I've had this file for some time, it has most major marks of operational WW2 fighters arranged in order of drag, both CdA and pure CD. The formatting isn't very good. Sorry. The calculations are made by using the usual formulas and data, details on request. I assume prop efficiency to be 80% and I give a little consideration to exhaust thrust where appropriate. Posting this by pages from the original document, firstly Cd0A (so-called equivalent flat plate area, in notional square feet) which is the absolute parasite drag, in ascending order. Secondly, the same list ordered by drag coefficient, with a couple of bombers added for interest at the end.
Calculated drag numbers for fighters
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GregP
Major
I appreciate your sharing of the data.
Could you add a sample calculation? Just one, any one? Your favorite?
Could you add a sample calculation? Just one, any one? Your favorite?
Or side by side calculation of F4U-1A and P-47D
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- #7
ch1 is an important NASA paper by Laurence K. Loftin, Director of Aeronautical Research at NASA Langley.
Appendix C has all the necessary formulae. I use 0.8 for the Oswald factor, e. I use the best speed/altitude/power combination I can find as the input numbers. Normal references for wing area and span. I don't have a way to take mach effects into account so I go for the lowest rated altitude to minimize that problem. I use 80% for propeller efficiency. That's not ideal but its a reasonable ballpark. Note that some later variants (109K, P-38L maybe) were prop-limited and suffered for it.
I'll write up the procedure as requested and include some examples.
(The listings are really for comparison between types. The methodology does not allow for absolute accuracy nor sophisticated drag analysis)
Appendix C has all the necessary formulae. I use 0.8 for the Oswald factor, e. I use the best speed/altitude/power combination I can find as the input numbers. Normal references for wing area and span. I don't have a way to take mach effects into account so I go for the lowest rated altitude to minimize that problem. I use 80% for propeller efficiency. That's not ideal but its a reasonable ballpark. Note that some later variants (109K, P-38L maybe) were prop-limited and suffered for it.
I'll write up the procedure as requested and include some examples.
(The listings are really for comparison between types. The methodology does not allow for absolute accuracy nor sophisticated drag analysis)
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- #8
This is the step-by-step calculation. For any WW2 fighter with a piston engine you can get a reasonable estimate of the drag. This does require a familiarity with arithmetic but no math more advanced than that.
First, you need the data. Using a decent reference book get the span and wing area. Take a compromise figure for the weight. You can't use empty weight or max take-off weight, the first is unreasonable and the second penalizes the aircraft which were loaded up with fuel and stores. Either normal loaded or empty plus 1500 pounds is what I use.
Find a good max speed figure, preferably from a flight test report by a proper flight test organization such as Farnborough, Wright Field or Rechlin. The best place to find them is of course at http://www.wwiiaircraftperformance.org where most important fighters are covered. It's best if you have speed, altitude and whatever engine settings were in use. A good engine reference helps to get an accurate horsepower figure for that set of conditions.
A word about rated altitude. That is the altitude at which the engine is rated to produce full power in whatever supercharger gear applies. At top speed any aircraft with a ram air intake (that's most of them) will have a full throttle height greater than rated altitude. It gets a free boost from ram air. Some flight test reports list speeds at rated rather than full throttle height. The calculation will still work but the aircraft may have a faster speed a couple of thousand feet higher up
I use Imperial units, UK style with lbs weight not slugs. G is 32.2 ft/sec^2, convert mph to ft/sec by multiplying by 88/60.
So, to start.
Step one, calculate the aspect ratio, that is span squared divided by wing area. It will almost always be around 6 for this class of aircraft. (Biplanes require a little extra work which I won't trouble you with here, just be aware that it's different.
I'm going to do the Typhoon 1b, late production, taking top speed as 412 mph at 19000ft with 1860hp from the Sabre in FS gear.
Span is 41.5 ft, wing area 278 sq ft. so aspect ratio is 6.2. Weight is 11000 lbs.
Air density at 19000ft is 0.042 lbs/cubic foot. You can look this up in a table of standard atmosphere, of which there are many. Many standards, I mean, all a little bit different. Personally I use a formula for sigma, the density ratio, and multiple by 0.07651 the air density at sea level in lb/ft^3 in the standard I use.
