Aces High Bulletin Board
General Forums => Aircraft and Vehicles => Topic started by: drgondog on December 30, 2010, 10:20:11 AM
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I posted this question in another thread but it has value in this one.
The rudder/elevator combination creates some interesting (potential)theoretical holes in applying Oswald efficiency factors to the manuevering models at low speeds versus cruising conditions in perfect trim.
It is the substance of my taking opposite sides from 'well known' applications of the Induced Drag portion of the Thrust=Drag equations. Raymer for example, is a well known author of Aerodynamics texts and uses (and applies) 'e' as an extension of lifting line theory but as near as I can tell does not account for the effects of the fuselage on spanwise lift distribution, nor does he extend to high AoA and the effects of changes in AoA increasing Viscous drag due to lift.
Oswald, when he developed and published his NACA paper NACA-TR-408 on this subject in 1932, clearly states "k (1/pi*e*AR) describes the variation of parasite drag with AoA and the increase of parasite drag over the minimum case of a wing with elliptcal lift"
His paper suggests various values for 'e' to apply to high and low wing a/c without futher delving into high AoA approaching stall - or complex asymmetric .flight conditions.
Beginning with Oswald's 1932 NACA papers through various texts applying 'e' to account for increased viscous/paraite drag due to planform and Lift Coefficient (changes to AoA) through many different derivations to better model theory to wind tunnel tests - I have never seen an extension to better account for an airplane in asymmetrical flight conditions at high AoA... such as a 360 turn.
Dick Shevell in "Fundamentals of Flight" does apply empirically developed contributions to both sweep and fuselage and planform efficiency as well as fuselage interference effects - for level flight. The charts he presented from Douglas Aircraft studies on the Douglas family of commercial air liners in level flight vs flight tests are impressive.
Using the Shevell approach and values for the P-51 yields an 'e' of .87 for example - which I believe might be good assuming no propeller 'complications'
Krishnamurti, Prabahudasar and Panda presented a paper "On the Upper Limits of the Oswald Efficiency Factor for an Airplane" in the Journal of Aero Society of India, Vol 53, number 4 which extrapolated very well the importance of the ratios and contributions of the horizontal stabilizer to the wing in the calculation of 'e' - proving that e may actually exceed 1.0
I suspect that increasing contributions of lift and drag surrounding a tail at a different AoA from the wing (immersed in downwash), which must change as the required deflections increase to maintain equilibrium, and are further subjected to the increasing turbulence (beyond normal flight rotating stream tube of prop wash) as separation inevitably increases on the wing - and vortex drag increases behind the wing - must change any previously accepted value of 'e' for that wing/body combination in level flight.
Ergo - applying 'e' = .8 or .85 (or any value), so many use to start a Performance discussion, has roots in level flight dash or cruise but must be carefully re-examined starting with level flight stall and really questions in high G asymmetric flight conditions.
Probably a better approach is to first examine 'e' when Thrust is supplied by jet engine so that variations in propeller efficiency may be eliminated at the beginning of the study.
Any thoughts?
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What do you speculate the practical consequences, in % or actual values, would be in modeling high G turns?
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What do you speculate the practical consequences, in % or actual values, would be in modeling high G turns?
Apparently I am limited to 8o char in the 'reply' mode. Simply, I expect that 'e' decreases considerably as a result of the elevator deflections and parasite drag increases beyond CDo
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What do you speculate the practical consequences, in % or actual values, would be in modeling high G turns?
[Specifically - CDtotal = CDparasite + CDlift related +CDcompressibility
CDparasite not only contains the zero lift profile drag of the wing but also friction and pressure drag of the fuselage, tail surfaces and any other component exposed to the airflow.
CDlift releated contains not only the expression for drag due to the lift of the wing but also drag due to planform and twist.
CDcompressibility is wave drag which is present only for speeds aat or above drag-divergence number (few examples in WWII but do include top speed examples of P-51, Ta 152, P-47M/N, etc where you need to be careful in calculating CDo.
