# Why doesn’t the minimum drag point meet the point at the best Cl/CD polar?

On the first picture, it shows that the minimum drag point is exactly the same as the maximum L/D ratio where it also supposes to be at The point as it shows on the second picture. However, according to the third picture, it doesn’t look like the minimum drag point is at the maximum L/D on the second picture. Why does this happen?

• simple answer: because lift is not constant and also increases with aoa Dec 5 '17 at 16:34

You need to distinguish between coefficients and the forces.

Coefficients are dimensionless numbers which are used to compare flow phenomena at different speeds, scales or atmospheric conditions. The force coefficients are the forces normalised by a reference area (to remove size effects), speed squared divided by two (to remove speed effects) and air density (to remove atmospheric effects). Mathematically speaking: $$c_X = \frac{X}{\rho\cdot\frac{v^2}{2}\cdot S_{ref}}$$ with $X$ being the force, $\rho$ being density, $v$ being air speed and $S_{ref}$ the reference area.

The lowest drag coefficient is at rather a low lift coefficient. In order to create enough lift to carry its own weight, the aircraft has to fly rather fast there. This high dynamic pressure at this polar point means that the total drag force is higher than at a polar point slightly above it.

At the point of best L/D the ratio between both forces reaches a minimum, so actually the lowest drag for a given lift is at this point.

Only if you disregard lift (say, in a vertical dive where the goal is to reach the highest speed) will the point of the lowest drag coefficient also be the point of lowest drag (at the same dynamic pressure). To generalize: The minimum drag point shifts between the maximum L/D polar point and the minimum $c_D$ polar point with the cosine of the flight path angle. Between 90° and 270° flight path angle (inverted flight) it shifts between the minimum $c_D$ point and the minimum L/D polar point (where lift is negative).

• Sorry, I couldn’t get you. Do you mean if it is a level flight, then the minimum drag point (only for level flight) is also the maximum L/D and CL/CD while in any other condition, the minimum drag point is at the point of minimum CD which was decided when the aero foil was made? Dec 6 '17 at 3:41
• I guess the 4.3 picture makes this confusing when it says ‘Minimum Drag’ and points at the blue 2deg. It should say : Minimum Drag Coeficient. Actual drag Force at 2deg is higher than at 6 deg ( for a given unaccelerated flight) Dec 6 '17 at 7:47
• @RobertLo: Yes in case of level flight, but for climb and descent it is still very near the maximum L/D point since sufficient lift needs to be created. Only in the vertical dive is no lift required, and only then will minimum drag be where minimum $c_D$ is. The minimum drag point shifts between optimum L/D and minimum $c_D$ with the cosine of the flight path angle. Dec 6 '17 at 8:03

"Minimum drag" at 2deg in image 4.3 is just minimum Cd (coefficient not force). It is the most streamlined position and you will generate the least amount of drag here, for a given speed. You won't generate much lift though, if any. In fact, if you want to fly level (ie. at 1g) you will have to fly so fast that the drag will still become expensively large. Larger that if you were to fly at a more optimum angle of 5deg.

If Cl was constant over aoa, the best Cl/Cd would indeed occur at minimum Cd. ( a fixed quantity divided by the smallest ammount), alas Cl and Cd both increase with increasing aoa, reaching a best Cl/Cd ratio (in your example) at 5deg in image 4.7.

After that point, the Cd will increase a lot more than Cl, thus making the ratio worse.

• What I confuse is that on the third picture,why is the minimum drag point not at maximum L/D and CL/CD? Dec 6 '17 at 3:43
• forget that this is about aerodynamics, just focus on the graph: on any polar graph like the left image 4.3 , the highest ratio Y / X value will occur on the tangent from origin (ie 5 deg in your image, see 4.7) not on the apex of MinX ( at 2 deg in your image).This is true for any polar graph; It’s math, not aerodynamics Dec 6 '17 at 7:35
• Re "Minimum drag is just that, minimum Cd." -- actually, not. For a different view see aviation.stackexchange.com/a/86327/34686. Apr 3 at 19:54
• Yeah, should have put that in quotes as I was referring to the label on the graph. I tried my best at rephrasing it Apr 3 at 20:26

On the first picture, it shows that the minimum drag point is exactly the same as the maximum L/D ratio

This makes sense, at least in the context of linear constant-altitude flight, because Lift must equal Weight and therefore is constant, so if we maximize L/D, we also must minimize Drag. (Also, if we're maximizing L/D, we're simultaneously maximizing Cl/Cd.)

But since airspeed is not fixed as angle-of-attack varies, it does not follow that by minimizing Drag, we are also minimizing Cd-- and that is the key to solving your quandry.

Despite the misleading label on the third graph, the flagged point is actually not the point of minimum Drag. Rather, it is the point where the drag coefficient is minimized. In the context of linear constant-altitude flight, where L = W but airspeed is free to vary as angle-of-attack varies, the point of minimum drag coefficient is completely different from the point of minimum Drag.