To convert horsepower to thrust, at prop efficiency 80%, multiply 0.8 by the horsepower figure 1860 and multiply by 375 then divide by the speed in mph. In this case the result is 1354.37 lbs force. (I've ignored exhaust thrust here, but you can add it in. Probably around 200 lbs for a 2000hp engine)
The coefficient of lift at the example speed is 2 x weight x 32.2 divided by (air_density x (speed_mph x 88/60 )squared x wingarea)
I get 0.1649. Sanity check, the max CL of a WW2 fighter is usually around 1.4 and CL at top speed is going to be a tenth of that, roughly.
The drag coefficient due to lift is calculated next. It is CL squared divided by (pi x oswald_number x aspect_ratio). Here it works out to 0.0017.
The total drag coefficient Cdtot is conveniently CL x thrust divided by weight. 0.0203 in this case.
The drag coefficient due to non-lift-dependent drag CD0 (that is friction, parasite, trim, everything but induced drag) is Cdtot minus Cdi
So that's 0.0203 minus 0.0017 which is 0.0186. Which can be used to check the ratio of lift and non-lift drag at the given speed. Normally Cdi will be less than 10% of CD0. A lightly loaded aircraft at high speed in denser air will be have a better figure. An aside. If you want to know how your favourite aircraft will perform in regard to induced drag the best comparison data is span loading, not wing loading. Short stubby wings on heavy aircraft have to work hard.
Why do these figures matter? Well, within limits they can be used to estimate the total drag at any condition of speed, altitude, weight and load factor. As the aircraft slows, the proportion of lift-induced drag increases as the inverse square of the speed. The non-lift-dependent drag decreases directly with the square of the speed. Minimum drag is where the two drag components are equal, and that is close to where the best climb speed will be for a prop aircraft. Because that is where the excess power available to climb is highest. We can easily calculate how hard the aircraft can turn without losing height, or how much height it will lose at max turn rate.
That's it. I tried not to be too technical or too simple for the varied audience here. Any good book about aerodynamics will have the formulae in proper math notation.
First, you need the data. Using a decent reference book get the span and wing area. Take a compromise figure for the weight. You can't use empty weight or max take-off weight, the first is unreasonable and the second penalizes the aircraft which were loaded up with fuel and stores. Either normal loaded or empty plus 1500 pounds is what I use.
Find a good max speed figure, preferably from a flight test report by a proper flight test organization such as Farnborough, Wright Field or Rechlin. The best place to find them is of course at http://www.wwiiaircraftperformance.org where most important fighters are covered. It's best if you have speed, altitude and whatever engine settings were in use. A good engine reference helps to get an accurate horsepower figure for that set of conditions.
A word about rated altitude. That is the altitude at which the engine is rated to produce full power in whatever supercharger gear applies. At top speed any aircraft with a ram air intake (that's most of them) will have a full throttle height greater than rated altitude. It gets a free boost from ram air. Some flight test reports list speeds at rated rather than full throttle height. The calculation will still work but the aircraft may have a faster speed a couple of thousand feet higher up
I use Imperial units, UK style with lbs weight not slugs. G is 32.2 ft/sec^2, convert mph to ft/sec by multiplying by 88/60.
So, to start.
Step one, calculate the aspect ratio, that is span squared divided by wing area. It will almost always be around 6 for this class of aircraft. (Biplanes require a little extra work which I won't trouble you with here, just be aware that it's different.
I'm going to do the Typhoon 1b, late production, taking top speed as 412 mph at 19000ft with 1860hp from the Sabre in FS gear.
Span is 41.5 ft, wing area 278 sq ft. so aspect ratio is 6.2. Weight is 11000 lbs.
Air density at 19000ft is 0.042 lbs/cubic foot. You can look this up in a table of standard atmosphere, of which there are many. Many standards, I mean, all a little bit different. Personally I use a formula for sigma, the density ratio, and multiple by 0.07651 the air density at sea level in lb/ft^3 in the standard I use.