So, CDp=CDo+r*CL>>2 ----> where CDo is zero lift parasite drag of the airframe and r=empirically determined constant to account for lift releated VISCOUS drag which varies with AoA and includes the change to friction and pressure drag as the AoA increases.
CDi=(CL>>2)/(e*pi*AR) which is what we frequently use in our calculations ---> where e is Oswald Efficiency factor , first estimted (.7-.9) then validated via wind tunnel and flight tests. This 'e' and associated values of ".8" is an Ok factor and generally accounts for the lift and drag of a horizontal stabilizer in trimmed flight.
So, now CDtotal = CDo + (r+ 1/(pi*e*AR))*CL>>2
Lets go to next post below..to finish this out./quote]
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What do you speculate the practical consequences, in % or actual values, would be in modeling high G turns?
[ So, to continue
CDi=(CL>>2)/(e*pi*AR) which is what we frequently use in our calculations ---> where e is Oswald Efficiency factor , first estimted (.7-.9) then validated via wind tunnel and flight tests. This 'e' and associated values of ".8" is an Ok factor and generally accounts for the lift and drag of a horizontal stabilizer in trimmed flight.
So, now CDtotal = CDo + (r+ 1/(pi*e*AR))*CL>>2 (familiar) + CDcompressibility.
Classic Aero 'redefines' e to now account for 'r' and tells us to 'assume' a value of .8 (or thereabouts). My question is "May we assume that wind tunnel tests and the associated design value for 'e' extrapolated as a function of CL vs AoA are in fact valid for high deflection angle elevator and rudder inputs, in which the relative AoA (and associated CL for both the wing will be presumed different in a banked high G turn near Stall?
Lets go to next post below..to finish this out./quote]
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Finally - I just don't have a real grasp of a % change which should be made.
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Finally - I just don't have a real grasp of a % change which should be made.
This is an excellent reference with sound approaches to develop values of 'e' which can be extrapolated when AR, CDo are known - pay particular attention to the plots and equations under Induced Drag chapter for Lift Dependent Drag Items.
http://adg.stanford.edu/aa241/AircraftDesign.html
Short answer to my belief system -
'e' for Mustang with extrememly low Drag wing/airframe and correspondingly low CDparasite must be significantly better than a Me 109G with similar AR for symmetrical flight conditions.. and that little research has been performed to estimate change in 'e' due to high angles of attack and asymmetrical flight conditions - while immersed in a turbulent/rotational stream tube behind the prop disk.
Next - never strive for a CDo based on high speed/high altitude speed runs for such a/c as Mustang and Ta 152 etc - which are all creeping into .65+M range at 25K+ altitude... until you have eliminated contribution due to drag rise. Meaning go for SL data to develop CDo... then you can go back to 25K to find CDm.
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The 80 character limit only applies to the subject line.
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The rudder/elevator combination creates some interesting (potential)theoretical holes in applying Oswald efficiency factors to the manuevering models at low speeds versus cruising conditions in perfect trim....
Ergo - applying 'e' = .8 or .85 (or any value), so many use to start a Performance discussion, has roots in level flight dash or cruise but must be carefully re-examined starting with level flight stall and really questions in high G asymmetric flight conditions.
Probably a better approach is to first examine 'e' when Thrust is supplied by jet engine so that variations in propeller efficiency may be eliminated at the beginning of the study.
Any thoughts?
Several thoughts for you:
1) Accepted values for e between .7-.9 applies for low angle of attack flight. At higher angles of attack viscous induced separation becomes a factor. At higher aoa using these e values without accounting for increases in lift dependent viscous drag is not appropriate. The result on drag is that lift dependent drag due to viscous forces increases with increasing aoa.
2) Because this lift dependent drag is a function of viscous induced separation of the boundary layer I don't know of a way to mathematically estimate this without some serious CFD analysis because details of what's happening in the boundary layer would need to be modeled.
3) Variation of e or lift dependent viscous drag can be obtained however from drag polars of wind tunnel or flight tests for specific aircraft.
