# Wake Wash

Discussion in 'Hydrodynamics and Aerodynamics' started by tspeer, Oct 20, 2013.

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### tspeerSenior Member

I've taken these quotes from the AC34 multihulls thread because I think this topic deserves its own thread. Instead of "downwash" I'll call it "wake wash" because it's not necessarily in the downward direction.

It's true that the wake wash in the Trefftz plane is a quantity that is peculiar to the Trefftz plane, being far enough away from the lifting line that the bound vortex has no effect and yet not having the wake rolled up, either.

However, if you look at the criteria qualitatively, it says for minimum drag, the wake should be shed as though it were a rigid sheet of metal, being extruded out the trailing edge at a uniform speed all along the span. Any shearing or wrinkling of the sheet means higher drag than necessary. For minimum drag with a specified moment, it should come off like a rigid sheet that has a rigid body rotation imposed on it. And for lifting line theory, although the magnitude of the wash at the lifting line is half what it is in the Trefftz plane, the shape of the wash in the near field is the same as the shape in the Trefftz plane.

This has had me wondering if the same qualitative principle would apply to the near field of a real flow, even though the near-field velocities normal to the plane of the wake will have both bound vortex and shed vortex contributions. The limited CFD data I've looked at for configurations that had been fairly well optimized for minimum drag have pretty linear velocity distributions, consistent with the idea.

So I have a hypothesis that making the velocity normal to the plane of the freestream or lifting surface have a linear spanwise distribution when measured in close proximity to the trailing edge, will result in minimum induced drag for a given spanwise location of the center of effort.

The immediate intuitive reaction of people I've approached with this idea is that the hypothesis is not true. The argument typically revolves around the fact that the downwash calculated in the Trefftz plane is not the same as the wake velocities that would be measured anywhere in a physical wake. Near the trailing edge, the contribution of the bound vorticity would contaminate the contributions from the trailing vortices, and away from the trailing edge, the wake would be rolled up making the velocity distribution quite different from the Trefftz plane. However, I don't know of any analysis that went beyond the intuitive reaction.

If the hypothesis were to prove true, it would have huge implications for aviation as well as sailing. So it's one of those high-risk/high-payoff items. It may well be false on the face of it. But it's also possible that just by looking into it, it could point the way to a method that would minimize the drag through wake measurements. One of the main attractions for me is that the hypothesis is a necessary condition for minimum drag, but it doesn't require actually calculating what the drag (or the spanwise lift distribution) actually is.

Another attraction is that, if true, one can look at a wash distribution and immediately say whether it is optimal - is it a straight line or not? One cannot look at a spanwise lift distribution and say whether it is optimal or not. The very same lift distribution may be minimum drag for some situations but not for others. For example, an elliptical lift distribution may have minimum drag with no interference, but for a sail rig with only a small gap between it and the ground, the minimum drag lift distribution will be more egg-shaped, and the minimum drag span load becomes a semi-ellipse when the gap disappears. So even if one can achieve the daunting task of measuring the spanwise lift distribution, it's not so obvious as to how to use that information to improve the performance.

So with that, I'll open up the discussion with the question, "Can anyone prove or disprove my hypothesis?"

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### markdrelaSenior Member

This can't be right. I can think of two reasons:

1) Induced drag is the kinetic energy (per unit streamwise distance) of the transverse velocities in the disturbed flow behind the wing:
Di = Int Int 0.5 rho (v^2 + w^2) dy dz
The integral is over the entire y-z plane anywhere behind the wing (even close to the TE where there is no rollup yet), provided that the transverse velocities v,w are only those associated with the trailing vorticity. The contributions to u,v,w from the bound vorticity must be excluded from the integral as written.

The reason is that the exact integral, including pressure-work terms and the streamwise perturbation velocity u is
Di = Int Int 0.5 rho (v^2 + w^2 - u^2) dy dz.
For the bound-vorticity contributions to u,v,w the v^2+w^2 part gets perfectly cancelled by the -u^2 term, so these contributions can be simply omitted. Indeed, the second exact Di integral above is exactly zero when evaluated behind a 2D airfoil, as expected.

