Bjorn’s Corner: Aircraft drag reduction, Part 13

By Bjorn Fehrm

January 19, 2018, ©. Leeham Co: In the last Corner, we described how the boundary layer theory lead to the understanding of Friction drag for aircraft. The mechanisms behind Induced drag was understood about the same time.

Once again Prandtl was involved, but it was an English person who first postulated the physical root of induced drag, Fredrick Lanchester.

Figure 1. Focke-Wulf Condor, a high aspect ratio aircraft from the 1930s. Source: Wikipedia.

Induced drag

Lanchester observed how the air flowed around a flat plate, Figure 2. He saw the air turning up ahead of the plate, then down on top and bottom sides and finally straighten out.

Figure 2. Airflow around a flat plate can be seen as a combination of the stream and and a circulation flow. Source: John D. Anderson, History of Aerodynamics.

From a theoretical point, he argues in his 1907 book, Aerodynamics, this can be thought of as a linear flow combined with a circular flow. On the topside, the flows augment each other; the air flows faster in a downward curve and the pressure drops (according to Bernoulli’s equation, which was accepted at the time).

On the bottom side of the plate the flows are opposed and the air slows and turns down. At the trailing edge, the two streams meet and at some length behind the plate the circular motion is finished, the air follows the free stream again.

This basic idea was used by Zhukovsky and Kutta to independently formulate the circulation theory of lift (the plate’s contribution to the free stream is adding the circulating motion, by it creating lift).

Lanchester also understood what happened at the tips of the plate/wing. He observed tip vortices and understood these were created by a flow which went from the higher pressure on the bottom of the wing to the lower pressure on the top of the wing.

This circular motion combined with the free stream created “vortex trunks”, shedding from the wing tips, Figure 3.

Figure 3. Lanchester’s “Vortex trunks” as described in his 1907 book Aerodynamics. Source: Google images.

Lanchester didn’t formulate any theory how to estimate the effect of these circulations. It was left to Kutta-Zhukovsky for their infinite wing circulation theory and to Prandtl for the finite wing circulation theory.

Induced drag is not a tip phenomenon.

It’s important to understand, induced drag is coming from the global spanwise change of direction of the air around a wing, Figure 4.

Figure 4. Spanwise flow around and aircraft. Source: Leeham Co.

The global spanwise circulation around the wingtips is the source of induced drag, not the very visible wingtip vortices. The air behind a wing creating lift is forced down into a giant vortex sheet (Figure 5) which continuous down behind the aircraft.

Figure 5. Vortex sheet behind aircraft wing. Source: Wing tip devices. McLean.

In air where the low pressure of the high-speed air causes condensation or at cloud tops, one can see this gigantic vortex sheet left by the aircraft, Figure 6.

Figure 5. Vortex sheet behind aircraft. Source: Google images.

The importance of induced drag.

Induced drag is highest at low speed, friction drag at high speed, Figure 7. A high aspect ratio wing is therefore important for take-off and initial climb performance.

Figure 7. Drag at different speeds. Source: Leeham Co.

Here how the different drag components affect a typical airliner flight:

  • At the start of take-off, induced and friction drag is zero.
  • When the aircraft accelerates down the runway, induced drag is nil as there is no lift produced. Friction drag is low as the speed is low.
  • When the aircraft rotates, induced drag shoots up as lift is produced by giving the wing an angle of attack (the angle alfa) against the air. Induced drag is now 80% or more of total drag. Friction drag is low as the speed is still comparatively low. There is some Form drag beside the Friction drag, as the high lift devices (Slats, flaps) create local separations.
  • The engine thrust is now fighting induced drag. Should one engine go inoperative (OEI), the thrust of the remaining engine must guarantee a climb angle of 2.4m for every 100m of flight (minimal regulatory climb gradient for a twin-engine airliner).
  • During the climb, the speed is gradually increased, first to 250kts, which is the maximum before passing 10,000ft inside controlled airspace, then gradually to about 0.2M below the cruise speed for the final climb to initial cruise altitude. By now induced drag is less than the friction drag.
  • Friction drag is the dominant drag component during cruise and descent. Induced drag is reduced to around 30-45% of total drag in these phases.