4) AH models this variation of lift dependent viscous drag. In fact for the P-51 Pyro has posted the data he has already:
(http://beta.hitechcreations.com/pyro/p51ddrag.jpg)
5) One way to account for all of this is to vary e. Variation of e with aoa applies for all aircraft including jets.
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Several thoughts for you:
1) Accepted values for e between .7-.9 applies for low angle of attack flight. At higher angles of attack viscous induced separation becomes a factor. At higher aoa using these e values without accounting for increases in lift dependent viscous drag is not appropriate. The result on drag is that lift dependent drag due to viscous forces increases with increasing aoa.
We are pretty much in agreement - having said that, there is a body of literature (Raymer and others) that stipulate that the viscous lift due to drag is in fact accounted for in 'e' as I expressed it above...I have an MS in Aero but candidly this is something I never questioned either in academics or industry until I started playing with 'performance models' of WWII a/c. It was then that I started playing with horizontal stab area/wing area and really started thinging about lift/drag due a significantly deflected elevator (and rudder) as well as started questioning a 'constant' e.
2) Because this lift dependent drag is a function of viscous induced separation of the boundary layer I don't know of a way to mathematically estimate this without some serious CFD analysis because details of what's happening in the boundary layer would need to be modeled.
Agreed again -and the complexity of asymmetric flight immersed in a turbulent and rotational prop stream tube near stall is another model that I doubt can be set up for a Navier Stokes solution
3) Variation of e or lift dependent viscous drag can be obtained however from drag polars of wind tunnel or flight tests for specific aircraft.
I did a literature search to see if I could find papers on tests and data regarding asymmetric flight profiles in high AoA but could not find them. Everything I did flind was 'normal AoA variations - doesn't mean the tests aren't out there, just that I was unsuccessful!
4) AH models this variation of lift dependent viscous drag. In fact for the P-51 Pyro has posted the data he has already:
(http://beta.hitechcreations.com/pyro/p51ddrag.jpg)
5) One way to account for all of this is to vary e. Variation of e with aoa applies for all aircraft including jets.
I absolutely buy the concept, just don't have an immediate approach at hand to apply to the WWII aircraft of interest. Good to chat!
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I absolutely buy the concept, just don't have an immediate approach at hand to apply to the WWII aircraft of interest. Good to chat!
Brain fart - I meant to say 'viscous drag due to lift' not 'viscous lift due to drag'
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But what amount of variation would we be looking at here, or perhaps I should say, what would be the perceptible difference in performance? Large, small, or unknown? Obviously, some sort of approximation has to be made for some characteristics that are beyond the scope of a flight model. As long as the context of those approximations are always stated when presenting theoretical performance data...
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It's significant at high aoa Stoney. For example just look at the P-51 graph above. AHT has CD for the P-51D at .0176 for level flight. According to Pyro's data at Cl of 1.0 the incremental drag due to viscous separation is about .016 by itself so the viscous effects by itself nearly doubles CD.
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It's significant at high aoa Stoney. For example just look at the P-51 graph above. AHT has CD for the P-51D at .0176 for level flight. According to Pyro's data at Cl of 1.0 the incremental drag due to viscous separation is about .016 by itself so the viscous effects by itself nearly doubles CD.
Dtango - do you have access to the written part of NA-46-130? (and thanks for posting that).
To Stoney - just a point of reference to dtango's comment about parasite drag doubling at CL=1.0, the approximate CLmax for a 9600 pound P51D is ~ 1.7 so extension of the drag polar from 1.0 to 1.7 that he presented takes it far above 'delta' .016.
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drgondog: I'd be happy to send it to you but in this case I don't have it :). I wish I did.
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drgondog: I'd be happy to send it to you but in this case I don't have it :). I wish I did.
LOL - For a long time I had access to NAA 'stuff' via Al White... now it is nigh impossible to get access to NAA archives from Boeing..
Thanks for posting the June 46 report on the effect of AoA in CDp - it is the first time I have seen it and I am curious whether that was a NACA result or perhaps some arcane Schmeud methodology.
