# The numerical method used by F.X Wortmann for inverse design

Apologies in advance for a long-winded question.

Can someone who understands some airfoil theory and German language please help me understand the method outlined in this paper from Prof. FX Wortmann.

I have back-quoted what I've got so far with Google translate but it doesn't appear to be accurate especially with the given context.

If you reduce the work of E. Truckenbrodt [8] to calculate the profile shape from the speed distribution to the mere calculation, you come to the following series of formulas: 1. At the n positions of the dimensionless profile depth

Table 3 contains the constants bm for N24. The following applies to the pth approximation step

Table 3 contains the constants is the slope of the tangent to the profile shape and results from

The values ​​amn are listed in Tables 4 and 5. The values ​​amn for even-numbered m - n disappear. The table amn with m = 2, 4, ... is therefore used to calculate the values ​​y, 'for n = 1, 3, ... A good estimate of the correction values ​​x, which greatly abbreviates the approximation, can be obtained if one starts from the examples already calculated. Eq. (A2) includes (N-1) 2 232 529 operations that can be performed in about two hours with a simple electrical calculator. 2. The skeleton line of a curved profile, in which the speeds on the top of the profile (index o) and underside (index u) are different, is calculated

Table 3 contains the constants dmn. The value for z is already known from the above calculation. This skeletal line still needs a minor correction, which is reflected in

The constants pa, qn are given in Table 6. The values ​​y, and are according to Eq. (A2) and (A5) known. The angle of attack a is calculated

The values ​​d are also shown in Table 3. The final coordinates of a curved profile then result from (A11) = y (s) 4ys) + y (). The positive sign applies to the top of the profile.

Also with my very limited understanding with possibly incorrect translation, I've got the following questions so far

1. What is the $$(t)$$ in $$\bar{y}_n^{(t)}$$ in (A-2)
2. what is the $$p^{th}$$ approximation in (A-3)
3. What is the $$(s)$$ in $$\bar{y}_n^{(s)}$$ in (A-2)
4. Does (A-6) refers to the camber line calculation?

What is the $$(t)$$ in $$\bar{y}_n^{(t)}$$ in (A-2)

The text doesn't say. t in German normally denotes chord (Tiefe), but since we later see $$\bar{y}_n^{(s)}$$ for the camber line (Skelettlinie), it here stands for Tropfen, denoting the thickness distribution of the uncambered airfoil.

I would also interpret the long dash between $$\bar{y}_n^{(t)}$$ and the sum in A2 as a misprinted equal sign. Otherways the whole sequence will not add up.

what is the $$p^{th}$$ approximation in (A-3)

You mean $$p$$th, don't you? p is used as the index of a stepwise approximation. $$f_m^{(p)}$$ is the $$p$$th iteration of $$f_m$$.

What is the $$(s)$$ in $$\bar{y}_n^{(s)}$$ in (A-2)

That is only in A 6 and indicates that $$\bar{y}_n$$ this time denotes the ordinate values for the camber line (see above).

Does (A-6) refer to the camber line calculation?

Yes.

• "...someone who understands some airfoil theory and German language..." It's like he was looking for you! ;) Mar 18, 2020 at 13:37