add physical components,
add motion components,
and compute forces acting on the system.

Analytic modelling framework
\[\zeta = \frac{P}{\Pw S} = \CR\left[1+\left(\frac{L}{D}\right)^2\right] f \left(\cos\beta\cos\phi - f\right)^2\]

Stagnant wind

Kite velocity
\[\vvk = \vvkx + \vvkz = \const\]

Glide ratio
\[E = \frac{\vkx}{\vkz} = \frac{1}{\tan\gamma} \quad \to \quad \text{glide angle } \gamma\]

Apparent wind velocity
\[\vva = \vvw -\vvk = -\vvk, \quad \text{with} \qquad \vw=0\]

Steady force equilibrium
\[m\vg + \vFa = 0\]

Geometric similarity of kinematic and force triangles because \(\vvkz \parallel \vFa\) and \(\vvk \parallel \vec{D}\)
\[E = \frac{\vkx}{\vkz} = \frac{L}{D}\]

Geometric similarity explained

Uniform and constant wind

Effect of additional wind velocity \(\vw > 0\)

If wing flies against wind, glide angle \(\gamma\) gets steeper

Apparent wind velocity
\[\begin{align} \vva & = \vvw -\vvk\\ \vax & = \vw -\vkx \\ \vaz & = -\vkz \end{align}\]

Geometric similarity of kinematic and force triangles
\[E = \frac{\vax}{\vaz} = \frac{L}{D}\]

Application in AWE:
limiting case of retraction in pumping cycle with
vanishing tether tension.

Massless kite

Kite velocity
\[\vvk = 0\]

Apparent wind velocity
\[\vva = \vvw - \vvk = \vvw\]

Static force equilibrium
\[\vFt + \vFa = 0\]

\[\Ft = \Fa = \sqrt{L^2 + D^2}\]

\[\begin{align} \tan\beta & = \frac{L}{D} \\ \beta & = \arctan\frac{L}{D} \end{align}\]

\[\lim_{\frac{L}{D}\to\infty} \beta = \frac{\pi}{2}\]

Kite with mass

Static force equilibrium
\[\vFt + \vFa + m\vg = 0\]

\[\Ft = \sqrt{\left(L-mg\right)^2 + D^2}\]

\[\begin{align} \tan\beta & = \frac{L-mg}{D} \\ & = \frac{L}{D}\left(1-\frac{mg}{L}\right)\\ & = \frac{L}{D}\left(1-\frac{1}{\mu}\right), \quad \text{with} \qquad \mu=\frac{\rho \CL S \vwexp2}{2 m g} \end{align}\]

Limiting cases
\[\begin{align} mg=0\,~ (\mu\to\infty) & \quad \to \quad \beta=\arctan{\frac{L}{D}}\\ mg=L ~ (\mu=1)\quad & \quad \to \quad \beta=0 \end{align}\]

Tether sag

Ideal crosswind kite

Ideal crosswind kite - theory

Reeling factor \(\to\) degree of freedom
\[f = \frac{\vkx}{\vw}\]

Tangential velocity factor
\[\lambda = \frac{\vky}{\vw}\]

Apparent wind velocity
\(\begin{aligned} \vva & = \vvw - \vvk \\ & = \left( \vw - \vkx \right)\vex + \left( - \vky \right)\vey \end{aligned}\)

Geometric similarity of velocity and force triangles
\[\frac{L}{D} = \frac{\vky}{\vw-\vkx} \qquad \to \qquad \frac{L}{D}\left( \vw - \vkx \right) = \vky \\ \lambda = \frac{L}{D} \left( 1-f \right) = E \left( 1-f \right) \quad \to \quad \text{not a degree of freedom!}\]

Ideal crosswind kite - theory

Apparent wind velocity
\(\begin{aligned} \vaexp{2} & = \left( \vw-\vkx \right)^2 + \vky^2 \qquad \to \qquad \left(\frac{\va}{\vw}\right)^2 = \left( 1-f \right)^2 + \lambda^2\\ \frac{\va}{\vw} & = \left( 1-f \right) \sqrt{1+E^2} \end{aligned}\)

Aerodynamic force
\[\Fa = \frac{1}{2}\rho S \CR \vaexp{2} = \frac{1}{2}\rho S \CL \sqrt{1+\frac{1}{E^2}} \left(\frac{\va}{\vw}\right)^2 \vwexp{2}\]

