Spitfire's drag - what was good, and what was not?

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Spitfire was found lacking two things: boundary layer separator (or tunel, playing the same role?) for the radiator(s), while the inner shape of the radiator was ill suited to fully harvest the Meredith effect. Maybe the (under) wing placement was restricting the convenient radiator height, in order to sufficiently slow the air flow after it enters the radiator casing? Also, the radiator casing was restricted in possible length, since it was located between U/C leg space (when retracted) and flap - not allowing to the Meredith effect to happen at a desirable level..
A boundary layer separator wasn't possible, since the "roof" of the radiator housing climbed up immediately after the intake lip, so that the top of the radiator matrix was just under the top skin of the wing's upper surface (this was the reason for erks being told to keep off the starboard wing.) I don't have the measurements, but the face of the radiator, in area, was roughly double that of the mouth of the housing, and the air expanded in four directions.
contrary to the radiator's exit flap of the P-51, that was progressively adjustable, the exit flap from the Spit (IX is mentioned in the article) have had two positions: fully open and partially closed, again messing with the M. effect. P-51s
The exit flap was only opened when the climbing temperature of the coolant required it; in the IX, it was fully automatic, though the pilot could override it if he needed to.
 
Guess riacrato was thinkering of radiator's smaller area, rather than of whole plane.

Further on Spitfire: the Mk.III is stated as being equipped with 'wheel well flap', presumable a wheel well cover? However, the 'flap' is not easy to spot on the photos. Wonder if someone has some details about it?

Exactly. The radiator of the Spitfire seems to have been superior efficiency-wise to its contemporaries, achieving appr. same cooling at appr. same weight but less frontal area than for example the Bf 109. The rest of the airframe is a different matter.
A boundary layer separator wasn't possible, since the "roof" of the radiator housing climbed up immediately after the intake lip, so that the top of the radiator matrix was just under the top skin of the wing's upper surface (this was the reason for erks being told to keep off the starboard wing.) I don't have the measurements, but the face of the radiator, in area, was roughly double that of the mouth of the housing, and the air expanded in four directions.
But does that really mean no boundary layer seperation is possible? Isn't the "roof" of the P-51 radiator also higher than the intake lip? Even if not, I don't see how this fact which is also true for the 109 and probably a lot of '30s aircraft designs eliminates the possility of having a "scoop" instead of a simpler opening for the intake. I'd rather think this simply wasn't a common practice in the thirties and not seen as a worthwhile cleanup later. Maybe the redesign and the added complexity simply wasn't worth the effort. I seem to recall even for the P-51 it took some time until they figured out the best dimensions and "offset" for the boundary layer seperator.
 
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Exactly. But does that really mean no boundary layer seperation is possible? Isn't the "roof" of the P-51 radiator also higher than the intake lip? Even if not, I don't see how this fact which is also true for the 109 and probably a lot of '30s aircraft designs eliminates the possility of having a "scoop" instead of a simpler opening for the intake. I'd rather think this simply wasn't a common practice in the thirties and not seen as a worthwhile cleanup later. Maybe the redesign and the added complexity simply wasn't worth the effort. I seem to recall even for the P-51 it took some time until they figured out the best dimensions and "offset" for the boundary layer seperator.

It did take time not only for the dimensions but also for the shape - and focused on the Reynolds number of the 'rumble' experienced as noise within a speed range.

Curiously the P-51H geometry and location was closer to the P-51A than B/C/D - as well as a much longer plenum aft of the radiator.

I suspect that the problem (aft wing location) with both the Spit and 109 versus the 51 is that the boundary layer buildup increases chord wise on a wing and may be subject to worsened adverse pressure gradients on the wing than centerline fuselage on the Mustang.
 
The Fw 190 was a much heavier aircraft but also had smaller wing area: 18.5sqm as opposed to the 22.5sqm of the lighter spitfire. The Fw 190 also had NACA 5 digit airfoils as opposed to the older NACA 4 digit airfoils, these newer airfoils generated slightly better lift and also L/D ratios.

The Spitfire had a fairly messy underside which caused considerable drag, tidying up this area surely would have paid dividentds. Its so called Meredith radiator system was not particularly unique nor more effective than others despit the myths sorounding it; everyone including Messerchmitt, Curtiss and the French fighters seemed to have the technology.

