# Is (L/D)max achieved at a specific speed or AOA?

I'm little bit confused about the (L/D)max.

So I've read this blog: https://joeclarksblog.com/?p=4155 and the author shows this graph:

So if I understand it correctly the (L/D)max is at the intersection of parasitical drag and lift-induced drag. Thus, at the point which has the minimum total amount of drag. Which is achieved at a specific speed.

Skybrary states the following:

The maximum lift/drag ratio occurs at one specific CL (Lift Coefficient) and AOA (Angle of Attack (AOA))

https://www.skybrary.aero/index.php/AP4ATCO_-_Lift/Drag_Ratio,_Forces_Interaction_and_Use

Here is is the lift coefficient:

The graph shows that the airfoil is the most efficient at a specific AOA. Thus in the ranges between 15° and 18°.

sciencedirect.com states the following:

The L/D is not constant for a given wing, however, but changes with the angle of attack. Researchers thus normally use the maximum lift-to-drag ratio, L/Dmax, to characterize a given wing.

https://www.sciencedirect.com/topics/earth-and-planetary-sciences/angle-of-attack

I'm confused because one source tells me that (L/D)max is achieved at a specific speed. The other source tells me that (L/D)max is influenced by the AOA.

Can you clarify this thought twist?

• "Researchers thus normally use the maximum lift-to-drag ratio, L/Dmax, to characterize a given wing." Researches do this, pilots also look at "hw abrut is the stall" (fundamental for a nice flare), "how does this profile behaves when dirty/wet?", "what's the L/D when flying not at the exact best L/d AOA?", "How steep is the alpha-momentum curve?" – Caterpillaraoz Jul 5 '19 at 17:43
• You're confusing maximum lift with maximum lift to drag ratio. Maximum lift is the largest lift force that an aerodynamic surface can provide at a specific speed (more accurately, dynamic pressure). Lift to drag ratio is a measure of how efficient the surface is at generating lift at the expense of drag (for an airplane, drag means more propulsion required). – JZYL Jul 6 '19 at 3:00

Induced drag is dependent on Angle of Attack while parasitic drag is dependent on airspeed. However there is a correlation between airspeed and angle of attack so the L/D graphs are usually given using airspeed. The airspeed is also more useful to pilots than angle of attack as most airplanes don't have a AoA sensor.

So really both sources are correct here but from a technical standpoint L/D is mostly dependent on Angle of Attack although parasitic drag is completely dependent on airspeed.

• Can you explain the correlation between airspeed and AOA? – Julian Jul 9 '19 at 14:53
• @Julian: This website explains it pretty well: av8n.com/how/htm/aoa.html#sec-ias-aoa – DLH Jul 9 '19 at 15:35
• I chose this answer as the best answer because the link clarifies the thought twist I had. – Julian Jul 15 '19 at 13:10
• The induced drag also depends on airspeed, right? Don't both induced drag and parasitic drag vary as the square of the airspeed? (And then, in straight and level flight, angle of attack varies with airspeed, which makes everything confusing.) – Terran Swett Aug 5 '19 at 21:01

I'm confused because one source tells me that (L/D)max is achieved at a specific speed. The other source tells me that (L/D)max is influenced by the AOA.

The speed is only correct for a specific wing loading and straight flight at 1g. Only then will the optimum lift coefficient be reached in level flight at that speed.

The angle of attack version is the more generic and better way. Actually, optimum L/D is achieved at a specific lift coefficient, which is closely related to the angle of attack via the lift curve slope.

The graph shows that the airfoil is the most efficient at a specific AOA. Thus in the ranges between 15° and 18°.

No, it only achieves the highest lift coefficient in that range. This is right before stall and certainly not the position of optimum L/D. Drag will have grown disproportionally over the previous angle of attack range. Without a drag polar it is impossible to say where that specific airfoil has its best L/D.

• wing loading and unaccelerated flight – Caterpillaraoz Jul 12 '19 at 9:34

"Eta is geometric".

L/D is referred to the angle of attack.

This is way easier understood looking at gliders: a glider with e.g. a best L/D of 50 will, in calm air and when gliding at the AOA where such L/D is obtained, advance 50 miles for every mile of altitude lost.

The speed at which such gliding flight will happen is depending on the loading: the heavier it is, the faster the speed. For this specific reason, water ballast are common in gliders: if the pilots wants to fly faster he can just "point the nose down" and follow a steeper path but in doing this the wing will be working at an AOA where the L/D is gets farther from the optimal one whilst adding mass the the aircraft he will be flying faster without having the wing working at AOAs with lower L/D. Same concept applies to power aircraft but it's harder to mentally visualize hence the choice of a glider for this answer.