To convert horsepower to thrust, at prop efficiency 80%, multiply 0.8 by the horsepower figure 1860 and multiply by 375 then divide by the speed in mph. In this case the result is 1354.37 lbs force. (I've ignored exhaust thrust here, but you can add it in. Probably around 200 lbs for a 2000hp engine)
The coefficient of lift at the example speed is 2 x weight x 32.2 divided by (air_density x (speed_mph x 88/60 )squared x wingarea)
I get 0.1649. Sanity check, the max CL of a WW2 fighter is usually around 1.4 and CL at top speed is going to be a tenth of that, roughly.
The drag coefficient due to lift is calculated next. It is CL squared divided by (pi x oswald_number x aspect_ratio). Here it works out to 0.0017.
The total drag coefficient Cdtot is conveniently CL x thrust divided by weight. 0.0203 in this case.
The drag coefficient due to non-lift-dependent drag CD0 (that is friction, parasite, trim, everything but induced drag) is Cdtot minus Cdi
So that's 0.0203 minus 0.0017 which is 0.0186. Which can be used to check the ratio of lift and non-lift drag at the given speed. Normally Cdi will be less than 10% of CD0. A lightly loaded aircraft at high speed in denser air will be have a better figure. An aside. If you want to know how your favourite aircraft will perform in regard to induced drag the best comparison data is span loading, not wing loading. Short stubby wings on heavy aircraft have to work hard.
Why do these figures matter? Well, within limits they can be used to estimate the total drag at any condition of speed, altitude, weight and load factor. As the aircraft slows, the proportion of lift-induced drag increases as the inverse square of the speed. The non-lift-dependent drag decreases directly with the square of the speed. Minimum drag is where the two drag components are equal, and that is close to where the best climb speed will be for a prop aircraft. Because that is where the excess power available to climb is highest. We can easily calculate how hard the aircraft can turn without losing height, or how much height it will lose at max turn rate.
That's it. I tried not to be too technical or too simple for the varied audience here. Any good book about aerodynamics will have the formulae in proper math notation.
Hi,
Thanks for sharing all your stuff. This looks very interesting. Out of curiosity I have some Oswald numbers for US WWII airplanes that I think came from the book "America's 100,000" by Francis Dean (but I don't have the book handy now and can't check to make certain. Anyway these values range from 0.7 (for the P-40) to 1.02 (for the P-47). Out of curiosity, have you done any calcs to determine how sensitive your results might be to your inputs, like your estimated Oswald Efficiency, etc?
Pat
PS, here is a list of the numbers that I do have.
Thanks for sharing all your stuff. This looks very interesting. Out of curiosity I have some Oswald numbers for US WWII airplanes that I think came from the book "America's 100,000" by Francis Dean (but I don't have the book handy now and can't check to make certain. Anyway these values range from 0.7 (for the P-40) to 1.02 (for the P-47). Out of curiosity, have you done any calcs to determine how sensitive your results might be to your inputs, like your estimated Oswald Efficiency, etc?
Pat
PS, here is a list of the numbers that I do have.
GregP
Major
I came in here earlier tonight and the links to the pages he went to for the drag numbers were there.
Now they're GONE! What happened?
Now they're GONE! What happened?
- Thread starter
- #12
PFVA, I have that book and my edition has that table but without e numbers. My inclination is that Oswald isn't particularly influential at high speed. I only expect to use speeds between max and best climb, but even there, roughly half top speed, the numbers are beginning to fall apart because drag coefficients don't stay the same for different Reynolds numbers or AoA.
I check the sanity two ways. By calculating the best climb rate, finding the drag at that speed by taking the two components at max and multiplying by v squared for the profile and 1/v^2 for the induced. Find the power required to fly at that speed and subtract from the power available, reduce by prop efficiency and using the horsepower formula you can get a climb rate. If it's not TOO far off you have a sound set of figures. You can get the ceiling too but that tends to be pretty variable. As it would be with two fighters from the same squadron.
