Article Index
The Spitfire Wing - A Mathematical Model
PART 2
Part 3
All Pages

Page 1 of 3

The Spitfire Wing – A Mathematical Model

Part 1 Spitfire Wing As Per Factory Drawings

WING PLANFORM

Whenever I hear of an elliptical wing it’s always the Spitfire, famous WWII fighter plane that comes to mind. Over the years I made a few attempts to produce a technical drawing of the Spitfire’s wing. Naturally, I first tried fitting an ellipse through a few cardinal points but that did not work. The wing plan-form shape is not elliptical but an arbitrary hand-drawn curve. It is, however, quite close to an ellipse. Had it been elliptical it would have been only about an inch wider near the tips (Figure 5). A lot of questions on various forums are being asked about this wing and no one seems to know the right answers. The object of this exercise is to sort out these ambiguities and set the record straight once and for all. You might be a modeler, building a scale aircraft, building a full scale replica, doing restoration, or perhaps you just want to do a 3D model. In any event I hope you find this information useful.

An ellipse would have been easier to apply as it is a curve represented by a simple mathematical equation. The one actually used is far from an ellipse. In fact there is nothing elliptical on this wing, not even the thickness distribution, this being a straight line followed by a curve resembling a cubic parabola. Whoever decided to adopt these ‘fair curves’ must have been very shrude. The Spitfire original wing design was of a trapezoidal planform with a thin airfoil. It was too thin to fit the guns, ammunition boxes, the retracted undercarriage and the split flaps along with their extension mechanism. The design team increased the root chord which kept the wing proportionally thin but with a similar depth as those of the Hurricane and the Messerschmitt. This long chord had to be maintained for most of the span and this could be achieved by an elliptical shape.

The only recorded account of Mitchell’s choice of an elliptical wing for the Spitfire, according to an autobiography by his Canadian aerodynamicist, Beverly Shenstone, is “I don’t care a damn what shape it is as long as we can get the guns in!” According to Mitchell’s Deputy Designer, Joseph Smith, Mitchell had a thick pencil and drew freehand large-scale curves, softly at first, then repeatedly over, harder and thicker until he got a fair curve. This was a practical and overworked man who had no time for aesthetics or analytical design. In November 1934 Shenstone went to the Paris Air Show where the Germans had unveiled their Heinkel 70 aircraft with elliptical wings. On November the 16th the RAF published the requirement for eight guns. It is believed that the decision for an elliptical wing on the Spitfire was made in the second half of that month.

The Spitfire original design drawings were done by a company Supermarine Aviation Works, and there survive only a few original drawings and design notes. Unfortunately, in those days the engineering culture was such that anything that did not have a border with a title did not survive and history suffers. Vickers-Armstrongs, the manufacturers, a much larger company, later continued with the production drawings. These were eventually deposited with the Royal Air Force Museum at Hendon and are only about 30 per cent complete. The original drawing for the Spitfire elliptical wing was drawn in the beginning of December 1934 with the drawing number 30000 sheet 12. This was a modification of an earlier straight tapered planform. Unfortunately this drawing is lost forever, probably during a bombing raid.

I got hold of a CD with drawings for the Spitfire. I could not find amongst some 2000 scans any information on the planform. After lengthy scratching around on the net, I eventually managed to get hold of a scan, Drawing No. 33708 sheet 8, titled ‘Wing Geometry’ (see Figure 1). This was truly the ‘Rosetta stone’ of the spitfire wing. The wing is for an extended wing-tip, High-Altitude Mark Spitfire. However, it was a standard wing up to Station 21 as the wing tip is removable and attached with two bolts. At first it looked hopeless to decipher anything as the scan was of poor quality and only about 20% was legible (see Figure 2). It took some time to figure out the essential values but worthwhile the effort. Chords for the remaining Stations 22 and 23 were indirectly obtained, for example from standard wing-tip drawings (Figure 3).

Figure 1

Figure 2

Figure 3

AIRFOIL

Ahead of the spar, the thick-skinned leading edge of the wing formed a strong and very rigid D-shaped box, which took most of the wing loads. At the time the wing was designed, this D-shaped leading edge was intended to house steam condensers for the evaporative cooling system intended for the PV XII engine.

