# lift without downwash?

Discussion in 'Hydrodynamics and Aerodynamics' started by lunatic, Oct 4, 2012.

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### markdrelaSenior Member

I'm not sure if this is a rhetorical question, but the answer is well defined:

You need to measure the streamwise vortex sheet strength gamma(y), which can be defined via the potential-velocity jump across the vortex sheet (same as the viscous wake sheet). This gamma(y) can also be obtained from the spanwise lift distribution L'(y) via the relation rho V gamma = -dL'/dy

You next assume that this gamma(y) and the flat sheet persists into the Trefftz Plane without rollup, where it determines the "downwash", or more precisely the entire transverse flowfield in the y-z Treftz Plane. The kinetic energy of this flowfield per unit streamwise length x is defined as the induced drag. So we have a well-defined calculation path from the lift distribution L'(y) to the induced drag Di, as expected.

For a non-planar wing, e.g. with winglets, then you'd use the arc length s along the sheet as the independent variable instead of y, and the required measurements are gamma(s), and the sheet geometry y(s),z(s). Everything is still well defined.

In this procedure the rollup can be neglected for a simple and rational reason: Rollup cannot change the kinetic energy of the transverse flow. So whether you keep track of the rollup or neglect it, you should get the same energy and hence the same Di. It's vastly simpler to neglect it. Kroo has made an alternative argument justifying rollup neglect based on force/momentum rather than kinetic energy.

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### markdrelaSenior Member

Addendum to last post.

The first PDF shows the Trefftz Plane potential velocity field grad(phi) in light blue, whose kinetic energy per unit x is the induced drag Di. This "downwash" velocity field is set up by the sheet strength gamma(s) which is determined from the body's lift distribution (not shown).

The second PDF shows a refinement of this model for the case where there is a fat fuselage of other large body which contracts the average streamlines from the trailing edge to the Trefftz Plane. This contraction does not change the sheet potential jump value Gamma(s) which is the same as the wing's local circulation, but it does redestribute it slightly. i.e. the y(s),z(s) coordinates in the Trefftz Plane are modified slightly from the wing's y(s),z(s) coordinates via the fuselage's contraction. The sheet strength is still gamma(s) = -dGamma/ds , but the derivative is now in the Trefftz Plane, not on the wing. The effect is to increase Di, because the effective span in the Trefftz Plane is reduced.

The same contraction effect will occur on a keel/hull combination. The effective semispan is not the depth of the keel tip, but the depth of its wake after the hull's contraction is accounted for.

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### daiquiriEngineering and Design

This is becoming an excellent discussion, I love it.

Thank you Mr. Drela and Mr. Speer for your contributions. These illustrations are so well-made and very clear.

First you have to define what do you intend by "an ideal fixed wing" and then how do you calculate the torque in the world of inviscid fluids, which is being treated here...

Apart that, a lifting circular cylinder and an airfoil are the same thing when it comes to the far field, if the circulation around them is the same. It stems from the Kutta-Joukowski Theorem, which is one of the pillars of mathematical fluid dynamics. You can find the enunciation and the proof of the KJT here: http://en.wikipedia.org/wiki/Kutta–Joukowski_theorem

So, given a total circulation Г around a 2-D body immersed in a uniform flow with velocity Uinf, the lift acting on that body will be equal to
L = ρ Uinf Г​
regardless of the form of the body. The total circulation Г can even be a sum of infinite number of infinitesimal vortex flows, like in case of a thin airfoil modelled as a vortex sheet. The flow field just doesn't care who or what is creating the circulation Г. The only things that count is that there are a uniform velocity Uinf and a circulation Г. If these are nonzero values, we get a lift force.

Kutta-Joukowski Theorem is a wonderful thing. It allows us to make huge simplifications when mathematically analyzing aerodynamic forces around bodies. It tells us that when we move far away from the body, so far away that the body becomes just a tiny dot, all lifting bodies become a superposition of one vortex flow and one uniform flow. Great stuff, isn't it?

Cheers

Last edited: Oct 9, 2012
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### daiquiriEngineering and Design

By the way... Don't know why, but I keep reading the title of this thread as "Life without downwash".

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### philSweetSenior Member

Thanks for hanging in there. Being an ME, I have a very hard time letting go of energy analyses. We tend to worry about turning big rocks into little rocks and pushing things around- All energy intensive activities.

I think I've finally gotten it though my head (for the forth or fifth time) that in a body centered reference frame, no work is being done. There for there is no energy transfer between the body and the fluid. ie. No change in farfield flow.

I seem to go through this once a year or so. Sooner or later it will stick.

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### jehardimanSenior Member

Yep, and that is one of my touchstone questions for young NAs...Is circulation real and how to you calculate Г for an arbitrary lifting surface? You can learn a lot about how they think about flow from how they answer those questions.

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### tspeerSenior Member

The beauty of the Trefftz plane is the induced velocity distribution ("downwash") provides a sufficient condition for minimizing the induced drag without having to actually determine the induced drag. This makes it possible to do inverse design by specifying the induced velocity distribution, calculating the spanwise lift distribution that created it, and then calculating the planform shape/twist/camber that will generate the lift distribution.

The papers I've read regarding wake surveys to determine lift and induced drag are typically formulated to estimate the moment of lift, which goes to zero at the root. This is useful for minimizing the uncertainties in a half-model wind tunnel test and the interference due to an aircraft's fuselage. However, it doesn't help with regard to computing the induced velocities.

I have a suspicion (hypothesis?) that it may be sufficient to use the actual sideways velocity distribution close to the trailing edge instead of the velocity distribution in the Trefftz plane, even though the near-field velocities include the influence of the bound vorticity as well as the trailing vorticity of the wake. But I've not been able to actually show that.