(I had a few email exchanges with Dizzy Dean when his book came out. What a gentleman, very helpful with setting up these calculations. Anybody with this sort of interest needs to get his book if they can.)
drgondog
Major
As for 0.1 x Thp being a ball park for in-line exhaust thrust. The P-51B Jet thrust component at 444mph at 29,000 feet, 61"/3000 RPM and Thp of Engine of 1019Thrust HP, jet velocity of 2610 ft/sec ,
= 303 pounds x cos 35 degrees = 288 pounds for the angle of the exhaust from CL. This is exactly why one can't rely on either Dean or Hoerner when considering Performance Calcs with respect to 'accurate vs wandering ballpark'.
The Jet thrust depends on external pressure (Po) at altitude examined, the manifold pressure (61" MP for 1650-3 engine at 29,000 feet w/ram air calculated) and the fuel flow at those settings.
Reference NAA "Report NA-5534 Performance Calculations for the P-51B-1 Airplane".
Further complications regarding calculating drag include:
Prop Vortex pressure drag along the fuselage plus ~ 33% of the wing, and the Hp losses due to cooling drag and pressure recovery of the carburetor scoop.
During WWII those calcs were expressed as Thrust Horsepower Losses, while the Prop and Jet Thrust components were combined for Thrust Horsepower Available.
Next, but not least, the prop efficiency of the P-51B/D Ham Standard 4 blade prop at 3000 RPM and 29K for velocities in range of 425-444mph was about 0.848 and 0.839 respectively.
Last, at the same GW, same engine, same boost/RPM the Total Drag of the P-51D and B were the same. The only reason the B quoted top speed varies from the D is that the 1650-3 had a critical altitude 4500 higher than a D with 1650-7 Critical Altitude.
That said, I really appreciate the work you put in this subject. My head hurt just focusing on the P-51.
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- #14
Jet thrust from the exhaust. What I didn't write was that it depends on altitude and on the indicated horse power of the engine, not the brake horse power. A merlin at high altitude would have an indicated HP of (say) 2000+. Around 150 for mechanical losses and 150 for each supercharger gear. That guess at the IHP would be what I took 10% of. And of course at altitude there is a reduction of back pressure and that increases the thrust too. Now, for a well-documented type such as P-51D you can find more detail. Other types are not so easy. Other exhausts are not so efficient. In some cases the designers didn't bother at all, with turbos they have already got as much energy as they can out of the exhaust. P-47 did try to get something, P-38 did not.
drgondog
Major
All the charts presented refer to Brake Hp, then fuel consumption, then RPM at both Low and High blower to pick the Charge Me in slugs/sec. Of course, altitude is part of the calculation but based on ambient pressure not height.
Then based on Charge, you need a cross plot on velocity and a range of Po/Pm to pick the Ve of the exhaust gasses at 90 degrees, then finish the Ve calc by multiplying Cos 35 degrees (=.82) to account for exhaust stack angle to get the component in line with thrust angle of prop.
There were no instruments available in the cockpit or elsewhere that I am aware of that would calculate 'indicated Hp;. Both analysis and flight Test reporting were based on Brake Hp as f(altitude) to provide common reference, before such calculations for ram air were applied.
The get reasonable values for even flat plate drag, you have to normalize every calculated value to a common airspeed and altitude for comparative purposes, which Dean did in his AOHT. That said his reference CDo was flawed in that not every CDo for every aircraft was tailored to Re equivalency.
As such, interesting but... I had several conversations (written) right after he published his book. He agreed my issues and pointed out his caveats and both of us agreed that his presentation was as good as could be reasonable expected when so little drag values of plots of CD vs Re were available.
To recapitulate, I am not criticizing your efforts, only commenting on the extreme difficulty at arriving with fact based comparisons - including the NASA discussions. In this case I am a critic, not a doer, save for P-51 discussions in great detail.. One last comment, the CD vs Re between the P-51D and B are virtually the same and both are very close to the preceding NA-73 through NA-99. The primary distinction are a.) slightly greater CDo for the earlier Mustangs due to the differences in the carb intake and less efficient scoop/cowl/exit scoop design before the XP-51B wrapped up at Ames during the Rumble Tests. Also, the cooling drag was not 'net zero' on the earlier Mustangs for high speed level flight.
Then the Tjet= Me*Ve and THp = T*V/375 to bring the calculations into Horsepower Available to add to the Prop Hp Available.
