Airborne Wind Energy
Basic mechanics of crosswind kites
26 September 2025
Outline
- Definitions and assumptions
- Kite kinematics
- Kite aerodynamics
- Tethered flight
- Steady gliding flight
- Tethered static flight
- Ideal crosswind flight
- Radial flight
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.
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.
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 properties
Flat wing surface area (unrolled)
Wing planform area (projected)
\[S, \quad {\rm m}^2\]
Wing span
\[b, \quad {\rm m}\]
Wing side area (projected)
Wing maximum chord
\[\cmax, \quad {\rm m}\]
Airfoil geometry
Kite mass¹
\[\mk, \quad {\rm kg}\]
1 Often also just denoted as \(m\), without subscript.
Example: TU Delft V3 kite
Kite kinematics
Kite velocity
\[\vvk\]
Apparent wind velocity
\[\vva = \vvw - \vvk\]
Resultant aerodynamic force
\[\Fa = \frac{1}{2}\rho\CR S \vaexp{2}, \quad
\text{with} \quad \vaexp{2} = \vva\cdot\vva\]
Apparent wind velocity
Kite velocity
\[\vvk\]
Apparent wind velocity
\[\vva = \vvw - \vvk = \vvw + (-
\vvk)\]
Determining aerodynamic coefficients
Wind tunnel measurement
- Rigid scale model with 1.28 m span
- Unloaded design shape
- Variable angle of attack & side slip angle
- Reynolds number lower than real kite
Computational Fluid Dynamics
- CAD model before meshing
- Unloaded design shape
- Variable angle of attack & side slip angle
- Uncertainty about captured physics
Aerodynamic characteristics
- Data for TU Delft V3 kite with empirical corrections for different flight phases: powered, depowered, steered.
Aerodynamic lift devices
Airfoil
- Full range of \(L/D\)
- \(D\) can be minimised
- Control by angle of attack
Rotating cylinder
- Limited \(L/D\)
- \(D\) can not be modified
- Control by angular velocity
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
\[E = \frac{\vkx}{\vkz} =
\frac{1}{\tan\gamma} \quad ⤳ \quad \text{glide angle }
\gamma\]
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
Same shape but different scale, in fact, different units (N vs m/s).
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.
Static massless kite
Kite velocity
\[\vvk = 0\]
Apparent wind
velocity
\[\vva = \vvw - \vvk =
\vvw\]
Static force
equilibrium
\[\vFt + \vFa = 0\]
\[\Ft = \Fa = \sqrt{L^2 + D^2}\]
\[\begin{align} \tan\beta & = \frac{L}{D} \\ \beta & = \arctan\frac{L}{D} \end{align}\]
\[\lim_{\frac{L}{D}⤳\infty} \beta = \frac{\pi}{2}\]
Effect of gravity
Static force equilibrium
\[\vFt + \vFa + m\vg = 0\]
\[\Ft = \sqrt{\left(L-mg\right)^2 + D^2}\]
\[\begin{align} \tan\beta & = \frac{L-mg}{D} \\ & = \frac{L}{D} \left[1-\frac{mg}{L}\right] \\ & = E\,\left[1-\frac{2gm}{\rho\CL S \vwexp2}\right] \end{align}\]
\[\begin{align} \phantom{\tan\beta} & = E\,\left[1-\frac{\color{red}{\frac{2g}{\rho\CL}\frac{m}{S}}}{\vwexp2}\right] \end{align}\]
But what is the physical meaning of \(\color{red}{\frac{2g}{\rho\CL}\frac{m}{S}}\)?
Static take-off limit
Generated lift just balances the effect of gravity:
\[\frac{1}{2}\rho\CL S
\vwexp2=mg\]
Equivalent to \(\beta = 0\) (horizontal tether).
Solving for \(\vw\) leads to static take-off wind
speed:
\[\vwsto =
\sqrt{\frac{2g}{\rho\CL}\frac{m}{S}}\]
Mass-to-area ratio \(m/S\) is an important scaling parameter affecting take-off limit
Definition identical to the stall speed \(\vstall\) of an aircraft (Anderson 2016).