As the aircraft increases speed toward cruise Mach, transonic drag sets in. This we will cover in the next Corner.

20 Comments on “Bjorn’s Corner: Aircraft drag reduction, Part 13

  1. Bjorn,

    Well said about the vortex sheet and induced drag etc.
    One thing to point out:

    The induced drag theory is called Prandtl’s Lifting Line Theory, because it replaces the wing by a bound vortex with varying circulation along the span, which in turn creates the vortex sheet, which winds up downstream to form the two trailing vortices. The Lifting line theory has been further extended to modern lifting surface theory, which makes use of computers to more accurately calculate various quantities. So far so good. But …

    There has been a vigorous debate about who gets the credit for the finite wing aerodynamics. It was part of the British-German clash for proper credit for major advances in Aerodynamics before the Second World War. Prandtl of course gets full and sole credit for his Nobel-quality boundary layer theory. But finite wing aerodynamics, there has been a debate. It is likely Prandtl knew about how the flow behaves on a finite wing and he was not initially aware of Lanchester’s work, until Lanchester gave a talk at Gottingen. What happened there is not clear, because of the language barrier – it is not clear whether Prandtl knew English well enough to understand Lanchester. Prandtl did come up with his lifting line theory a bit later, another breakthrough in aerodynamics and Lanchester did NOT. Perhaps he lacked the math skills of Prandtl, who was essentially an applied mathematician. So often people refer to the theory as Prandtl-Lanchester theory (or if British, Lanchester-Prandtl theory). But it is more than likely, given the genius Prandtl was, he did not need Lanchester to tell him about how flow behaved on a finite wing. And he was the ONE, who put abstract ideas into a mathematical model that was very helpful in understanding induced drag (often called penalty for producing lift on a finite wing) and its role in aircraft drag behavior.

    Just an interesting tidbit of history …

    • To suggest that Lanchesters earlier work published in English ( Aerodynamics 1906 and Aerodonetics 1908) but developed earlier and published in German in 1909 ‘wasnt understood’ by Prandtl is very shabby.
      The visits by Lanchester to Gottingen in 1908 and 1909 as an invited lecturer would have been expressly to discuss his work ( and likely his work had been translated for the use of the Aerodynamischer Versuchsanstalt staff and students).
      As Otto Foppl, a student at the time and later Prandtls assistant later wrote that Prandtl had trouble with Lanchesters uncoventional mathematics and terminology and of course didnt understand English and Lanchester didnt understand German. But Carl Runge , whos mother was English and could speak fluently was able to converse with Lanchester ( Runge was professor of Applied mathematics at Gottingen), so there was
      certainly understanding of the importance of Lanchesters work in Gottingen

      As indeed Prandtl said to the Royal Aeronautical Society ‘Wilbur Smith memorial lecture’ in 1927, its quite rightly called the Lanchester -Prandtl theory and discussed how the Germans had a far better understanding of his work than was the case in England at the time. Of course he still maintained that his ideas occurred to him independently but he didnt work on a useful theory in this area till 1911.
      The ‘ History of Aerodynamics’ by John Anderson

      • “Shabby” – how British indeed! Look, Prandtl, the genius he was, had no need to “steal” some one else’s idea. If he said, he thought of it independently, he was telling the truth. Whether he understood Lanchester’s talk (that is what I meant) because of language issues, is not clear. But it is noteworthy that it was Prandtl, who worked out the equations for the induced drag and not Lanchester, who had considerable lead on him. A lot of literature refers to the lifting line theory as just Prandtl’s lifting line theory! No need to shoot the messenger!

        • Lanchester was an engineer and inventor better known in the automotive field. Not a theoretician like Prandtl who didnt have a patent to his name.
          Its clearly untrue , that some one of his stature, didnt understand the essentials of what a guest lecturer ( asked to Gottingen x2) was talking about. He just had to refer to Lanchesters published work or discuss it with his colleagues
          and a peer like Runge who did.
          Prandtl was clearly the person to develop the theory , but the timelines dont support anyone saying he developed his theory without any knowledge of Lanchesters work, whose prior publication means history should credit both.

