Multi-Element Airfoils

Parameterization for optimization

When optimizing a multiple degree-of-freedom system like the positioning of elements in a multi-element airfoil, it can be important to have a smartly chosen set of independent variables. For example, if you change the height of only the 2nd element in a 3-element airfoil, you'll likely hurt the overall performance (due to choking a slot gap) even if the optimum position has the 2nd element placed higher - for example, if the entire multi-element airfoil should be moved higher. This method isn't impossible to optimize, just very inefficient to do so. Instead of defining the element locations globally, defining them relative to each other lets us use meaningful independent variables like slot gap and slot overlap. For adjusting angles of attack, I prefer to define the angle for each element individually. When defining angles relative to each other, increasing the angle of the 1st element will rotate the entire multi-element airfoil. While specifying the relative angles can speed up the initial optimization, I feel it isn't efficient for fine-tuning, which is where most of ones time is spent.

Setting up this parameterization can be a bit of a geometric brainteaser, which is why I'm sharing it:

Given elements $i = 1$ to $N$, import them all into your software of choice, whether you're optimizing them in CFD, or recreating the final design in CAD. All elements must start with their leading edge on the coordinate system origin, and with zero angle of attack.

First, pick an element to act as the 'anchor' for the multi-element airfoil. Your anchor element will define the one point that you have global control over, usually for something like placing the leading or trailing edge near a bounding box in the regulations.

Given the variables

\begin{itemize}

 \item Global anchor position, $x\textsubscript{anchor}$ and $z\textsubscript{anchor}$
 \item Element angle of attack, $\alpha\textsubscript{i}$
 \item Element cord, $c\textsubscript{i}$
 \item Slot gap height and overlap, $h\textsubscript{i}$ and $l\textsubscript{i}$
 \begin{itemize}
 \item Be aware that this will not be the true slot gap due to element thickness
 \end{itemize}
 \item Element pivot position, $x\textsubscript{pivot,i}$ and $z\textsubscript{pivot,i}$
 \begin{itemize}
 \item This is eyeballed as the location on an element that is closest to the trailing edge of the preceding element, which allows us to adjust element angles without altering the true slot gap in the process.
 \end{itemize}

\end{itemize}

Now to position the multi-element airfoil, starting from the back:

For each element downstream of the anchor element, starting at the back (elements $j = N$ to $i\textsubscript{anchor}+1$):

\begin{enumerate}

 \item Rotate elements ($N$ to $j$) by $(\alpha\textsubscript{j}-\alpha\textsubscript{j-1})$ about $(x\textsubscript{anchor,j}, 0,  z\textsubscript{anchor,j})$
 \item Translate elements ($N$ to $j$) by $(c\textsubscript{j-1} - l\textsubscript{j}, 0, h\textsubscript{j})$

\end{enumerate}

Now, for each element upstream of the anchor element, starting at the front (elements $k = 1$ to $i\textsubscript{anchor}-1$):

\begin{enumerate}

 \item Translate elements ($1$ to $k$) by $(l\textsubscript{k+1}-c\textsubscript{k}, 0, -h\textsubscript{k+1})$
 \item Rotate elements ($1$ to $k$) by ($\alpha\textsubscript{k}$-$\alpha\textsubscript{k+1}$) about $(x\textsubscript{anchor,k+1}, 0, z\textsubscript{anchor,k+1})$

\end{enumerate}

Finally, for the entire multi-element airfoil:

\begin{enumerate}

 \item Translate the multi-element airfoil such that the anchor point is at the origin
 \item Rotate the multi-element airfoil by $\alpha\textsubscript{anchor}$
 \item Translate the multi-element airfoil by ($x\textsubscript{anchor}, 0, z\textsubscript{anchor}$) into its final position

\end{enumerate}