2) The total normalwash velocities v,w close to the wing at some specified location can be easily changed by merely redistributing the chordwise loading. For example, you can easily change the normalwash velocity just behind the wing TE, at x/c = 1.1 say, by shifting the chordwise loading rearwards, by e.g. drooping the airfoil nose a lot and adding some rear camber. If the local cl is maintained in this load redistribution then surely you haven't changed the induced drag, even though the near-field normalwash has been significantly changed.

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### Ben GJunior Member

Interesting theory, if it is true I think it gives a lot of power to 'intuitive' aerodynamic design, which is so much of what design is.
I'm an intuitive thinker, and usually create a design by 'assembling' theoretical components to get the best result.

So while I only understand a little bit of lifting line theory I think I follow what you mean. It would be handy in assessing discontinuities in the lifting surface, for eg elements at different angles of attack, tip shapes and so on.

As well as the velocity normal near the trailing edge (say, a small distance dx just after the trailing edge) I'd have thought you would need to add the spanwise velocity as well.

Say, the take spanwise velocity just before the trailing edge - as this would contribute to the strength of the vortex. Practically this spanwise velocity may need to be measured at the top of the boundary layer or some finite distance away from the wing surface, I guess the velocity has to be zero at the trailing edge, on the wing surface. With high AR surfaces this could probably be neglected, but at lower AR surfaces or at discontinuities probably not. For example at the top of a square top mainsail. The head can be a leading edge, trailing edge, wingtip (ie spanwise flow only), or a combination of all three, depending on shape...
And that edge would be a critical variable for vorticity and drag creation, which is where I see this hypothesis to be most useful.

Am I on the right track?
Another version of your hypothesis would be to take the velocity normal to the edge regardless of whether it's a trailing edge or not. Ie the angle of the trailing edge relative to the free stream needs to be accounted for.

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### Mikko BrummerSenior Member

1) Thank you for this definition of induced drag, I think I understand it better now. As I recall from textbooks, I was only familiar with the 2D-definition.

2) This is even easier to grasp than the rather esoteric integral equations. It sort of hints to my perception about Tom's hypothesis being linked to the lifting line theory (?) - after all, lifting line theory ignores the chordwise distribution of lift.

I ran a couple of tests before reading your reply - the 1/4 ellipse wing and rectangular wing sport a flat wake (as expected no camber-no twist), while the real sail shows a much more complex wake, with plenty more source for induced drag.

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### markdrelaSenior Member

This might be a good time to write down some definitions of the various terms being thrown around here.

Trefftz Plane theory:
The expression of all the forces (viscous and inviscid) on a body only in terms of the velocities and pressures in its downstream wake.

This is 100% rigorous regardless of the shape of the body. But in order to put the TP expressions into forms which can be evaluated, some idealization of the wake is required. The simplest idealization is to neglect roll-up and to assume that the viscous thickness of the wake is very small. Only then does the resulting simplified expression for the total drag decompose into profile-drag and induced-drag components. Without some idealization of the wake, this drag decomposition cannot be done.

This requirement for wake idealization is less onerous than it might seem. The TP force expressions, if carefully defined, are valid even if the TP is placed closely behind the trailing edge, before any significant rollup or viscous wake spreading has occurred.

Lifting Line theory:
The assumption that the flow at each spanwise location of a wing is the same as a 2D airfoil flow, but with a modified freestream velocity.

The modification is due to the "downwash" of the wing's trailing vorticity (but not the bound vorticity!) at that spanwise location. One by-product of LL theory is a workable definition of induced drag: The local lift force is tilted aft by the local downwash angle, and the spanwise integral of the aft lift force component is the overall wing induced drag.

LL theory is valid only in the limit of very large aspect ratio. And even with a large aspect ratio, LL is still inaccurate near a wingtip and near the root of a swept wing.