Often flight envelope characteristics such as stall speed best L/D speed etc are expressed well as.... speed for pilot's convenience and making a few assumptions (MTOW load and level flight for stall speed, unballasted weight of a glider for best L/D) but in reality there are no stall speed nor best L/D speed: there are stall AOA and best L/D AOA)

Let's look at a glider's polar and L/D, referred to the L/D measured on the whole aircraft and not only on a wind tested in a wind tunnel (but similar results would be expected thee):

L/D

Polar

images from second answer of this interesting question

As you can see the curves have the same shape but are shifted depending on the wing loading, especially in the first graph can be seen that the same best L/D of about 47 is obtained at a higher speed the higher the wing loading (and thus the total weight since they all refer to the same aircraft), that's the speed where the wing flying at its optimal AOA gives a lifting force equalling the total weight thus making for an unaccelerated descent.

As for what is causing the wing's polar to have that precise shape: yes, the various forms of drag enter into account and this would be an interesting question you should ask: why polars have their characteristic shape?

• What is the vertical axis of the second plot? – JZYL Jul 6 '19 at 3:07
• @jimmy: sink speed in m/s – Caterpillaraoz Jul 6 '19 at 13:06
• I would recommend adding explanation on why the second plot matters (for example, best glide/climb speed coincides with L/Dmax for that weight, and why). I also don't think the second plot could be called a polar. Good data. – JZYL Jul 7 '19 at 4:32
• @Jimmy -- "A polar curve is a graph which contrasts the sink rate of an aircraft (typically a glider) with its horizontal speed." en.wikipedia.org/wiki/Polar_curve_(aerodynamics) – quiet flyer Jul 7 '19 at 16:39
• @quietflyer In airplane design world, polar means something a little different, although it would share the exact same characteristic as the plot shown in the answer. I will defer the nomenclature to those of you who are more familiar. – JZYL Jul 7 '19 at 16:43

For any aerodynamic surface, lift and drag are: $$L=qSC_L$$ $$D=qSC_D$$

where $$q$$ is dynamic pressure and $$S$$ is reference area.

Assuming a thin, high aspect ratio, non-swept lifting surface at low Mach, like the wing of a glider, we can divide its drag coefficient into two components, as the Wikipedia entry described: parasitic drag, $$C_{D_0}$$, which we can assume to be a constant, and induced drag, $$C_{D_i}=KC_L^2$$, where $$K$$ is a constant dependent on the shape of the surface (aspect ratio, taper, etc.).

At $$L/D_{max}$$,

$$\frac{d(L/D)}{dV}=\frac{d(C_L/C_D)}{dV}$$ $$=\frac{1}{C_{D_0}+KC_L^2}\frac{dC_L}{dV}-\frac{2KC_L^2}{(C_{D_0}+KC_L^2)^2}\frac{dC_L}{dV}$$ $$=1-\frac{2KC_L^2}{C_{D_0}+KC_L^2}=0$$

So, $$C_{L_{L/D_{max}}}=\sqrt{\frac{C_{D_0}}{K}}$$

There is a unique AOA on the lift coefficient curve, like the one you presented, that corresponds to this $$C_L$$ (pre-stall), so the statement:

The maximum lift/drag ratio occurs at one specific CL (Lift Coefficient) and AOA (Angle of Attack (AOA))

is indeed correct and there is a unique $$L/D_{max}$$ for a given lifting surface (see the first plot presented by Caterpillaraoz, notice how remarkably similar the maximum $$L/D$$ is for different wing loadings).

This seems to directly contradict with your second source:

The L/D is not constant for a given wing, however, but changes with the angle of attack. Researchers thus normally use the maximum lift-to-drag ratio, L/Dmax, to characterize a given wing.

Notice I was only talking about lifting surfaces. For a whole aircraft, there is also trim drag, which is AOA dependent. And if we deviate from our assumption of thin, high aspect ratio, non-swept, low Mach, then $$C_{D_0}$$ is no longer a constant and becomes a function of Reynolds number, AOA, AOS and Mach. For high speed airplanes, or airplanes with flaps/slats, our simple model breaks down, and $$L/D_{max}$$ is indeed, AOA dependent.

• +1 for specifying "Assuming a thin, high aspect ratio, non-swept lifting surface at low Mach". Simplifications are cool and telling the conditions where such simplifications are valid is even cooler. – Caterpillaraoz Jul 6 '19 at 13:08

"So if I understand it correctly the (L/D) max is at the intersection of parasitical drag and lift-induced drag. Thus, at the point which has the minimum total amount of drag. Which is achieved at a specific speed"

Actually, speaking in the most general terms, neither L/D max nor maximum Lift nor minimum Drag are achieved at a specific speed. The graph tying the L/D ratio to a specific airspeed is ASSUMING one specific weight, and also is ASSUMING 1-G flight. In many cases this is a perfectly reasonable assumption, but not always. Fundamentally speaking, L/D ratio and lift coefficient and drag coefficient are all functions of angle-of-attack, NOT airspeed.