Retractable tailwheels came in with the Spitifre Mk VIII but this airframe was little produced.

The Me 109F's systems was every bit as sophisticated as the excellent P-51 system; it also had a boundary layer bypass. Of course being much smaller its dimensions were far less optimal. Oddly the Me 109G gave up on the boundary layer system. I've heard it said the Spitfire's radiatorsthemselves were more effective than the Me 109in terms of frontal area but that this can be attributed to the use of copper in the Spitfire over Aluminium in the Me 109 since copper had better thermal conductivity and it might possibly be cast more fine. This was a close run thing and I don't think it was ever resolved. Copper was a critical material to the Germans.

Modern Aluminium automotive radiators are actually hybrid plastic-aluminium systems of very fine construction.

Frontal area in a radiator is not everything, low pressure loss is perhaps more important, this is why the Germans went to anular radiators even on the Fw 190D as it allowed increased area and therefore reduced pressure loss. Napier wanted to do the same on the Sabre installations on Tempest.

The boundary layer intake seperation became an issue on split or side intake jet aircraft which the Luftwaffe wanted in order to provide for increase protection against gun fire and to allow for area ruling and as was seen on the Lockheed P-80. To do this they actually wanted to use active broundary layer control. The Messerchmitt P.1112 http://www.luft46.com/mess/mep1112.html actually was to use a 200hp compressor driven of the jet engine accesories to draw in the boundary layer. A half intake model was succesfully wind tunnel tested.

That gives an idea of its importance.

I rather expect that the Hawker Hurricane's system would have been quite effective bar the relatively low speed.
 
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...
The Me 109F's systems was every bit as sophisticated as the excellent P-51 system; it also had a boundary layer bypass. Of course being much smaller its dimensions were far less optimal. Oddly the Me 109G gave up on the boundary layer system...

I'll just cover this tidbit.
The 'every bit as sophisticated' cooling system of the 109F was abandoned for the 109G, since it gave no practical advantage vs. the system with no bypass. Nothing 'oddly' here, it just was not worth it. FWIW:
http://www.luftwaffe-experten.org/forums/index.php?showtopic=16738
 
The Spitfire had a fairly messy underside which caused considerable drag, .
You have figures, of course?
tidying up this area surely would have paid dividentds
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you mean we'd have won the Battle of Britain by an even bigger margin?
Its so called Meredith radiator system was not particularly unique nor more effective than others despit the myths sorounding it; everyone including Messerchmitt, Curtiss and the French fighters seemed to have the technology
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Who claimed that it was unique? And, if you check, the Meredith effect is air entering an intake, then expanding, which is what happened in the Spitfire radiator(s,) so let's have a bit less of this "so-called," please.
Retractable tailwheels came in with the Spitifre Mk VIII but this airframe was little produced.
It would be nice if you did everyone the courtesy of carrying out some research, before pontificating on a subject you obviously know little about; the Spitfire VII was the first service airframe to have a retractable tailwheel, though it was trialled first on the Mk.III. Also your "little produced" VIII ran to 1650 airframes, in just two years, more than the Mk.I (1530,) Mk.II (920,) Va (1367,) Vc (635,) VI (100,) VII (140,) XII (100,) XIV (957,) XVI (1054,) in fact only the Vb IX were built in greater numbers. Even the whole lot of Merlin-powered Seafires, I, II, III only totalled 1792.
 
You have figures, of course?

I agree, figures would be interesting, but even without figures it is clearcut that semi-faired undercarriage, a large boxy radiator or lots of casing ejector openings is bad from the drag point of view. Most aircraft in the 1930s looked like it, but most did move away from that and started using more streamlined radiators and fully faired undercarriage.

you mean we'd have won the Battle of Britain by an even bigger margin?

British and German losses in the Battle of Britain were in fact very similar in absolute numbers, and the Spitfire in any case contributed much less to that than the numerous Hurricane. The Spitfire's major problem during development was high drag, that actually got worse as this was being compensated by more powerful engines (which ironically added even more drag with the new systems added like bigger radiators and intercoolers) and it was increasingly difficult to increase it's performance. Cleaning up the design a bit would have easily gained 30 and perhaps as much as 40 km/h.

Who claimed that it was unique? And, if you check, the Meredith effect is air entering an intake, then expanding, which is what happened in the Spitfire radiator(s,) so let's have a bit less of this "so-called," please.

