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, it isn't efficient for fine-tuning, which is where most of ones time is spent.

Below is how to set up this parameterization:

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

- Global anchor position, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{anchor}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{anchor}}

- Element angle of attack, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_i}

- Element cord, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i}

- item Slot gap height and overlap, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_i} (Be aware that this will not be the true slot gap due to element thickness)

- Element pivot position, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{pivot,i}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{pivot,i}} (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)
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i_{anchor}} +1):

1. Rotate elements (N to j) by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\alpha_j-\alpha_{j-1}} ) about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_{anchor,j}, 0, z_{anchor,j})}

2. Translate elements (N to j) by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (c_{j-1} - l_j, 0, h_j)}

Now, for each element upstream of the anchor element, starting at the front (elements k = 1 to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i_{anchor}} -1):

1. Translate elements (1 to k) by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (l_{k+1}-c_k, 0, -h_{k+1})}

2. Rotate elements (1 to k) by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\alpha_k-\alpha_{k+1})} about Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_{anchor,k+1}, 0, z_{anchor,k+1})}

Finally, for the entire multi-element airfoil:

1. Translate the multi-element airfoil such that the anchor point is at the origin

2. Rotate the multi-element airfoil by $\alpha _{anchor}$

3. Translate the multi-element airfoil by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_{anchor}, 0, z_{anchor})} into its final position

Example of anchoring by the front element

Example of anchoring by the rear element