Effect of gravity
Static force equilibrium
\[\vFt + \vFa + m\vg = 0\]
\[\Ft = \sqrt{\left(L-mg\right)^2 + D^2}\]
\[\begin{align} \tan\beta & = \frac{L-mg}{D} \\ & = \frac{L}{D} \left[1-\frac{mg}{L}\right] \\ & = E\,\left[1-\frac{2gm}{\rho\CL S \vwexp2}\right]\\ & = E\,\left[1-\frac{\color{red}{\frac{2g}{\rho\CL}\frac{m}{S}}}{\vwexp2}\right] \end{align}\]
\[\begin{align} \phantom{\tan\beta} & = E\,\left[1-\left(\frac{\color{red}{\vwsto}}{\vw}\right)^2\right], \quad \text{with} ~ \vwsto = \sqrt{\frac{2g}{\rho\CL}\frac{m}{S}} \end{align}\]
Ideal crosswind flight
TU Delft V3 kite flying on launch mast in 2012
Massless crosswind kite
Reeling factor ⤳
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
\[\frac{L}{D} = \frac{\vky}{\vw-\vkx} \qquad
⤳ \qquad \frac{L}{D}\left( \vw - \vkx \right) = \vky \\
\lambda = \frac{L}{D} \left( 1-f \right) = E \left( 1-f \right)
\quad \quad \text{dependent variable, i.e. not a degree of
freedom!}\]
Massless crosswind kite
Apparent wind velocity
\(\begin{aligned}
\vaexp{2} & = \left( \vw-\vkx \right)^2 + \vky^2 \qquad ⤳ \qquad
\left(\frac{\va}{\vw}\right)^2 = \left( 1-f \right)^2 + \lambda^2\\
\frac{\va}{\vw} & = \left( 1-f \right) \sqrt{1+E^2}
\end{aligned}\)
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
\[\zeta = \frac{P}{\Pw S} = C_L
\sqrt{1+\frac{1}{E^2}} f \left( 1-f \right)^2 \left( 1+E^2 \right),
\quad \text{with} \qquad \Pw =
\frac{1}{2}\rho\vwexp{3}\]
Maximum reel-out factor?
- At \(f=1\) the apparent wind speed
vanishes
⤳ kite can not produce any lift. - But what happens for \(f>1~(\vt>\vw)\)?
⤳ gliding in downwind direction,
⤳ vanishing tether tension.
- Practical application is step towing,
by
alternating downwind gliding with \(f>1\), and
upwind retraction of the kite with \(f<0\). - Technique well known from winch
launching of paragliders.
Note: for \(\beta>0\) and/or \(\phi \ne 0\), \(\va\) vanishes for \(f<1\).
Crosswind kite with mass
Static force equilibrium
\[\vFt + \vL + \vD + m\vg = 0\]
Horizontal lift force
component
\[|\vec{L} + m\vec{g}| = \sqrt{L^2 -
\left(mg\right)^2}\]
Geometric
similarity
\[\frac{\vky}{\vw-\vkx} = \frac{\sqrt{L^2 -
\left(mg\right)^2}}{D}\]
Tangential velocity
factor
\[\lambda = (1-f)E\sqrt{1 - \left(
\frac{mg}{L} \right)^2}\]
2025-04-10-resit
Crosswind kite with mass
Apparent wind speed
\[\begin{align}
\left( \frac{\va}{\vw} \right)^2 & = (1-f)^2 + \lambda^2 \\
& = \left( 1 - f \right)^2 \left\{
1 + E^2\left[ 1-\left(\frac{mg}{L}\right)^2 \right] \right\} \\
& = \left( 1 - f \right)^2 \left\{
1 + E^2\left[
1-\left(\frac{\frac{2g}{\rho\CL}\frac{m}{S}}{\vaexp{2}}\right)^2 \right]
\right\}
\end{align}\]
Substitute
\[\color{red}{x} = \left( \frac{\va}{\vw}
\right)^2\quad ⤳\]
\[\color{red}{x} = \left( 1 - f \right)^2 \left\{ 1 + E^2\left[ 1-\left(\frac{\vwsto}{\vw}\right)^4 \frac{1}{\color{red}{x^2}}\right]\right\}\]
2025-04-10-resit
Crosswind kite with mass
\[\color{red}{x} = \left( 1 - f \right)^2 \left\{ 1 + E^2 - E^2\left(\frac{\vwsto}{\vw}\right)^4 \frac{1}{\color{red}{x^2}}\right\}\]
Cubic equation, with missing linear term
\[\color{red}{x^3} - \left( 1 - f \right)^2
\left( 1 + E^2 \right) \color{red}{x^2} + \left( 1 - f \right)^2 E^2
\left(\frac{\vwsto}{\vw}\right)^4 = 0\]
Cubic equation in standard form
\[a x^3 + b x^2 + c x + d = 0, \quad
\text{with}\]
\[\begin{align} a & = 1, \\ b & = - \left( 1 - f \right)^2 \left( 1 + E^2 \right), \\ c & = 0, \\[-15px] d & = \left( 1 - f \right)^2 E^2 \left(\frac{\vwsto}{\vw}\right)^4. \\ \end{align}\]
2025-04-10-resit
Cubic polynomial
Polynomial function and derivatives
\[\begin{alignat*}{2}
&p(x) &&= x^3 - \left( 1 - f \right)^2 \left( 1 + E^2
\right) x^2 + \left( 1 - f \right)^2 E^2
\left(\frac{\vwsto}{\vw}\right)^4, \\[-6mm]
&p'(x) &&= 3x^2 - 2\left( 1 - f \right)^2 \left( 1 +
E^2 \right) x, \\
&p''(x) &&= 6x - 2\left( 1 - f \right)^2 \left( 1 +
E^2 \right).
