Commit 0c1d63da authored by Friedrich Beckmann's avatar Friedrich Beckmann
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......@@ -135,7 +135,7 @@ frametitle={#1},
%\subject{Hochschule Augsburg}
%\titlehead{Hochschule Augsburg}
\author{Prof. Dr.-Ing. Friedrich Beckmann}
\date{June 2020 - Version 1} % Activate to display a given date or no date
\date{November 2021 - Version 1.0} % Activate to display a given date or no date
\KOMAoptions{titlepage=no}
%Requirementscounter
......@@ -283,29 +283,22 @@ Figure \ref{fig:prandtlca} shows the measurement results of lift coefficients fo
\label{fig:prandtl389}
\end{figure}
The lift coefficient for a given angle is reduced when the aspect ratio reduces. An increased aspect ratio will give a higher lift at a given angle, i.e. the slope of the lift curve is reduced. The wing with the aspect ratio 1:3 in figure \ref{fig:prandtlca} also shows that the maximum achievable lift is reduced. The basic behaviour is however very similar to the 2D wing, i.e. there is a more or less linear relation between lift coefficient and angle. The aspect ratio of a Boing 747-400 is 6.96 while for a motor glider Stemmer S10 it is 28.2 \cite[p. 203]{sadraey}.
The slope of the lift curve for an ideal infinite thin wing is $2 \pi$. Ludwig Prandtl estimates the reduced slope for a finite wing in equation \ref{equ:liftprand} \cite[p. 263]{drela} \cite{prandtl1}.
The lift coefficient for a given angle is reduced when the aspect ratio reduces. An increased aspect ratio will give a higher lift at a given angle, i.e. the slope of the lift curve is reduced. The wing with the aspect ratio 1:3 in figure \ref{fig:prandtlca} also shows that the maximum achievable lift is reduced. The basic behaviour is however very similar to the 2D wing, i.e. there is a more or less linear relation between lift coefficient and angle. The aspect ratio of a Boing 747-400 is 6.96 while for a motor glider Stemmer S10 it is 28.2 \cite[p. 203]{sadraey}. The slope of the lift curve for an ideal infinite thin wing is $2 \pi$ when the angle is given in radians. Ludwig Prandtl estimates the reduced slope for a finite wing in equation \ref{equ:liftprand} \cite[p. 263]{drela} \cite{prandtl1}.
\begin{align}
\frac{\partial C_l}{\partial \alpha}&= \frac{1}{1 + 2/AR} \\ \label{equ:liftprand}
\end{align}
An equivalent view of this behaviour is that the finite wing experiences an airflow which is bend by an additional induced angle $\alpha_i$. So the equivalent angle of attack is reduced compared to the angle of attack for an infinite 2D wing. To achieve the lift of the 2D infinite wing one has to add an additional induced angle $\alpha_i$ to the angle of attack $\alpha_{2D}$ of the infinite wing \cite{prandtl1}[p. 37].
\begin{figure}[!htb]
\centering
\includegraphics[width=0.7\textwidth]{prandtl-induced-angle}
\caption{Airflow showing the induced angle $\alpha_i$ due to induced drag $W_i$ (from \cite{prandtl1}[p. 37]) }
\label{fig:prandtl-ai}
\end{figure}
Figure \ref{fig:prandtl-ai} shows the bend airflow with the induced drag $W_i$. The angle of attack for the infinite wing $\alpha_{2D}$ is related to the finite wing $\alpha_{3D}$ according to equation \ref{equ:prandtl-ai} \cite{prandtl1}[p. 37].
\begin{equation}
\alpha_{2D} = \alpha_{3D} - \frac{c_l}{\pi \cdot AR} \label{equ:prandtl-ai}
\end{equation}
Note that the angles are computed in radians.
\subsubsection{Lift for the XUAV-Minitalon}
......
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