I think Siegfried is referring to that since Hugo Junkers described and used such a system many years before Meredith described it, the general attribution to Meredith is somewhat unfair.
 
I agree, figures would be interesting, but even without figures it is clearcut that semi-faired undercarriage, a large boxy radiator or lots of casing ejector openings is bad from the drag point of view. Most aircraft in the 1930s looked like it, but most did move away from that and started using more streamlined radiators and fully faired undercarriage.
Like the 109 190? And the ejector/muzzle openings were sealed shut, until the pilot opened fire.
British and German losses in the Battle of Britain were in fact very similar in absolute numbers, and the Spitfire in any case contributed much less to that than the numerous Hurricane.
125 fewer, in fact - 530 to 655, achieved by 19 Squadrons against 33 Hurricane Squadrons. And if you add aircrew into your "absolute numbers," the Germans lost a lot more, so much that at least one German general said that it was their single most telling defeat.
The Spitfire's major problem during development was high drag, that actually got worse as this was being compensated by more powerful engines (which ironically added even more drag with the new systems added like bigger radiators and intercoolers) and it was increasingly difficult to increase it's performance. Cleaning up the design a bit would have easily gained 30 and perhaps as much as 40 km/h.
Spitfire I flew at 362mph, and the XIV at 442, an increase of 80mph (128kph) or 22%, with an increase of 13000' in the service ceiling as well.
 
Try figuring the sea level speeds for a MK I and later aircraft (MK IX and MK XIV for instance) and seeing what power was used for what speed. Use the cube law to figure the power required and see how close they come out.

According to the cube law a MK I going 336mph at sea level needs 1520hp if I have done the math right and 1870hp to do 360mph at sea level. How much power did a MK IX have at SEA level and how much did a MK XIV have.

MK I with 880hp with constant speed prop and a sea level speed of 280mph used as base line.
 
Yes TJ be sure the Bf109E was as smooth as a baby's rear end. Even later 109s had all kinds of bumps, open wheel wells and tail wheels sticking out into the air stream.
 
Or, in other words, try to work out some way to fiddle the figures to get the result you want; play your games on your own.

We may disagree on the Whirlwind but the procedure I outlined actually works out rather well for the Spitfire.

While the Merlin 65 or 66 in the MK IX was good for 1705 at 5750ft it was a lot closer to 1600hp at sea level (18lb limit). Theory says 80-100hp less is needed. MK I didn't have the external guns and certainly had smaller radiators, The MK I in the test didn't have pilot armor bullet proof windscreen is unknown but the plane only weighed in at 6,050lb so induced drag was less than the MK IX. I am seeing a bit more drag but not the big increase others are claiming. The MK XIV numbers are even more interesting, Theory says 1870hp needed, (for the light plane with no external gun barrels), While a Griffon 65 was good for around 2000hp at full thottle height in low gear it was good for around 1850hp at sea level at 18lbs boost (boost used for the 360mph speed in test). The MK XIV weighed 8400lbs in the test. this gives us a much heavier plane, more drag due to guns, yet the power for speed is coming out almost right on or just few % off, I am not seeing any big increase in drag for the later mark Spitfire here.

I had done this exercise before and knew how it came out. Spitfire bashers are welcome to poke holes at it ( I am sure there are a few flaws) but I think it is better than "I think, it must have" and so on without even an attempt to show how the "extra" drag affects things.
 
You have figures, of course?
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you mean we'd have won the Battle of Britain by an even bigger margin?
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Who claimed that it was unique? And, if you check, the Meredith effect is air entering an intake, then expanding, which is what happened in the Spitfire radiator(s,) so let's have a bit less of this "so-called," please.

It would be nice if you did everyone the courtesy of carrying out some research, before pontificating on a subject you obviously know little about; the Spitfire VII was the first service airframe to have a retractable tailwheel, though it was trialled first on the Mk.III. Also your "little produced" VIII ran to 1650 airframes, in just two years, more than the Mk.I (1530,) Mk.II (920,) Va (1367,) Vc (635,) VI (100,) VII (140,) XII (100,) XIV (957,) XVI (1054,) in fact only the Vb IX were built in greater numbers. Even the whole lot of Merlin-powered Seafires, I, II, III only totalled 1792.

Let me answer your 'protestations' one by one.

