%% $Date: 2012-07-06 17:42:54#$
%% $Revision: 148 $
\index{wind\_vector\_3d\_raf}
\algdesc{3D Wind Vectors}
{ %%%%%% Algorithm name %%%%%%
wind\_vector\_3d\_raf
}
{ %%%%%% Algorithm summary %%%%%%
This algorithm applies vector transformations using aircraft speed, angle of attack and sideslip to calculate the three-dimensional wind vector
components. 
}
{%%%%%% Category %%%%%%%
Thermodynamics
}  
{ %%%%%% Inputs %%%%%%
$U_{a}$ & Vector & Corrected true air speed [m/s] \\
$\alpha$ & Vector & Aircraft angle of attack [rad] \\
$\beta$ & Vector & Aircraft sideslip [rad] \\
$u_{p}$ & Vector & Easterly aircraft velocity from INS [m/s] \\
$v_{p}$ & Vector & Northerly aircraft velocity from INS [m/s] \\
$w_{p}$ & Vector & Upward aircraft velocity from INS [m/s] \\
$\phi$ & Vector & Roll [rad] \\
$\theta$ & Vector & Pitch [rad] \\
$\psi$ & Vector & True Heading [rad] \\
$\dot{\theta}$ & Vector & Pitch rate [rad/sec] \\
$\dot{\psi}$ & Vector & Yaw rate [rad/sec] \\
$L$ & Vector & Distance separating INS and gust probe\\
& & along aircraft center line [m]
}
{ %%%%%% Outputs %%%%%%
$u$ & Vector & Easterly wind velocity component [m/s] \\
$v$ & Vector & Northerly wind velocity component [m/s] \\
$w$ & Vector & Upwards wind velocity component (positive up) [m/s] \\ 
}
{ %%%%%% Formula %%%%%%
\begin{align*}
D &= \sqrt{(1 + \tan^{2} \alpha + \tan^{2} \beta)} \\
u &= - U_{a} D^{-1} \left[ \sin \psi \cos \theta  + \tan \beta (\cos \psi \cos \phi + \sin \psi \sin \theta \sin \phi) + \tan \alpha (\sin \psi \sin \theta \cos \phi - \cos \psi \sin \phi) \right] \\
   & \qquad{} + u_{p} - L (\dot{\theta} \sin \theta \sin \psi - \dot{\psi} \cos \psi \cos \theta) \\
v &= - U_{a} D^{-1}  \left[ \cos \psi \cos \theta - \tan \beta (\sin \psi \cos \phi - \cos \psi \sin \theta \sin \phi) + \tan \alpha ( \cos \psi \sin \theta \cos \phi + \sin \psi \sin \phi) \right] \\
    & \qquad{} + v_{p} - L (\dot{\psi} \sin \psi \cos \theta + \dot{\theta} \cos \psi \sin \theta) \\
w &= - U_{a} D^{-1} ( \sin \theta - \tan \beta \cos \theta \sin \phi - \tan \alpha \cos \theta \cos \phi) + w_{p} + L \dot{\theta} \cos \theta 
\end{align*}
}
{ %%%%%% Author %%%%%%

}
{ %%%%%% References %%%%%% 
NCAR-RAF Bulletin \#23 \cite{NCAR23} 
}


