Around 2010 Execuform issued a 1/72 scale vacform model of the Postjager. A 1/72 resin model is expected before the end of 2012. I initially planned to build the Execuform vacform, and started researching the subject. For some time, the black and white photos confused me thoroughly. But patterns emerged slowly, and it is now possible to analyze the colors with just five photos. The result was slightly surprising.
There is not much data available on the technical design of the Postjager. One of the questions is the wing design, more specifically the wing profiles. The NVM drawings state that the center section was 'M12 17%' and the tip 'M12 11%', and it contains a table of profile ordinates (coordinates) for both. The use of the M12 profile is confirmed by one of the old newspaper articles listed at the end of the Postjager colors and markings page. But I found the use of M12 in two different thicknesses strange and confusing - I hade never heard of a drastic rescaling of a profile. To answer that question, I looked into the design of the M12 profile. Normally, the ordinates of a profile are all you need. But in this case, to understand the 'thickened' version', the camber line and thickness distribution must be known, and preferably their mathematical equations known. A long search followed..
In the 'National Archief' I found a RLS report about wind tunnel measurements on a 1/20 scale wing, dated 4 April 1933. The measurements focused on the aileron and flap effectivess, but luckily the report also included a detailed measurement of the wing itself. No drawing or photo of the wing model were included. The reports lists a centerline chord 175 mm, span 819 mm, area 0.1157 m^2. X and Y are expressed as percentage of the centerline chord, Z is expressed as percentage of the span. Therefore the measurements correspond to profiles 2.572 m from the centerline which is very close to the wing kink, and 7.387 m from the centerline which corresponds to the outboard edge of the flap. The measurements are copied below.
The first three profile measurements should be identical. After correcting for small incidence angle differences, and the y-position difference due to the model's dihedral, the three measurements agreed pretty well.
The average profile was calculated (blue), and it showed a thickness of 15.8%. Next I added a 15.8% scaled M12 profile (red), see the analysis further down the page. The agreement is excellent I think.
Therefore I conclude the wind tunnel model had a 15.8% M12 profile for the area between the outboard engines. Since construction of the real aircraft started soon after, I would guess the designer retained this profile choice.
I copied the 15.8% M12 over the profiles shown in the NVM drawing. Since the drawing states that the profile is 17% thick, I was surprised the match was quite good.
I copied the 15.8% M12 over the preliminary drawing that I found in the 'Nationaal Archief', and again the match was good.
The 'M12' profile is the 12th Munk-designed profile, in a series of 27, for NACA. Their profile data and performance are all listed in NACA Report 221: Model tests with a systematic series of 27 wing sections at full Reynolds number. Table XXVIII of that report lists which combinations of camber lines and thickness distributions were used. Table XXIX list the ordinates of all 27 profiles. It includes an obvious typing error for the 80% chord lower side ordinate. Furthermore, I found that M12 has a Göttingen equivalent, the Go 676 profile, but I could not find out which came first.
Unfortunately, NACA Report 221 does not contain the mathematical models for the three camber lines and the three thickness distributions, it only identifies them as "straight, a and b" and "I, II and III" respectively. Therefore I entered the data of six profiles in Excel, and starting doing various analyses to try to understand the way these profiles were constructed.
The good news is that I was able to establish the camber line equation of M12 reasonably well. Note that M10 and M11, with the same 'b' camber line, don't have their start and end point on the chord line - welcome to the world of old profiles. Seeing the slight reflex at the rear, and knowing the German background of Munk, I remembered the equations that were used to design the Me-163B wing profile, using a P-curve and an S-curve combination for the camber line (Skeletlinie), see Göttingen 765: attempt 6: camber line plot.
yp = h * ( 1 - (1 - 2x) 2n ) called P-curve
ys = lambda * ( (1 - 2x) - (1 - 2x) 2m + 1 ) called S-curve
And indeed, the combination of n = 1 and h = 1.75 for the P-curve and m = 1 and lambda = 1.5 gave a fairly close approximation of the camber line calculated from the profile data.
I think that the thickness distributions I, II and III are identical except for their thickness, but I have not yet identified a model that describes them. It's not the NACA 4 distribution, I checked that. Considering Munk's German background, it's most likely one developed in Göttingen. Also, there information on the leading edge radius is also missing.
Luckily I got expert help from Keith Pickering, who provided the excellent report 'Recovered equations and ordinates of the Göttingen 765 airfoil' for my Me 163B Komet website. He used the same method here, optimised the results with a least-squares method, and his analysis resulted in the following model for the thickness distribtion.
The thickness distribution ahead of the maximum thickness point x = m is described by:
± yt = T /.2 * (a0 * x0.5 + a1 * x + a22 * x2 + a3 * x3)
m = 0.323075441, so the maximum thickness is a bit aft of the reported 30% chord
T = 0.119434915
a0 = 0.296308751
a1 = -0.185708798
a2 = -0.01012475
a3 = -0.218443619
The thickness distribution aft of of the maximum thickness point x = m is described by:
± yt = d0 + d1 * (1 - x) + d2 * (1 - x)2 + d3 * (1 - x)3
d0 = 0.002
d1 = 0.147610807
d2 = -0.058247106
d3 = -0.050013811
I tried to find more information on the Gõttingen 676, which is the M12 equivalent, although I still don't know which came first. I consulted 'Aerodynamische Profile - Windkanal-Messergebnisse, theoretische Unterlagen' by Friedrich Wilhelm Riegels from 1958. On page 150, the ordinates are listed, and they do not agree with the numbers in NACA Report 221. On page 128 it lists performance data of 676, and quotes 'E IV' as the source. Unfortunately no list of references is provided, so the trace ended there.
Worth mentioning is Parametric airfoil catalog Part II Goettingen 673 to YS930 by Thomas Melin. Melin presents a method to represent airfoils by four Bezier curves, including the Go 676 (equivalent to M12) profile. It might be useful for a 3D model.
From NACA Report 221 I conclude that Munk himself combined various thickness distributions with a camber line. Since thickness distributions I, II and III appear identical in shape, you could also say that he scaled the thickness as desired. In that sense, a '15.8% M12' profile would fit in Munk's approach. But the drawback is that no aerodynamic characteristics are know for a self-defined thickness. It would seem more logical to me to use profile M15 then, that was an 18% profile using the same camber line, and for which all aerodynamic characteristics were measured and report in NACA Report 221. Therefore I still don't think I understand the reasoning for the '15.8% M12'.