One.
The battle of Britain wasn't won by the "iconic" Spitfire which in itself would have need the numbers of the Hurricane anyway, it was "thrown" by the strategic need to withdraw forces and put limited German resources into operation Barbarossa which itself was about cutting of a long term Soviet threat while securing the materials and strategic posture needed to form an defensible and economically autonomous Imperium Europa that might be able to take on the inevitable US/UK axis. Put simply Kriegsmarine was far too small, the Heer didn't have ANY amphibious assault capacity and the Luftwaffe had been prepared primarily as a tactical air force to support the most important German defence arm, the Heer and its need to deal with bordering countries.

Dealing with bordering French, Polish and Russian even Italian armies almost totally occupied German planning. Britain was a distant concern. In addition there were no strategies, significant plans or preparations for an air war against Britain let alone invading Britain. The only thing that might have made a difference is the proper preparation of drop tanks for the Me 109. There was more chance of invading the Soviet Union in short order than getting a force across the channel and defeating the RN in a short time scale.

Two
"Meredith" invented nothing. Hence the term "So called Meredith effect". All he did was write an non-academic explanatory paper on heat recovery that became popular in the UK and perhaps US. The idea was reasonably known and started to make sense with the appearance of pressurised cooling loops which raised coolant temperature enough so as to allow smaller radiators.

Three.
Sorry for answering the question "what was good and what was bad" re Spitfire aerodynamics. A picture is worth a thousand words. Any Google image search of "Spitfire underside" versus "P-51 underside" will show how messy the Spitfire was in comparison. I would put the Fw 190 definitely in the much cleaner underside category. The Me 109 had its own issues caused by enlarged wheels and machine gun bulges (eventually rectified)

The retractable tail wheel versions (mainly Mk VII,VIII,XIV) were little produced in terms of overall numbers of 25000 spitfires. I stand by that statement. On the other hand all P-51 and Fw 190, P-47 and P-40 had retractable tail wheels as did all Me 109F, some Me 109G, Most Me 109K. Gun bulges, partially uncovered wheels, no less than 3 separate coolers, various antenna, non flush rivets(I believe) all marred the Spitfire underside. Never mind the P-51's radiator system or laminar flow wings. Look at the underside of the Spitfire to reveal what is likely 10-20 mph of speed losses due to the superior construction values of the P-51 and more modern aircraft. Of course these things were likely known but production demands meant what appear to be trivial changes were not be made.
 
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Like the 109 190? And the ejector/muzzle openings were sealed shut, until the pilot opened fire.

IMHO the 109/190 should deserve it's own thread, but in short the 109 improved considerably through the war from it's early state that represented the technology of the mid-1930s. The late 109s underside is effectively flush for example. The 190 did not evolve much, but it should be noted that it incorporated all the latest developments since it's birth.

125 fewer, in fact - 530 to 655, achieved by 19 Squadrons against 33 Hurricane Squadrons. And if you add aircrew into your "absolute numbers," the Germans lost a lot more, so much that at least one German general said that it was their single most telling defeat.

Well that's grand, and how does that view fit into the question of the Spitfire's drag exactly? I fail to see the relevance.

Spitfire I flew at 362mph, and the XIV at 442, an increase of 80mph (128kph) or 22%, with an increase of 13000' in the service ceiling as well.

Yes, it improved very well, chiefly because they managed to put larger, heavier and more powerful engines to it. Well let's see the numbers.

The Spitfire I achieved 282 mph with about 880 HP, the Spitfire XIV achieved about 358 mph with 1840 HP at SL ('paper' figures, invidual planes may have flew faster or slower)

Since power requirement for speed increases on the cube, to drive a 282 mph plane to 358 mph you need (352/282)@3 = 1.94 the times of power, if all things are similar. That is 1.94*880 HP, or about 1710 HP, the amount of power a Spitfire I airframe would require to go as fast as the real life Mark XIV. As we can see by the XIV the increase in drag was equivalent to about 130 HP. BTW that 1710 HP is almost precisely what the Merlin 66 Spits had, so but those were only good for about 330 mph at SL.

From that it follows that the drag increase during the development of the Spitfire was equivalent to about 25-30 mph speed loss. That's hardly insignificant.
 