The June 1946 Report No TSCEP5E-1908 by Gentile has the most representative flight values that I use for the 51D
so far..
http://www.wwiiaircraftperformance.org/mustang/p51d-15342.html
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lol I know what you mean with about Boeing. I tried to get some docs from them but they no longer had a way to transfer from microfiche to any other format. I almost bought them a digital camera as a work around. Pretty pathetic. The Boeing NAA history dept is really crappy. I don't know, maybe I should try connecting them up with the NASM to get some gov't funds to get something that can read microfiche into PDF or something. Given all the $ Boeing is sinking into the Dreamliner I doubt it's going to improve anytime soon.
As to P-51 drag polars NACA ACR 4K02 has a compilation of flight test and wind tunnel data.
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930092458_1993092458.pdf
Flight Test Drag Polar P-51B
(http://brauncomustangs.org/upload/files/p51dragpolar-flight.jpg)
Wind Tunnel Drag Polars P-51B
(http://brauncomustangs.org/upload/files/p51dragpolar-wt.jpg)
I haven't ever attempted comparing NAA-46-130 with the NACA data though for various reasons.
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lol I know what you mean with about Boeing. I tried to get some docs from them but they no longer had a way to transfer from microfiche to any other format. I almost bought them a digital camera as a work around. Pretty pathetic. The Boeing NAA history dept is really crappy. I don't know, maybe I should try connecting them up with the NASM to get some gov't funds to get something that can read microfiche into PDF or something. Given all the $ Boeing is sinking into the Dreamliner I doubt it's going to improve anytime soon.
As to P-51 drag polars NACA ACR 4K02 has a compilation of flight test and wind tunnel data.
I haven't ever attempted comparing NAA-46-130 with the NACA data though for various reasons.
IIRC, NAA a.) stated the flight test results were in close agreement with the wind tunnel tests, but the approximation of the desert 'dust' was not as close to an attempt to reproduce a similar surface roughness to get true RN equivalency?
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It's significant at high aoa Stoney. For example just look at the P-51 graph above. AHT has CD for the P-51D at .0176 for level flight. According to Pyro's data at Cl of 1.0 the incremental drag due to viscous separation is about .016 by itself so the viscous effects by itself nearly doubles CD.
No, I understand the relationship of AoA to Cd...I was curious about the original post regarding estimating "e". Perhaps I should have worded it differently. Raymer states that Oswald's technique is only valid at moderate AoA. He recommends the suction method as more accurate for high-speeds and to account for viscous separation...etc...
My question was how much variation in "e" do we see, away from the oft used estimate of .8, in high AoA maneuvers and whether or not it was significant. Perhaps I'm misunderstanding something.
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No, I understand the relationship of AoA to Cd...I was curious about the original post regarding estimating "e". Perhaps I should have worded it differently. Raymer states that Oswald's technique is only valid at moderate AoA. He recommends the suction method as more accurate for high-speeds and to account for viscous separation...etc...
My question was how much variation in "e" do we see, away from the oft used estimate of .8, in high AoA maneuvers and whether or not it was significant. Perhaps I'm misunderstanding something.
Ah I see. Using the same P-51 data from Pyro above I get a value of e~ .6 at Cl=1.0 vs. e=.87 at level flight.
For reference here's a comparison of e variation for modern jets at load factors of up to 3 g's so the variation of e from low aoa can be significant as evidenced.
(http://thetongsweb.net/images/e-values.jpg)
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Ah I see. Using the same P-51 data from Pyro above I get a value of e~ .6 at Cl=1.0 vs. e=.87 at level flight.
For reference here's a comparison of e variation for modern jets at load factors of up to 3 g's so the variation of e from low aoa can be significant as evidenced.
(http://thetongsweb.net/images/e-values.jpg)
Interesting Tango - I find it interesting that L/D as a function of g is linear, particularly with relatively low aspect wings.. but then at .6M I guess I should expect Induced Drag to be relatively low.