In 1933 a family of new airfoil shapes were developed by N.A.C.A. based on a systematic study of aerodynamic characteristics with variations in airfoil thickness and mean-line form. These airfoils were ideal for lofting wings with variable thickness to chord ratios as they were easy to derive mathematically and/or graphically. The 2200 series airfoil gave the thickest nose for the D-shaped leading edge. I suspect the Canadian had a lot to do with the wing design, particularly with airfoil selection.

Drawing No. 33708 sheet 8 has tabulated values for the airfoil sections in standard NACA form (Figure 2), unlike the Mustang P 51-D were the airfoil ordinates readily delimit incidence.

WING TWIST

Wing twist, or washout is helical, unlike the Mustang wing which has a linear (lofted) twist.

WING GEOMETRY

Table 1 ‘Wing Geometry' and the accompanying notes completely define the Spitfire wing, resulting in the rear spar ordinates given under ‘Spar Geometry'. However, there is a problem with the rear spar. There is a discontinuity at the dihedral point resulting in a joggle as well as a twist in the spar web. More about this in Part 2. Table 1 is derived from Supermarine Drawing No. 33708 sheet 8 and is summarized in Figure 4.

Table 1

Figure 4

WING CHORD DISTRIBUTION

The wing plan-form of the Spitfire is not elliptical. Figure 5 gives a comparison between a true elliptical chord distribution and that from Table 1.

Figure 5

A good way to see how the chord distribution deviates from a true ellipse is to normalize the chord and semi-span axes to unity so that the ellipse becomes a circle (Figure 6).

We then convert the chord from Table 1 into polar coordinates (Table 2) and plot the deviation (Figure 7).

Figure 6

where

Table 2

Figure 7

We won't worry about the type of curve at this stage. The curve in Figure 7 has some noise and the first task is to smoothen it while keeping as close as possible

to the original ordinates. We use regression analysis (curve fitting) to achieve this.

The following formula gives a good fit.

where the coefficients in Table 3.

Table 3

Figure 8 and Table 4 show smoothened .

We now derive a smooth chord distribution.

Figure 8

Table 4

Table 5 gives a smoothened chord distribution to three decimal places.

Table 5

Chord distribution is not elliptical. However, the slopes of the chord distribution curve in Figure 6 should be 0° at centre of aircraft and -90° at the tip.

In order to meet these conditions, the Curve in Figure 7 should have 0 slope at = 0 and = /2 .

We now manually (using CAD)draw a spline through points in Table 2 such that the ends of the spline have zero slope. We then add new points near the ends of the spline and read off the resultant points (Table 6).

Table 7 and Figure 9 give the corrected .

Table 6

Figure 9

Table 7

The following formula is used for curve fitting

with coefficients in Table 8.

Table 8

Now we derive a smooth chord distribution.

This is tabulated in Table 9 and plotted in Figure 10.

Table 9

This is a good fit shown by the residuals (Table 9) of one hundredths of an inch, that's one quarter of a millimeter. I don't think that the Spitfire wing was manufactured within these limits.

Figure 10

Table 10 gives a smoothened chord distribution to three decimal places.

Table 10

Brutal power is used to smoothen the curves. The type of curve-fitting equations used here are unimportant at this stage. The main object is to have smooth curves to work with in Part 2 to build mathematical models. Figure 10a shows the residuals of chord distribution and the noise is evident so Part 1 here deals with 'cleaning up' the curves.

Figure 10a

LEADING EDGE CURVE

We now convert the leading edge offsets from Table 1 into polar coordinates (Table 11) and plot the deviation (Figure 11).

where

Table 11

The following formula is used for curve fitting the bell shape

with coefficients in Table 12

Table 12

We now derive a smooth chord distribution.

This is plotted on Table 13 and shown in Figure 11.

Figure 11

Table 13

Table 14 gives smoothened L.E. offsets to three decimal places.

Table 14

WING TWIST

Wing twist (or washout) is helical along the front spar datum-line: 2° at 31" semi-span to -½° at wing-tip and is calculated with the

following formula

This is tabulated in Table 15.

Table 15

FRONT SPAR TO REAR SPAR ORDINATES

Distance B along the chord is given by the formula

The horizontal distance from the front spar is

The vertical distance from the front spar is given by