Understanding Induced Drag

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Alan_VA

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Apr 17, 2011
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Bristow, VA
I'm trying to understand the drag and thrust curves of the plane I'm building. Profile/form drag - no problem. My sticking point right now is induced drag. All of the graphics on the topic show induced drag decreasing rapidly with increasing speed. But the equations I find on the subject, as on page 3 of http://www.lancair.net/archives/Drag_Reduction_Part_1.pdf show induced drag as being proportional to the square of speed (i.e., increasing as speed increases). What am I missing ?

Thanks,
Alan
 

Victor Bravo

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Are you sure you are not seeing a graph of parasite drag or form drag instead of induced? The faster you go the less lift the wing needs to make, and the less lift it makes the less induced drag there is. At least that's my numbskull understanding of it.
 

wsimpso1

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The document is really clear. I suggest that you re-read Background section. Pages 2 and 3 are essential pieces and the plots on pages 5 & 6 are the result of that math.

I can do some math here if math helps you at all.

L = rho/2*V^2*Cl*S

where L is lift, rho is density of the fluid, V is velocity of the fluid, Cl is coefficient of lift, and S is foil plan form area. We can solve for Cl required at any given airspeed and Lift:

Cl = 2*L/(rho*V^2*S)

If L and S and rho stay the same, but we change V, the Cl changes. Double the airspeed and the Cl needed drops to a quarter of what it was...

Cdi = 0.318 * Cl^2/AR

Cdi is induced drag coefficient, and AR is Aspect Ratio. 0.318 has come out of the analysis of wing lifting theory, and, as nearly as we can tell from a lot of wind tunnel work, it fits really well.

Hmm, let's stick the earlier form of Cl in the Cdi equation:

Cdi = 0.202*(L/(rho*S))^2*(1/V^4)

so, all other stuff being the same, Cdi goes with 1/V^4. Yep, Induced drag coefficient goes down rapidly with speed.

Now to get Induced Drag from all this

Di = rho/2*V^2*Cdi*S

Di = rho/2*V^2*S*0.202*(L/(rho*S))^2*(1/V^4)

Simplifying terms

Di = 0.101*L^2/(rho*S*V^2)

So, if lift and rho and S remain constant, then induced drag goes with the inverse of V^2, just like it shows on the plots on pages 5 & 6.

Low AR wings thus pay little drag penalty at speeds substantially above the min total drag point. The ultimate on this is an airplane designed for speed between two points at low altitudes. Little time in climb, lots of airspeed, who needs aspect ratio. Some folks will say "oh, IF1 airplanes", but I say they spend a lot of time in a steeply banked turn on the course, so lift required is more like 2 times airplane weight, so, induced drag is bigger for them then you might think. On the other hand airplanes that climb at airspeeds near their min total drag point can benefit substantially if they climb to high altitudes or otherwise spend large amounts of their flight time at low indicated speeds. Think airliners and sailplanes here... Yes many airliners are big fractions of Mach, but air density (rho) is much lower at their preferred altitudes, and thus their q (rho/2*V^2) and indicated airspeed that goes with it are much lower than you might think. So, airliners and sailplanes do spend much of their time in places where induced drag matters a bunch.

Billski
 

Alan_VA

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Apr 17, 2011
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Bristow, VA
Cdi = 0.202*(L/(rho*S))^2*(1/V^4) is the key.

I was trying to figure out the induced drag for a steady state condition. And in that case, as everyone is taught in ground school Lift = Weight, and in the equation for Di, Cl is not a constant. So, I should insert the plane's weight into the equation
Di = rho/2*V^2*S*0.202*(L/(rho*S))^2*(1/V^4).

Thanks for helping me unwrap myself from around the equations.

Alan
 

cblink.007

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Jul 7, 2014
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57
Location
Texas, USA
I'm trying to understand the drag and thrust curves of the plane I'm building. Profile/form drag - no problem. My sticking point right now is induced drag. All of the graphics on the topic show induced drag decreasing rapidly with increasing speed. But the equations I find on the subject, as on page 3 of http://www.lancair.net/archives/Drag_Reduction_Part_1.pdf show induced drag as being proportional to the square of speed (i.e., increasing as speed increases). What am I missing ?

Thanks,
Alan
Alan, I strongly recommend the book "Fluid Dynamic Drag", by Hoerner. It is a great and easy read...not requiring a Calculus for Dummies book to follow along. As an engineer, I had no idea this text existed until I watched the Mike Arnold AR-5 videos (Tapes 1 & 3) long after I left academia! One of the best investments I ever made! You can order it here:


I've used the book extensively in optimizing the design of my canopy and engine cowling, and anxiously looking forward to comparing the actual results against simulations!
 

thjakits

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Jul 3, 2014
Messages
48
Location
Panama, Rep. of Panama
...okay - I just checked and I have TWO Hoerner books on file!
Fluid Dynamic Drag and Fluid Dynamic Lift!
I will try to get them on Dropbox ...for a while...

thjakits
okay - DONE - I have both books on my dropbox account - send me PM or email and I share the link.
thjakits@gmail.com

PS: IF you have other Hoerner books, I certainly would like a copy! ....or any other interesting books!
 
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