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No, I understand the relationship of AoA to Cd...I was curious about the original post regarding estimating "e". Perhaps I should have worded it differently. Raymer states that Oswald's technique is only valid at moderate AoA. He recommends the suction method as more accurate for high-speeds and to account for viscous separation...etc...
My question was how much variation in "e" do we see, away from the oft used estimate of .8, in high AoA maneuvers and whether or not it was significant. Perhaps I'm misunderstanding something.
Stoney - here is Raymer's doctoral thesis. I am going to spend more time understanding his approach to Design Optimization but have gone far enough to see that he will not attempt a 'closed form solution' via Calculus of Variations.. I also noticed he does not define or use 'e' in his Nomenclature/Index of terms but may have overlooked something.
I do recall in past cursory reading of one of his design texts that he used 'e' strictly as a relation to wing planform w/o taking into consideration a 'wing/body' effect to 'e'
http://www.aircraftdesign.com/RaymerThesisFinalRevLowRes.pdf
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A couple of thoughts on Raymer's & Shevell's e estimation methods as I understand them...
Raymer uses a semi-emperical approach to estimate K from leading edge suction. The issues with this approach are: a) you use a look up chart for S based on Cl but this is based on knowing the design Cl of the wing, b) the technique was geared more toward thin wing, low aspect ratio jets, c) it doesn't account for the entire plane including the fuselage.
Shevell's approach is also a semi-emperical approach based on data obtained from data from Douglas aircraft and accounts for fuselage affects. However in the past when I've run the calculations using Shevell's estimating method it estimates e much higher than what the P-51 data posted by Pyro shows.
I believe both of their estimation methods are intended for estimating initial designs. They are great for that purpose. However to reverse engineer and accurately estimate lift dependent viscous drag for an airplane I think we're stuck with getting into nasty Navier-Stokes world of CFD...
...or go find some good drag polars for specific aircraft based on flight tests or wind tunnel tests :).
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How much impact does the fuselage have at high AoA?
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How much impact does the fuselage have at high AoA?
It's all about what happens in the boundary layer and the boundary layer surrounds the surface of the entire plane, fuselage and all. The details are very important. Something simple as an open tail wheel well, the angle of the windscreen, curvature of the fuselage, etc. make big differences on the pressure gradient in the boundary layer. So qualitatively it has a signifcant impact. Quantitatively it's hard to estimate what that impact would be without some serious computation.
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Well, Oswald's original idea was that Cl^2/Cd relation is usually roughly linear (ie the Cl/Cd is parabolic) between Cl 0.1-1.0 and as this Cl range covers about pretty much entire flight envelope, it's been proven to work pretty well. As example Pyro's data from NAA report gives almost linear line at this Cl range. However, its probably a calculation.
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A couple of thoughts on Raymer's & Shevell's e estimation methods as I understand them...
Raymer uses a semi-emperical approach to estimate K from leading edge suction. The issues with this approach are: a) you use a look up chart for S based on Cl but this is based on knowing the design Cl of the wing, b) the technique was geared more toward thin wing, low aspect ratio jets, c) it doesn't account for the entire plane including the fuselage.
Shevell's approach is also a semi-emperical approach based on data obtained from data from Douglas aircraft and accounts for fuselage affects. However in the past when I've run the calculations using Shevell's estimating method it estimates e much higher than what the P-51 data posted by Pyro shows.
I believe both of their estimation methods are intended for estimating initial designs. They are great for that purpose. However to reverse engineer and accurately estimate lift dependent viscous drag for an airplane I think we're stuck with getting into nasty Navier-Stokes world of CFD...
...or go find some good drag polars for specific aircraft based on flight tests or wind tunnel tests :).
We appear to be in violent agreement..
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Well, Oswald's original idea was that Cl^2/Cd relation is usually roughly linear (ie the Cl/Cd is parabolic) between Cl 0.1-1.0 and as this Cl range covers about pretty much entire flight envelope, it's been proven to work pretty well. As example Pyro's data from NAA report gives almost linear line at this Cl range. However, its probably a calculation.