\end{alignat*}\]
⤳ Wind speed \(\vw\) affects vertical offset of polynomial
End behavior
\[\begin{alignat*}{3}
&x\to & -\infty &⤳ p(x)\to & ~ -\infty, \\
&x\to & \infty &⤳ p(x)\to & ~ \infty.
\end{alignat*}\]
Critical points: local
extrema \((p'=0)\)
\[\begin{alignat*}{4}
&x &&
=0 &&⤳
\text{local maximum} && (p'' < 0). \\
&\xc && =\frac{2}{3} \left( 1 - f \right)^2 \left( 1 + E^2
\right) &&⤳ \text{local minimum} \quad && (p''
> 0).
\end{alignat*}\]
Polynomials for the TU Delft V3 kite at \(f=0\).
Physical solution space
\(x \ge \xmin = (1-f)^2.\)
This lower limit follows from the definition of the
non-dimensional apparent wind speed as
\[x = (1-f)^2+\lambda^2,\]
and the fact that the crosswind contribution is always positive, i.e. \(\lambda^2 \ge 0\).
\(x \le \xmax = \left( 1 - f \right)^2 \left( 1 + E^2 \right).\)
This upper limit is the apparent wind speed for Loyd’s ideal crosswind case, where \(m=0\).
Interesting is also the observation
\[\xc = \frac{2}{3} \xmax.\]
Polynomials for the TU Delft V3 kite at \(f=0\).
Minimum flyable wind speed
- At \(\vw=\frac{1}{4}\vwsto\), the cubic equation has no real roots in the physical solution space.
- At \(\vw=\frac{1}{3}\vwsto\), the cubic equation has two real roots in the physical solution space.
- At what value of \(\vw\) does the cubic equation have a repeated real root, transitioning from no real root to two real roots?
- This transition is described mathematically by the discrimonant \(\Delta\) changing the sign from negative to positive.
Polynomials for the TU Delft V3 kite at \(f=0\).
Full wind speed range
Tangential velocity factor \(\lambda_\mathrm{max} = \lambda(m=0)= E\).
Radial flight
Massless radial kite
Reeling factor ⤳
degree of freedom
\[f = \frac{\vk}{\vw}\]
Apparent wind
velocity
\[\vva = \vvw - \vvk\]
Decomposition of kite
velocity
\[\vvk = \vec{b} + \vec{c}\]
\[\vva = \vvw - \vec{b} - \vec{c}\]
\[\vvw = \vva + \vec{c} + \vec{b}\]
\[\vw^2 = \left( \va - c \right)^2 + b^2\quad
\text{(positive vector magnitudes!)}\] \[\frac{\va}{\vw} =
\sqrt{1-\left(\frac{b}{\vw}\right)^2} +
\frac{c}{\vw}\]
Displayed is the case for reel in with \(f<0\)
Massless radial kite
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 \(f\)
Displayed is the case for reel in with \(f<0\)
Massless radial kite
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
\[\zeta = \CL
\frac{f\left[\sqrt{1+E^2(1-f^2)}-f\right]^2}{E\sqrt{1+E^2}}\]
Displayed is the case for reel in with \(f<0\)
Loyd’s theory - comparison
