# About the induced drag of sails

Discussion in 'Hydrodynamics and Aerodynamics' started by Mikko Brummer, May 18, 2020.

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### TeddyDiverGollywobbler

Steve Killing wrote samekind of nonsense in his book..

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### patzefranpatzefran

IMHO, you are saying the same thing, a directional change may be also quoted as a difference of speed (direction) !

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### Mikko BrummerSenior Member

Thank you Philsweet.
So I should look at horizontal cutplanes for speed gradients at the leech. The `horisontal rubbing`would only occur on the sailsurfaces within the boundary layers on each side, which then merge in the wake. Easy to understand how small differences would result into slight bending of the wake.
Do you agree then, that induced drag could be described as the result of the vertical direction jump? Easier to understand for a layman than for instance gradients in the spanwise lift distribution.

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### DCockeySenior Member

What do you mean by "induced drag"? Do you mean the definition which has been used for 90 years or so in aircraft aerodynamics? If so induced drag is the drag due to trailing vorticity aligned with the freestream, and that vorticity arises from spanwise variatin in the lift/load. The trailing vorticity is proportional to the difference in direction of the flow between the flow just outside of the boundary layers at the trailing edge. The result of the difference in flow direction is a vorticity sheet which downstream rolls up into two concentrated vortices. Added: The difference in flow direction is proportional to the difference in spanwise velocity component.

Induced drag (as the term is used in aircraft aerodynamics) is strictly a 3D phenomena due to variation in spanwise lift. A wing/sail/keel/etc of constant section, not twist and infinite span, ie 2D, has zero induced drag.

Confusion comes about when induced drag is described as the total change in drag due to lift. Induced drag by the conventional aircraft aerodynamics defintion is only one part of drag due to lift. A significant part of the change in drag of an airfoil/wing/keel/etc with lift is due to changes in the skin friction as the velocity over the foil changes with lift. This change in drag with lift occurs with 2D foils as well as 3D foils. See the drag vs lift curves for any 2D foil section for an example.

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### patzefranpatzefran

Best way to understand induced drag is to read some father of theoretical aeronautics, e.g; Ludwig Prandtl, NACA Rept N°116, Application of Modern Hydrodynamics to aeronautics ! This is true Physics
No need to make computational experiments with CFD codes !

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### JotMJunior Member

Except perhaps that in the real world the boat is pitching and rolling, which together with ever small changes to the speed and direction of the wind itself, is causing a constant stream of "starting" and "stopping" vertices being created at the trailing edge in order to restore the Kutta condition?

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### JotMJunior Member

Another way is taking (or in my case: trying) the class on flight vehicle aerodynamics by fellow forum member Mark Drela on edX: https://www.edx.org/course/flight-vehicle-aerodynamics (this course can be taken for free when choosing "audit this course").
I guess the induced drag Mikko is referring to is what prof Drela is explaining in these lectures:

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### JotMJunior Member

By the way, ever since having watched the video below of an address by Al Bowers a couple of years ago I have been fooling around with non-linear and increased twist (in sail7, as I have not yet been able to implement it in OpenFOAM and I do not have Fluent or the likes at my disposal), to see if I could minimise induced drag in a similar way. I can see I can get the vorticity roll up at a position lower up on the mast instead of at the mast tip. "Proverse yaw" would translate into "negative induced drag" at the tip of the mast, wouldn't it?

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### patzefranpatzefran

Yes, I watched Al bowers Video, on classical wings planes e.g. with elliptic loading, induced drag at the tips associated with bank to turn rolling flaps (at the tips) gives an adverse yawing moment. The pilot need to counteract this yawing moment using aft vertical rudder to keep yaw turning. Bell and Jones loadings gives negative induced drag at the tips and proverse yawing when using bank to turn rolling flaps, this moment provide automatically the needed yaw rotation, no need of a vertical aft rudder.

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### philSweetSenior Member

I took Drela's course online when it was first offered - good stuff.

JotM likes this.
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### philSweetSenior Member

Sorry Mikko, I missed your post earlier. The induced drag can be related to a number of macro features. I don't think the direction jump is a very intuitive way to understand it as a drag force. Especially since the jump is across a moving wake boundary. Viscosity eventually destroys these features, but in the case of induced drag, it happens very slowly. It takes ages for the induced drag features to die out for an aircraft. Induced drag is a matter of the streamwise momentum deficit in the nonviscous potential flow domain.

The easiest explanation is that as a lift force is generated by a wing, the fluid deflects downward near the body. This changes the incidence angle further along the cord. The total lift vector along the entire cord is rotated aft by half the trailing edge downwash angle. That immediately gets the basic relation that induced drag goes as lift squared (if the lift vector is twice as big, the downwash angle is also twice as big, so the drag is four times as big.) Secondly, if you double the speed at constant lift, the lift vector is the same length, but the downwash angle is halved, so induced drag is inversely proportional to speed.

Now if you add a downward velocity to a glob of fluid, and you want to conserve energy, and the wing isn't doing any work in this frame of reference, you have to source that energy from the fluid itself. The energy that appears added to the downwash, and to the entire fluid domain's transverse motion, exactly matches the energy taken from streamwise motion; and it turns out to be a bit easier to compute the added transverse fluid energies than to compute the streamwise deficit directly. But in basic F=MA terms, drag means you have reduced the streamwise energy of the fluid. It's just easier to see where the energy ended up than to see where it went missing from.

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### DCockeySenior Member

The explaination above completely ignores the fact that induced drag vanishes as the span becomes very large. Induced drag is not caused by lift along, but by the variation in lift along the span.

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### philSweetSenior Member

I mentioned the way induced drag changed due to a change in lift and due to a change in speed. If those are kept the same and the span is doubled instead, the induced drag is halved. F=MA - the force is the same, the fluid mass involved is doubled, everything else is similar, so the downwash angle is halved, just like it is halved when the speed is doubled, or it is halved when the fluid density is doubled, which both involve a doubling of the fluid mass.

The only quirk with this is whether you can keep everything else similar when you double the span. In order to keep the inviscid fluid domain similar, you need to scale the cord with the span, so the plan area is four times as large. This maintains the aspect ratio of the wing. But now there is change in the boundary layer and wake shape which has to be fixed. If you change the span, I need to scale the cord proportionally and scale the viscosity proportionally, the latter to keep the boundary layer shape similar.

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### Mikko BrummerSenior Member

Yes, I understand that, but with reference to my colleague Tom Whidden's in my view lousy attempt to explain what induced drag is about (my opening post), I was trying to find a way to describe it in layman's terms to regular sailors. They are not likely to read Prandtl, yet are more & more inclined to aerodynamic "term dropping" by using expressions like induced drag.

On the other hand, with all respect to the great Ludwig Prandtl, I don't think his inviscid theories are no more true Physics than the modern CFD codes... some might claim the contrary ;-).

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### Mikko BrummerSenior Member

I used to think so too, JotM, but simulations show no evidence of that happening. I'm sure there is one while tacking, but just pitching and rolling, I think the period is so long and the motion slow, compared to the wind speed, that the flow appears to be "continuous." See this one with a 470:

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