Go look at the charts (Oswald's Report 408, charts 24) on page 25 which are flight test results for various German a/c. Note the rapid change in CDp with respect to CL @ ~ 1.0 with no further extrapolation (other than a huge gradient) past CL=1 .
For the turning CL's experienced in high G manuevers, the CL's approach CLmax, which for most WWII fighters is between 1.4 and 1.7 for zero flap manuever.
It is clear from the charts that CDp is changing far more rapidly than CDi above CL=1.0
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Ok, but I guess where I'm hanging up on this is that it would seem that as the wing approaches extreme AoA, the drag from the fuselage isn't the rub, its the fact that the boundary layer on the wing is almost totally separated. Is the Cdi the only indication of the separation on the wing? Is the increase in drag primarily from the fuselage/empenage?
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Ok, but I guess where I'm hanging up on this is that it would seem that as the wing approaches extreme AoA, the drag from the fuselage isn't the rub, its the fact that the boundary layer on the wing is almost totally separated.
Is the Cdi the only indication of the separation on the wing?
No. CDi is solely associated with the Induced Drag of the wing and expressed as CL>>2/pi*AR
Is the increase in drag primarily from the fuselage/empenage?Mostly yes - in the form of incremental Parasite Drag, and further referred to as viscous drag related to lift - more below. It affects the fuselage and tail by immersing those components into a larger wake - but also affects friction drag on the wing aft of the separation
Stoney - The boundary layer growth and separation on a wing is caused by adverse pressure gradient. When it 'separates' the wake/form drag and even friction drag increases dramatically. These effects are Not limited to the wing but affects the fuselage/tail, etc which are immersed in the expanded wake.These components are frequently further defined within parasite drag as 'viscous drag related to lift' - but not part of induced drag.
'e' theoretically accounts for these incremental 'delta' drag components related to lift, and as such, is built into the Induced Drag component of Total Drag. Initially CDi was expressed only in context of wing, but the more sophisticated developments of 'e' also accounted for the wing/fuselage combinations (high/middle/low wing for example)
Induced drag, is the drag due to the lifting line discontinuity of the wing at the wing tip - which creates an infinite vortex and is a function of the variation from elliptical wing planform as well as aspect ration to further refine from two dimenional airfoils to 3-d realities..
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Go look at the charts (Oswald's Report 408, charts 24) on page 25 which are flight test results for various German a/c. Note the rapid change in CDp with respect to CL @ ~ 1.0 with no further extrapolation (other than a huge gradient) past CL=1 .
For the turning CL's experienced in high G manuevers, the CL's approach CLmax, which for most WWII fighters is between 1.4 and 1.7 for zero flap manuever.
It is clear from the charts that CDp is changing far more rapidly than CDi above CL=1.0
I suggest that you actually read the text of your source before forming an argument; the CDi in these charts is an ideal case value ie lift distribution is assumed to be elliptical and all the additional drag from what ever source (induduced or parasitic) is lumped to CDp. In other words the term CDp contains also all the induced drag beyond perfect elliptical lift distribution.
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I suggest that you actually read the text of your source before forming an argument; the CDi in these charts is an ideal case value ie lift distribution is assumed to be elliptical and all the additional drag from what ever source (induduced or parasitic) is lumped to CDp. In other words the term CDp contains also all the induced drag beyond perfect elliptical lift distribution.
I agree that CDi (not corrected by e as denoted by solid line) is a calculated value assuming an eliptical lift distribution - and that the CDp is derived by CD-CDi, where CD total is flight test data in each of the 8 plots represented. What I see is a sharp gradient of CDp as f(CL) and no further data points to help us understand data at say 1.3 to 1.6. This is the substance of my comment that you chose to interpret as a 'failure to read'
So, repeating the argument in more detail. 1.) CDp as presented by Oswald does include drag terms above and beyond induced drag. 2.) CDi in the solid line is in fact CDi calculated based on 'span efficiency' as a correction to a pure elliptical planform. 3.)His arguement, elegant for CL at or below 1.0, is that the CDp represented by the dotted line plot does include 'other drag terms due to Induced drag... and therein lies the 'theory of e'.
