Basic aerodynamic equations

I collect here some basic equatin wich we need e.g. for airfoil dimensioning.

Total lift force

\[ \begin{equation} F_{L}=\frac{1}{2}\rho v^2 c_{L} S=m_{plane} \cdot g \label{eq:liftForce1} \end{equation} \]

with
\(F_{L}\): lift force (N)
\(\rho\): denisty of air (we typ. use 1.1673 kg/m³ at 500m)
\(v\): velocity (of the plane, m/s)
\(c_{L}\): total lift coefficient
\(S\): projected surface (m²)

We use large letters as indices, e.g. \(F_{L}\) and \(c_{L}\) to indicate total (integrated over the whole wing) values in contrast to local values (e.g. given for a wing section).

Stall velocity

We regularly need to determine the lowest velocity, a plane can be flown without loosing height. Therefore, we resolve \(\eqref{eq:liftForce1}\) for \(v\):

\[ \begin{equation} v=\sqrt{\frac{2\cdot m_{plane}\cdot g}{\rho c_{L} S}}\label{eq:velocityFromMass} \end{equation} \]

Reynolds number

\[ \begin{equation} Re=\frac{\rho \cdot v \cdot ch}{\mu}=\frac{v\cdot ch}{\nu} \label{eq:reynolds} \end{equation} \]

with
\(\mu\): dynamic viscosity
\(\nu\): kinematic viscosity of air (we typ. use 1.52 E-05 m²/s at 500m)

ResqrtCl

Solving \(\eqref{eq:reynolds}\) for \(v\), insertion in \(\eqref{eq:liftForce1}\) and grouping the terms gives:

\[ \begin{equation} \underbrace{F_{L}=m_{plane\cdot g}}_{const_1} =\underbrace{\frac{1}{2}\rho \frac{\nu^2\cdot S}{ch^2}}_{const!} \cdot \underbrace{Re^2\cdot C_{L}}_{\Rightarrow const} \label{eq:resqrtcl1} \end{equation} \]

First, we note that, in horicontal flight, the lift force needs to be constant (\(const_1\)). For higher velocities \(v\) we require less \(C_l\) and vice versa. All the values aggregated as \(const!\) are definitely constant. From this follows that \(Re^2\cdot C_{L}\) needs to be constant, too. Hence:

\[ \begin{equation} Re\cdot\sqrt{C_{L}}=const \mid_{horicontal\ flight} \label{eq:resqrtcl2} \end{equation} \]

Note: at a given wing section we have local values (small indices)

\[ \begin{equation} Re\cdot\sqrt{C_{l}}=const \mid_{@wing\ section,\ horicontal\ flight} \label{eq:resqrtcl3} \end{equation} \]