When ventus 45 wrote of
I wasn’t sure if that was an invitation or a challenge, especially since I share jimmyg’s feelings a little.
You know when I think about I am not sure if our poster was setting bait on such a complex subject matter.
But since it was an interesting paper and generated some ideas that others here might also find interesting I thought I would follow it through although some illusions might be shattered. My apologies to those who know all this already!
The paper contains a lot of complicated and off-putting equations, but when one gets down to it one finds that for most of the time the authors are simply repeating very similar thoughts in sixteen different ways. Worse, they spend this time on what is probably the least important bit of the drag due to lift and they apply an empirical correction for Mach number that simply doesn’t work above about 0.8M
I find it easier to explain and understand the problem using the Limey version of the expression for drag due to lift : Cdi = k*CL^2/(pi*A)
where the “induced drag factor” (k) is the reciprocal of the Oswald efficiency (e).
That way an increase in the coefficient corresponds to an increase in drag.
Most researchers assume the drag due to lift to be composed of two parts – the inviscid (theoretical) vortex drag and the viscous (practical) drag. Using this concept the induced drag factor is:
k = (1+delta)+ (pi*A)*dCd/dCL^2
1 is the much discussed but never achieved ideal vortex drag that goes with an elliptical lift distribution
delta is the increment to be applied to that ideal to allow for taper ratio, fuselage effects etc.
dCd/dCL^2 is to account for all the bits of viscous dependent drag that vary with lift coefficient.
If you apply the equations given in the paper to the airplanes listed in the Appendix (leaving out the propeller driven machines and the two labelled unreliable) the average (1+delta) is 1.07 and the average “k” deduced from the tabulated values of “e” is 1.35.
In other words the additional lift dependent drag coming from viscous effects is four times greater than that from less than ideal lift distribution and makes the concentration on inviscid drag terms and the finer points of taper and sweep variations rather academic.
So when pugilistic animus writes
All in all aerodynamics is truly an art...because nature laughs at complex mathematics
one can only agree!
BTW underfire when you write
They reference specifically Boeing data, so it comes as no surprise that the paper shows the 737-800 having more drag reduction at cruise that the A320 Neo....
It is difficult to see how you arrive at this conclusion, since theA320 span efficiency is given as 0.783 and the B737-800 as 0.66
Consideration of the viscous effects is where the black art comes in, and where some of the classic assumptions start to go awry.
The paper mentions that “e” falls as wing t/c is increased, which is right because the flow over the rear part of the upper surface starts to separate earlier (starting from the TE) as CL increases. At the other end of the airfoil there is an area of very low pressure where the flow moves around from the stagnation point on the lower surface to go over the top (leading edge suction in the vernacular). If you can arrange to have a significant amount of forward facing surface in this zone you will get a “thrust” component of the aerodynamic force that lowers the effective drag coefficient. Airfoils with a large nose radius can exploit this, this wings with sharp leading edges cannot. If you examine the profile of modern supercritical wing sections you will find that they have large nose radii despite the fact that they are intended to fly at highish Mach numbers. Balancing the effects at either end of the airfoil and their influence on the structure is where the art comes in.
However, the fact that they are “supercritical” sections means that they have significant areas of supersonic flow over the upper surface, so the drag is not simply vortex plus some viscous effects – there is also wave drag to consider.
A modern airliner wing is designed using CFD methods to arrange camber, twist and thickness distributions to minimise drag at some defined design condition of Mach number and lift coefficient. That means balancing vortex drag, viscous drag and wave drag to get the best result. One consequence of this is that the sectional drag is not a minimum at zero lift coefficient. The old classic: Cd = Cdo + k*CL^2/pi.A is not accurate; instead one has to use Cd= Cdo + k*(CL-CLo)^2/pi.A. This in turn means that one cannot predict lift dependent drag using a simple value of “e” (or “k”); one must know the associated value of CLo as well.
Worth mentioning perhaps is that the intention is to design the best wing, not the best aerodynamic design. This means that it may be (usually is) better to back off a little from the ideal lift distribution to bring the centre of pressure further inboard which reduces the wing bending moment and weight.
The paper goes on to attempt a correction for Mach number effects, but if you calculate this using their expressions you will find that the span efficiency falls to zero at 0.825M for any airplane – not very useful if your design is meant to cruise at 0.84M!
BTW pugilistic animus, following up on your thoughts:
For some of their (the authors of the above paper) compressibility issues, they would probably have gotten a better data fit if they used the Karman-Tsien rather than the Prandtl-Glauert, just my opinion though..
I tried Karman-Tsien and although Prandtll-Glauert was hopeless as a basis for Mach number correction, K-T was worse!
All in all, you can have your choice of methods to estimate “e” for Mach numbers up to about 0.7 (the Douglas Aeroplane Company method described in Dick Shevell’s book is good and easy to use) but above that, and for modern designs, you need a more sophisticated technique.
All of which is just a complicated way of agreeing with jimmyg:
You would need more data and a specific computer design program to generate better conclusions.
Try and contact Boeing or Airbus design engineering departments for more information.
And the best of luck 