In other words, Oswald has no thesis for 'e' for high AoA/High CL and does not present data or conclusions to account for a 'correction factor e'prime' to further account for increases to viscous drag on the fuselage and other components immersed in an increasing wake influenced by boundary layer separation, adverse pressure gradients and increased friction drag.
My thesis is threefold. 1.) there is a departure point where a constant 'e' is no longer valid, and 2.) that that the increased 'drag due to lift' such as all the viscous and friction drag associated with increases to AoA beyond CL~1.0 is neither linear nor predictible by the methods suggested by Oswald, and 3.) that the use of 'e' in CDi calcs for high G manuevers for WWII fighters in 'games' is - simply - inaccurate .
You and I may agree to disagree but I will keep it civil.
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@gripen You bring up a good point regarding Pyro's NAA data on the linearity. I assumed that was additional drag beyond e assumed in the induced drag already. It could be that the chart is representation of linear e as you suggest. As drgondog says, it might be some schmeud methodology ;) for accounting for it.
@stoney drgondog answered your question already. The point is boundary layer separation occurs not only along the wing but other parts of the aircraft too. It's possible that the wing dominates with the lift dependent viscous drags, but it's also probable that fuselage & other parts can play significant enough of a factor as well. That's the tricky thing about the boundary layer.
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@gripen You bring up a good point regarding Pyro's NAA data on the linearity. I assumed that was additional drag beyond e assumed in the induced drag already. It could be that the chart is representation of linear e as you suggest.
My point was that we can't explain the additional drag as function of parasitic drag using the Pyro's chart, firstly because it gives almost linear response and secondly because the scale of the effect is far too small; even at Cl 1.0 it's below 15% of total drag.
However you have allready posted a chart which contains the needed information for changing the polars (Cd0, Clmax and e) according to mach number:
(http://brauncomustangs.org/upload/files/p51dragpolar-wt.jpg)
Oswald's system is basicly a stochastic model of drag where entire additional drag caused by increased lift coefficient (complex phenomena) is lumped together to a simple term of e. It's not a perfect model but it has been proven to work well at low speeds. Obviously it does not work well when the mach number increases but we can extend it by adding some additional factors like mach number etc.
The reasons for the increased drag are, of course, even more complicated than at low speeds and we can argue about the explanations until the end of the day. However, based on the flight and wind tunnel tests we have some idea how the polar behaves when the mach number increases. And based on this knowledge we can form a relatively simple model to simulate an extremely complicated phenomena. In other words, we have limited knowledge why all this happens but we know the outcome, so we can model the phenomena in some degree.
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My point was that we can't explain the additional drag as function of parasitic drag using the Pyro's chart, firstly because it gives almost linear response and secondly because the scale of the effect is far too small; even at Cl 1.0 it's below 15% of total drag.
The reasons for the increased drag are, of course, even more complicated than at low speeds and we can argue about the explanations until the end of the day. However, based on the flight and wind tunnel tests we have some idea how the polar behaves when the mach number increases. And based on this knowledge we can form a relatively simple model to simulate an extremely complicated phenomena. In other words, we have limited knowledge why all this happens but we know the outcome, so we can model the phenomena in some degree.
My points and debate have focused not on high mach numbers where compressibility (and drag divergence) occur, it has always been about lower range speeds in high G, asymmetric flight conditions.
Revisit what I posed in the first post -
Ergo - applying 'e' = .8 or .85 (or any value), so many use to start a Performance discussion, has roots in level flight dash or cruise but must be carefully re-examined starting with level flight stall and really questioned in high G asymmetric flight conditions. I debate that an Oswald factor derived and explained and correlated with level flight, 1 G, symmetric flight conditions is equally valid for the turn manuever models applied to so many game simulations.
Period. The end.