However, in 1-G flight at a given weight, there's a one-to-one correlation between angle-of-attack and airspeed. Once you understand this, your confusion will vanish.

If we DO assume 1-G flight at one specific weight, then it's fine to say that variables like L/D ratio, lift coefficient, and drag coefficient are functions of angle-of-attack, or of airspeed, whichever we prefer. When presenting the information to a pilot who will be interested in the aircraft's performance in 1-G flight at some given known or assumed weight, then obviously it makes more sense to present the information in terms of airspeed, because the pilot has an airspeed indicator but not an angle-of-attack indicator. This would pertain more to L/D ratio than to lift coefficient or drag coefficient, because the L/D ratio is tied directly to something the pilot is actually interested in optimizing (the glide ratio), while the lift coefficient and drag coefficient aren't tied directly to any one parameter that the pilot cares about. (The lift coefficient is NOT a measure of the "efficiency" of the wing, as the question suggests.) Hence the reason you are seeing the L/D curve presented with airspeed along the X axis, rather than with angle-of-attack along the X axis. It's a graph that's meant to tell the pilot something he can actually use in the cockpit, while flying in the 1-g condition at some known or assumed weight.

Here's another bit of trivia to muddy the waters:

If the L/D graph is drawn for horizontal powered flight, the point of max L/D will occur exactly at the point of minimum Drag (not the same as minimum drag coefficient), because in horizontal powered flight Lift = Weight, assuming no complications from a tilted thrust line, and Lift is therefore constant.

If the L/D graph is drawn for gliding flight, the point of max L/D will occur ALMOST at the point of minimum Drag, but not exactly, because in gliding flight Lift is not constant. It is slightly less than Weight, and more so at poor glide ratios than at large glide ratios. The difference will likely be too small to notice though-- for all practical purposes the gliding L/D curve and the horizontal L/D curve are interchangeable, except in aircraft with very poor glide ratios. Therefore for all practical purposes, the L/D curve can also be viewed as a Drag curve (not the same as a drag coefficient curve) in gliding flight as well as in horizontal flight, except in aircraft with very poor glide ratios.

In both cases (gliding flight or horizontal flight) the point of max L/D will occur at the point of max (lift coefficient / drag coefficient), which will occur NEITHER at the point of max lift coefficient nor at the point of minimum drag coefficient.

(By the way, all the above statements are true regardless of whether we are putting angle-of-attack, or airspeed, along the x axis of our graph.)

Always keep in mind that Drag is not the same as drag coefficient, and Lift is not the same as lift coefficient. However the Lift/ Drag ratio is always exactly equal to the ratio of (lift coefficient / drag coefficient).

• -1: Max L/D has nothing to do with minimum drag (for low speed, min drag would occur at zero lift), or whether it's powered or unpowered, unless your thrust line is offset from your waterline CG, which would introduce some trim drag. – JZYL Jul 7 '19 at 16:25
• @Jimmy -- the whole point of the "polar curve" (airspeed versus L/D) well-beloved of glider pilots is that you are assuming 1-G steady-state flight. Otherwise, there would be no way to assign an airspeed to any given angle-of-attack value, and all you could do would be to create a curve showing angle-of-attack versus L/D. And obviously, zero lift is not compatible with any form of steady-state flight except a vertical dive. A steady-state vertical dive may well yield the lowest drag COEFFICIENT, but it doesn't yield the lowest Drag VALUE, because the airspeed is very high, and the (ctd) – quiet flyer Jul 7 '19 at 16:32
• @Jimmy-- continuing-- A steady-state vertical dive may well yield the lowest drag COEFFICIENT, but it doesn't yield the lowest Drag VALUE, because the airspeed is very high, and the Drag force in pounds or whatever is related to both drag coefficient and AIRSPEED SQUARED. So, yes, absolutely max L/D has a great deal to do with minimum Drag, as I explained in the answer. – quiet flyer Jul 7 '19 at 16:34
• @Jimmy-- actually I've slightly mis-spoken. The graph of airspeed versus L/D is normally not called the "polar". That word is usually used for the graph of airspeed versus sink rate. Sorry about that. "A polar curve is a graph which contrasts the sink rate of an aircraft (typically a glider) with its horizontal speed." en.wikipedia.org/wiki/Polar_curve_(aerodynamics) – quiet flyer Jul 7 '19 at 16:38
• @Jimmy -- so to sum up, in steady-state horizontal flight, assuming no complications from a tilted thrust line, minimum Drag (as a force in pounds or Newtons or whatever) IS obtained at the point of the max L/D ratio. – quiet flyer Jul 7 '19 at 16:42