* * *

The TP expression for the induced drag happens to be identical to the LL expression for the induced drag, but their derivations are very different. The TP derivation is much stronger, and does not rely on any high aspect ratio or local-2D approximations whatsoever. So when a "lifting line" model gives you surprisingly good Di results even for very low aspect ratios, you shouldn't be surprised. You're really using TP theory without realizing it.

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### Leo LazauskasSenior Member

An idea of how LL compares to other inviscid theories can be seen
in the graph of lift coefficient slope as a function of aspect ratio
for flat rectangular wings.

At an AR of 10, there is about a 10% difference between LL and
lifting surface theory. For very small AR, e.g. 0.5, the differences
are much greater.

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### tspeerSenior Member

Is the color scheme for the Finn sail the same as for the 1/4 ellipse and rectanguar wing - the Z velocity component? Or do the colors of the Finn sail correspond to the total velocity magnitude?

Aside from whether or not such a change would be feasible for a membrane sail, do you get less drag at the same lift if you adjust the twist so as to make the crossflow velocity distribution more linear along the span for the Finn sail?

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### tspeerSenior Member

How do continuity considerations play into this? Do they put some constraints on v, given w?

What is the expression for lift, as determined by integration of velocities in the TP?

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### Mikko BrummerSenior Member

Excellent - thank you.

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### Mikko BrummerSenior Member

No, I'm afraid the color scheme is not the same - total velocity for the Finn. Also, the free stream speed is different, and the free stream for the Finn has a gradient & shear in it, and AWA is much bigger (5 m/s & 5 deg for the planars, 4,4 m/s & 26,5 deg at h=10 m for the Finn). I was simple referring to the shape of the wake, planar for the planar wings and complicated for the real, triangular sail not touching the deck. It's a light wind simulation for the Finn, with close to max. lift.

Yes, you probably could get less drag by adjusting the twist - you would have to go to negative numbers (not feasible for membranes), since in this light wind case the leech is practically straight - but separation would increase, so this is not far from optimal. Although the geometrical twist is about zero, effective twist from the zero lift angle, allowing for draft variation from foot to head, is clearly negative. This compensates for the triangular shape of the sail, vis a vis induced drag.

Attached a wake shot colored with Z-velocities, roughly in a similar scale to the planar sails.

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### Mikko BrummerSenior Member

The Finn case is practically the same as the ones featured in our you tube channel:

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### markdrelaSenior Member

Yes, but in this plot you're also in effect comparing the lift-distribution prediction accuracy between LL and lifting-surface (LS) theories, because the lift distribution alone is sufficient to determine induced drag (Di).

I was mainly discussing Di prediction accuracy obtained when the lift distribution from an LS (or panel, or Euler) calculation is plugged into LL's Di formula. In this case the accuracy is essentially perfect. The reason is that LL's Di formula is the same as TP's Di formula which doesn't rely on the LL approximations.

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### markdrelaSenior Member

In the TP formula derivations, the perturbation velocities in the TP are assumed to have zero divergence, as they must by continuity.

The attached PDF lays out all the assumptions and derivations. Continuity is applied between equations (5.10) and (5.13), and also between (5.42) and (5.43). The TP formulas for Di, L, Y are (5.47), (5.64), (5.65).

You can also compare the Di formulas:
(5.25) for LL
(5.47) for TP
They are the same because of the following relations used by LL:

L' = rho V Gamma
Gamma = delta phi
alpha_i = -(d phi/dn) / (2 V)

For alpha_i, compare figure 5.6 with 5.7,5.8

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### Leo LazauskasSenior Member

Yes, all I was trying to show was how large the aspect ratio must be before
LL starts giving predictions of lift comparable to other models. It is
surprisingly large, i.e. AR > 10.

I have posted induced drag (and leading-edge suction) predictions for some
planar wings in other posts. They were intended to show that there are
interesting methods other than TP for estimating induced drag of planar
wings.

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### markdrelaSenior Member

I think the TP method of Di calculation is inherently more trustworthy than all the nearfield methods which use an LE suction model. One reason is that for a 2D flow, the TP method gives the exact answer Di=0, no matter how coarse the paneling is. The LE suction methods that I've seen will not give exactly Di=0 for a 2D case.

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