Let me answer your 'protestations' one by one..
1/. My "protestations" are corrections to your fabrications.
One.
The battle of Britain wasn't won by the "iconic" Spitfire which in itself would have need the numbers of the Hurricane anyway, it was "thrown"
Dear Lord, is there no end to your manipulation of history? So, now, we're to believe that the Battle was deliberately lost, and hundreds of elite German aircrew thrown away on a whim? Truly that means that the German hierarchy were even more incompetent than we first thought. Oh, and the Spitfire persuaded thousands of people, as far apart as Japan, South Africa, America and Argentina, to raise £5,000 for each aircraft (when the average weekly wage, in the U.K., was £1.5) to a total of over 1400 aircraft, so your "iconic" is a compliment, not an insult.
Two
"Meredith" invented nothing. Hence the term "So called Meredith effect". All he did was write an non-academic explanatory paper on heat recovery that became popular in the UK and perhaps US. The idea was reasonably known and started to make sense with the appearance of pressurised cooling loops which raised coolant temperature enough so as to allow smaller radiators.
What Meredith wrote, in fact, was the germ of the idea for jet propulsion (and I have a copy of his letter, headed, "Invention relating to jet propulsion of aircraft," sent to the Air Ministry in 1935, taken from our National Archives, if you insist on arguing the point.)
Three.
Sorry for answering the question "what was good and what was bad" re Spitfire aerodynamics. A picture is worth a thousand words. Any Google image search of "Spitfire underside" versus "P-51 underside" will show how messy the Spitfire was in comparison.
But you didn't answer it, you simply made a rude comment about it. We have never-ending references to the "draggy" radiators, and I'm simply asking where are these references to be found? I've been through dozens of test reports, on the Spitfire (and other aircraft,) in our archives, and have never yet found one condemning the radiators; all I ask is where are these reports to be found? I have regular access to our National Archives, and should be able to find them, if you would just indicate where to look.
The retractable tail wheel versions (mainly Mk VII,VIII,XIV) were little produced in terms of overall numbers of 25000 spitfires. I stand by that statement.
Out of 20,334 Spitfires (do stop massaging figures, there's a good boy) and 2,408 Seafires, 4,255 Spitfires and 786 Seafires had retractable tailwheels, which is a little over 22%, so not as insignificant as you like to make out.
On the other hand all P-51 and Fw 190, P-47 and P-40 had retractable tail wheels as did all Me 109F, some Me 109G, Most Me 109K. Gun bulges, partially uncovered wheels, no less than 3 separate coolers, various antenna, non flush rivets(I believe) all marred the Spitfire underside.
Do show me where the three coolers are, under the Spitfire; in all my years, crawling under them, I've only ever found two. Your "belief" in non-flush rivets is another fabrication; the only area of domed rivets was on the fuselage, aft of the cockpit, and then only until June 1943. Allow me to tell you a little story, well-known among those who do bother to carry out research:-
K5054 was entirely flush-rivetted, but the process was expensive, so Supermarine glued half a dried pea onto every rivet, to see how a dome would affect the speed, then progressively removed lines of peas, checking as they went. It was found that domes made virtually no difference on the aft fuselage, but that was all, so the rest had to be flush. At the time (1936, remember) the technology for "blind" rivetting didn't exist, so every wing and tailplane rib had a strip of wood added underneath, to which the skins were fitted, using countersunk woodscrews, which were then filled and sanded smooth. When the ability to do so was available, rivets took over.
Look at the underside of the Spitfire to reveal what is likely 10-20 mph of speed losses due to the superior construction values of the P-51 and more modern aircraft
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You're guessing again; got any figures to back that up? Oh, and the Spitfire employed exactly the same "construction techniques," namely filled and smoothed wing surfaces, plus smooth synthetic, rather than rough matt cellulose, paints, from August 1942.
Of course these things were likely known but production demands meant what appear to be trivial changes were not be made
Yet another complete fabrication; I have a copy of the complete ledger of the modifications, carried out to the Spitfire and Seafire ( there were more than 2900,) which include production-shattering items like, ""To introduce adhesive tape for electrical cables," and "To secure egine data plate with bolts and nuts in place or rivets." Every single change had to go before, and be approved by, the Local Technical Committee (there was a war on, remember,) so stop talking nonsense, and learn.
In 1959 I went to work in a factory, which repaired Service vehicles, and, over every door, there was a sign "When in doubt, ask, don't guess," but I suppose the word "doubt" doesn't exist in your vocabulary?
 