          Betz’s law is an example of original work in 1919,which followed on from Lanchester work in 1915, again ignoring prior publication- although WW1 would have limited any academic discussions.
          It goes to the truism attributed to Mark Twain:
          It takes a thousand men to invent a telegraph, or a steam engine, or a phonograph, or a photograph, or a telephone or any other important thing—and the last man gets the credit and we forget the others.

      • Starting from an infinite wing a fitting conformal mapping should provide for a best fit transformation.
        my guess is the “receipt” results in a complex but simplifyable to elliptical planform.
        Conformal mapping as a solution tool was introduced by Riemann afair.

  2. You have stated: “When the aircraft accelerates down the runway, induced drag is nil as there is no lift produced.”

    A slight correction may be required here.
    Lift is produced on a wing any time airflow occurs over the wing, right?

    • Yes, the wing profile gives lift at 0 alfa angle. But we focus on the main issues, the induced drag from a runway rolling aircraft is negligible, especially if its an A330 🙂 (nose down ground rolling angle).

      • Would that not create reverse lift (not very efficient) and have an inducted drag of its own until they rotated and got to zero?

        When does Beronooilie get his day in the flow?

        And do we get into upside down flying and if I remember right, super sonic jets that had the wing curve on the bottom and flat on top ?

  3. Would a u-shaped wing profile (instead of the flat wings on figure 4) reduce the induced drag and help to minimize the vortex?

    Is that why newer Boeing planes have “flexible” wings that bend quite a bit upwards?

    • Induce drag comes from the creation of lift. It has to do with the strength of the downwash behind the aircraft. As a curved wingshape will create a stronger downwash for the same planform and angle of attack, induced drag will be higher but lift will be higher too. For the same lift it will create less drag (not necessarily induced drag) as Form drag will be less.

      • Maybe a good point to ask a question.

        As the wing is bending, how does that affect the controls and do the computers have to be mapped to adjust for that?

        I had read Boeing was not sure their 787 wings would not just keep bending if they tested them (also they stopped as they did not want to deal with the explosion of CRFP)

        That said if you loaded them enough in flight would you loose control anyway?

  4. Does frontal area have any affect on friction drag, or is it just a measure of the surface area of the skin?

    • Frontal area creates friction drag in an indirect way. The air is forced to speed up around an object with a frontal area, by it increasing airspeed and therefore friction drag. If there is no separation around the back of the object there is no Form drag, as described before.

  5. It could be interesting to know the performance effect of lamniar flow wings at airline speeds, how much would it reduce fuel consumption on a 787-9 and an A350-900. Airbus are doing tails with just the outer section on an A340, one side has a Saab Composite outer wing the other a GKN version.
    The lamniar flow normally easier can cause deattaced flow loosing lift on a big section that the FBW system must recognize.

    • 4.6% 🙂
      Those things are known.

      The BLADE demonstrator shows what issues remain.
      How to build ( 2 path shown).
      How to upkeep laminar flow properties.
      How to handle abrupt loss of laminar flow.

      • Uwe: Boeing gave it up on the tail of the 787 (Vertical I believe) . Couldn’t keep it clean enough to work.

        US commercial with little scrubber bubbles to keep it clean. ?

        • Well in scope of the engine nacelles Boeing also sold 25% ( or slightly more than previous designs ) as 100% laminar flow 🙂

          “could not keep it clean” could as well be “Pravda”. The real reason might be different ( high cost, low _monetizable_ gain )

          • I had forgotten about the engine cowls and the special paint

  6. Thanks for dispelling the very common notion that induced drag is a local phenomenon at the wingtip! McLean’s book is by far the best intuitive explanation for this. The misunderstanding has led to far too many ill-conceived gadgets which try to dispel wingtip vortices.

    My suggestion is that this column doesn’t really explain what induced drag is – you mention downwash but don’t tie it in to drag at the airplane level (lift induces downwash, which reduces local angle of attack, which rotates the lift vector rearwards). I hope that methods of reducing induced drag (aspect ratio, wingtip devices) will be addressed in a future column as well!

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