Non-dimensional tether force
\[\frac{\Ft}{qS} = \CL \sqrt{1+\frac{1}{E^2}} \left( 1-f \right)^2 \left( 1+E^2 \right), \quad \text{with} \qquad q=\frac{1}{2}\rho\vwexp{2}\]

Power harvesting factor
\[\zeta = \frac{P}{\Pw S} = C_L \sqrt{1+\frac{1}{E^2}} f \left( 1-f \right)^2 \left( 1+E^2 \right), \quad \text{with} \qquad \Pw = \frac{1}{2}\rho\vwexp{3}\]

Ideal crosswind kite - optimal reel-out speed

Maximize by extereme value analysis
\(f \left( 1-f \right)^2\)

Optimal reeling factor
\[\fopt=\frac{1}{3}\]

Maximum harvesting factor
\[\zetaopt = \frac{4}{27} C_L \sqrt{1+\frac{1}{E^2}}\left( 1+E^2 \right)\]

Maximum harvesting factor for high-performance kites
\[\lim_{E\to\infty} \zetaopt = \frac{4}{27} C_LE^2\]

Non-maneuvering kite

Non-maneuvering kite - theory

Reeling factor \(\to\) degree of freedom
\[f = \frac{\vk}{\vw}\]

Apparent wind velocity
\[\vva = \vvw - \vvk\]

Decomposition of kite velocity
\[\vvk = \vec{b} + \vec{c}\] \[\vva = \vvw - \vec{b} - \vec{c}\] \[\vvw = \vva + \vec{c} + \vec{b}\] \[\vw^2 = \left( \va - c \right)^2 + b^2\quad \text{(positive vector magnitudes!)}\] \[\frac{\va}{\vw} = \sqrt{1-\left(\frac{b}{\vw}\right)^2} + \frac{c}{\vw}\]

Non-maneuvering kite - theory

Geometric similarity
\[\frac{b}{\vk} = \frac{L}{\Fa} = \frac{1}{\sqrt{1+\frac{1}{E^2}}}\] \[\frac{c}{\vk} = \frac{D}{\Fa} = \frac{1}{\sqrt{1+E^2}}\]

Normalized to the wind speed
\[\frac{b}{\vw} = -\frac{f}{\sqrt{1+\frac{1}{E^2}}}\] \[\frac{c}{\vw} = -\frac{f}{\sqrt{1+E^2}}\]

Minus signs aacount for negative value of \(f\)

Non-maneuvering kite - theory

Apparent wind speed
\[\begin{align} \frac{\va}{\vw} & = \sqrt{1 - \frac{f^2}{1 + \frac{1}{E^2}}} - \frac{f}{\sqrt{1 + E^2}}, \\ & = \frac{\sqrt{1 + E^2\left( 1 - f^2 \right)} - f}{\sqrt{1 + E^2}} \end{align}\]

Non-dimensional tether force
\[\frac{\Ft}{qS} = \CL \frac{\left[\sqrt{1+E^2(1-f^2)}-f\right]^2}{E\sqrt{1+E^2}}\]

Power harvesting factor
\[\zeta = \CL \frac{f\left[\sqrt{1+E^2(1-f^2)}-f\right]^2}{E\sqrt{1+E^2}}\]

Non-maneuvering kite - theory

Elevation angle \(\to\) not a degree of freedom
\[\cos\beta = \frac{\sqrt{1+E^2(1-f^2)}+fE^2}{1+E^2}\]

Derivation of minimum reeling factor \(f_{\rm min}\)

Loyd’s theory - comparison

awesco.eu/awe-explained/

How to use these analytic models?

The wind speed \(\vw\) describes the available energy resource.
The lift-to-drag ratio \(E\) characterizes the specific kite design.

For the ideal crosswind kite prescribe

the tether reeling speed \(\vt\) to calculate the achievable kite crosswind speed \(\vkx\), or
the kite crosswind speed \(\vkx\) to calculate the required tether reeling speed \(\vt\).

For the non-maneuvering kite prescribe

the tether reeling speed \(\vt\) to calculate the achievable elevation angle \(\beta\), or
the elevation angle \(\beta\) to calculate the required tether reeling speed \(\vt\).