Well that's grand, and how does that view fit into the question of the Spitfire's drag exactly? I fail to see the relevance.
Do me the courtesy of reading the statement, to which I was responding, before indulging in your latest bout of sarcasm.
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The Spitfire I achieved 282 mph with about 880 HP, the Spitfire XIV achieved about 358 mph with 1840 HP at SL ('paper' figures, invidual planes may have flew faster or slower)
It would be nice if you didn't blindly follow your pro-German mate:-
Spitfire I max speeds were 295 @ 2,000', 328 @ 10,000', 362 @ 18,500 (height of most interceptions in 1940,) and 315 @ 30,000'. The Merlin II produced 990hp @ 12,000' and 1060 @ 17,000'.
Yes, it improved very well, chiefly because they managed to put larger, heavier and more powerful engines to it.
Which is a real (if not intended) compliment to Mitchell, because the 1935 airframe proved to be so adaptable, right through to 1950, (and, of course, the Germans never altered, or improved, their engines during the whole war.)
 
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Well let's see the numbers.

The Spitfire I achieved 282 mph with about 880 HP, the Spitfire XIV achieved about 358 mph with 1840 HP at SL ('paper' figures, invidual planes may have flew faster or slower)

Since power requirement for speed increases on the cube, to drive a 282 mph plane to 358 mph you need (352/282)@3 = 1.94 the times of power, if all things are similar. That is 1.94*880 HP, or about 1710 HP, the amount of power a Spitfire I airframe would require to go as fast as the real life Mark XIV. As we can see by the XIV the increase in drag was equivalent to about 130 HP. BTW that 1710 HP is almost precisely what the Merlin 66 Spits had, so but those were only good for about 330 mph at SL.

From that it follows that the drag increase during the development of the Spitfire was equivalent to about 25-30 mph speed loss. That's hardly insignificant.

Try again and see my post.
The Merlin 66 did not have 1710hp at sea level, it had 1705hp at 5750ft and had to use a partially closed throttle at sea level (much like the DB 605A puts out 1475ps at sea level but 1525-1550ps at a bit over 2000m). Merlin 66 was good for a lot closer to 1600hp at sea level.
The 20mm gun installation was worth a few mph, drag yes but little to do with the "dirty underside". the increase in weight of over a ton means higher induced drag. We can account for 8-12mph just in the induced drag and guns (minimum), Between just those losses from drag and the actual power being lower than you figured the increase in drag from the "dirty underside" is heading for "insignificant" really quick.
 
Do me the courtesy of reading the statement, to which I was responding, before indulging in your latest bout of sarcasm.

Yes I have read it. Siegfried said the Spitfire's underside could use a bit of cleanup and that would be increasingly useful in the long term, to which you replied with a sarcastic and nationalistic fit about how much greater RAF glory (as you perceive the events) would have been in 1940 then.

At this point I dared to point out that you are not making any sense. Did I miss something perhaps?
 
We may disagree on the Whirlwind but the procedure I outlined actually works out rather well for the Spitfire.
I had done this exercise before and knew how it came out. Spitfire bashers are welcome to poke holes at it ( I am sure there are a few flaws) but I think it is better than "I think, it must have" and so on without even an attempt to show how the "extra" drag affects things.
My apologies if I was a little (!!!) short, earlier, but I'm growing tired of what you call "Spitfire bashers," who never produce anything to back up their "findings," or massage the available figures to suit themselves. If, during my years of research, I'd found anything, at all, condemning the radiators, I would say so, since truth is more important than Jingoism (of whatever nation.) Always, in the back of mind, there hovers the saying, "There are lies, damned lies, and statistics."
 
Yes I have read it. Siegfried said the Spitfire's underside could use a bit of cleanup and that would be increasingly useful in the long term, to which you replied with a sarcastic and nationalistic fit about how much greater RAF glory (as you perceive the events) would have been in 1940 then.
At this point I dared to point out that you are not making any sense. Did I miss something perhaps?
Very definitely, since I was responding to your comment "and the Spitfire in any case contributed much less to that than the numerous Hurricane," not something written by our Australian-based pro-German, so try again.
As for the "RAF glory," it's not just how I see it, but all the non-German-lovers, of this country, see it that way, as well, and, if you think that statistics are sarcastic, it could well explain your misuse of them.
 

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