# What is the formula for induced drag?

I am looking to create one of these graphs for a simple model I tested in a wind tunnel.

I have managed to define the lift coefficient and thus I've been able to create the parasitic drag line. The problem is that I am having issues with finding the correct formula to use for the induced drag. I have been looking a while now and the only formula's I have found had the induced drag increase with a higher velocity.

the problem i'm running into is that the formula i see around is

$D_i = \frac{L^2}{\frac{1}{2}\rho V^2 \pi b^2 \epsilon}$

this formula has Lift squared above the divider and lift itself is dependant on velocity so whenever I use that formula the graph I get is of one that increases with speed instead of decreases.

What I'm asking for is the correct formula to use here

• you asked this already yesterday and we pointed you at the solution. what do you hope to accomplish by deleting the old post and posting it again? (apart from being banned by the system) – Federico Mar 6 '17 at 7:24
• I'm just totally "winging" it here, but isn't Lift also dependent on AoA, and constant for straight-and-level flight, regardless of speed? ie as V increases, you decrease AoA to keep L constant and the airplane in level flight? – Daniel Mar 6 '17 at 7:34
• also, you edited after it was closed, it could have been reopened, you just had to wait a little, the system does not like deleted questions. – Federico Mar 6 '17 at 7:34
• Additionally, parasite drag doesn't depend on lift?? I may be totally off base, but it sounds like you may be super confused about terminology and baseline assumptions. – Daniel Mar 6 '17 at 7:37
• You may also read this: What is wing inductance? to get a definition of the coefficient of induced drag. You can use this coefficient the same way than $C_L$ for the lift. – mins Mar 6 '17 at 8:03

It seem your graph of induced lift is not decreasing because you assume that the lift increases with velocity. This is generally not the case.

Typically, a drag vs velocity graph is made for unaccelerated level flight. Under these conditions the lift is equal to the weight of the aircraft.

$L = W = \frac{1}{2}\rho V^2 c_L S$

From this we can obtain the lift coefficient as a function of velocity:

$c_L = \frac{W }{\frac{1}{2} \rho V^2 S}$

The drag of the aircraft is the sum of the parasite drag and the induced drag:

$D = D_p + D_i$

With the parasite drag:

$D_p = c_{D,0}\frac{1}{2} \rho V^2 S$

And the induced drag:

$D_i = \frac{1}{2}\rho V^2 S \frac{c_L^2}{\pi AR \epsilon} = \frac{W^2}{\frac{1}{2}\rho V^2 S \pi AR \epsilon} = \frac{W^2}{\frac{1}{2}\rho V^2 \pi b^2 \epsilon}$

It is important to understand that this only holds when the lift is equal to the weight of the aircraft (e.g. straight & level flight)

Nomenclature:
$L\:\:\:\:\:$ lift
$W\:\:\:$ aircraft's weight
$\rho\:\:\:\:\:$ air density
$V\:\:\:\:$ velocity
$S\:\:\:\:$ wing surface area
$c_L \:\:\:$ lift coefficient
$c_{D0} \:$ zero-lift drag coefficient
$\pi \:\:\:\:\:$ 3.14159$\dots$
$AR \:\:$ aspect ratio of the wing
$\epsilon \:\:\:\:\:\:$ the wing's Oswald factor
$b \:\:\:\:\:\:$ wing span

• Thanks a lot for the answer, this should solve my problems and you probably saved my ass <3. – Patrick Mar 6 '17 at 10:50
• Dear @DeltaLima, I quoted this From this we can obtain the lift coefficient as a function of velocity, then the equation. CL = W/(0.5rho*V^2*S). I red many times saying that way. My question, should not the coefficient of lift is calculated in the wind tunnel test? How can we calculate coefficient of lift while we still don't have Lift force? The Coefficient of lift itself we require is to calculate the lift, not vice versa. – AirCraft Lover Dec 19 '18 at 4:46
• @AirCraftLover In this case, we know the lift force. When the aircraft is in horizontal, non-accelerated flight (i.e. no turns); the lift force is equal to the weight of the aircraft. So the lift coefficient follows from the weight, speed, air density and wing surface area. – DeltaLima Dec 19 '18 at 13:48