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### tspeerSenior Member

I think there is a change in the farfield flow. The wake from a lifting surface with finite span results in a change that persists clear back to the starting vortex, which is infinitely far away in the steady state. There is a drag on the surface, so work is being done on the flow.

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### markdrelaSenior Member

Correct, especially if you can ignore the body contraction effects which are usually very small anyway. Referring to the second PDF in my last post (dphiconv.pdf), the vortex sheet strength vector is

gamma = n x (V1 - V2)

where n is the unit normal to the sheet, and "x" is the usual cross product. This gamma vector is parallel to the angle-bisecting Va vector shown in the PDF, which for a flat wing is very nearly in the freestream direction in the absence of significant fuselage effects.

The physical requirement is that the wake sheet has equal pressures on top and bottom, p1 = p2, which via Bernoulli implies that the two velocities have equal magnitude.

|V1| = |V2| = V

The consequence is that gamma can be indirectly measured by having at the trailing edge...

1) A local static probe which gives V, and
2) A "differential alpha vane", which has one vane on top and one on the bottom of the trailing edge, and gives the angle between V1 and V2 velocity vectors. A pair of three-hole probes might be a more robust alternative to the differential alpha vane.

Using V and the angle, A bit of vector trigonometry then gives the magnitude of gamma, which in practice is sufficient:

|gamma| = gamma = 2 V sin(angle/2)

The spanwise circulation at some y location is the running integral of this gamma:

Gamma(y) = -Int_ytip^y gamma(y') dy'

Or this Gamma(y) could also be obtained from the local lift/span, if that's measured somehow (surface pressure taps?).

Using the usual Trefftz-Plane relations with gamma(y) and Gamma(y) then gives the downwash and also the induced drag.

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### Leo LazauskasSenior Member

I guess that's where we part ways, because I want to calculate the induced
drag, directly or indirectly, just for simple planar wings with curved leading
and trailing edges.
As you know, there is an "exact" solution for the lift-slope of a circular
planform wing, which means it is a great example with which to assess the
capabilities of various methods.
Why is it then, that most panel methods I have seen usually give poor
estimates of lift and induced drag for this simple case?

Now that I have some free time, I guess I should try AVL to see how it
performs, but I suspect it will have difficulties too.
Has anyone here tried AVL on circular planforms?

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### markdrelaSenior Member

Yep:

CL = 1.79075 * alpha (exact)
CL = 1.80480 * alpha (AVL 48 x 48)
CL = 1.81113 * alpha (AVL 32 x 32)
CL = 1.82400 * alpha (AVL 16 x 16)
CL = 1.86190 * alpha (AVL 8 x 8 )

Pretty good I think.

I also tried a square wing, which also has an effectively exact solution:

CL = 1.46023 * alpha (exact)
CL = 1.45780 * alpha (AVL 16x16)
CL = 1.45760 * alpha (AVL 8x8)
CL = 1.45640 * alpha (AVL 4x4)

The square wing is much less sensitive to vortex-panel resolution since there's no curved outline to resolve.

The very low sensitivity to panel resolution is one very curious thing about VL methods in general. They are more accurate than they should be.

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### Leo LazauskasSenior Member

Thanks, Mark.

Yes, VLM can exhibit "remarkable" accuracy as RM James wrote in his paper:
"On the Remarkable Accuracy of the Vortex Lattice Method" in 1972.

However, I do have some issues with your calculations for the circular wing.

That value of 1.79075 is definitely NOT the exact solution. It was due to
Hauptman and Miloh who eventually admitted that their wing was not
planar, but had twisted wingtips. It seems impossible to dispel the notion
that 1.79075 is the "exact" solution,
See for example, the attached preprint of a paper that appeared in J. Ship
Research.

Although the 1.79075 value is quite reasonable (for engineers), Hauptman
and Miloh's method yields much poorer estimates of the pitching moment
and it is pretty poor in describing the behaviour of vorticity production as we
approach the wingtips. But that's just what we want to get accurate
estimates of the induced drag.

The exact value of the lift-slope is 1.7900230... as calculated by Peter Jordan.
Incidentallly, that whole issue has a somewhat controversial and seedy
history. Boersma had great difficulty in even having his confirmation of
Jordan's exact result published, because (as he wrote to Tuck in about 1989)
he was repeatedly blocked by an editor/referee whose name I won't disclose
here.

I agree with your calculations and comments on the square wing. The
method I use is more accurate than VLM for square wings, but it only
becomes so when using a very large number of panels, typically greater

All the best,
Leo.

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### Leo LazauskasSenior Member

I am a bit loathe to show the attached internal report in its present draft
form (and because Drela will mock me mercilessly for using the LE suction
analogy in some parts), but the Section on Elliptical wings (page 14) has
quite a lot on various approximations for the circular planform that might be
of interest.

The lift-slope is the easiest quantity to calculate, and even that can present
great difficulties.
I'm happy to agree that AVL gets results of reasonable engineering accuracy
for the lift-slope, but more difficult to get accurately are the LE suction, the
strength of the LE singularity, and the behaviour of the vorticity as we
approach the wing tips.

How does AVL's estimates compare with the exact and values estimated by
other researchers?

My obsession with high accuracy is more than just nerdy foot-stamping.
These quantities are required for some planing problems, and at least one
digit of accuracy is lost in calculating required derivatives. Suddenly, AVL's
lift-slope estimate and my estimates don't look so good.

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### quequenSenior Member

May I say this thread is really amazing, thanks you all! By now it has tourned too high for me, but there's a lot of knowledge to digest just on the part inside my understanding.
Baekmo ilustrated my mistake and Daikiri explanation is really cool. By now I'm playing with the java solvers posted by Pi, which helps to illustrate some of this concepts.

lift of rotating cylinder
http://www.grc.nasa.gov/WWW/K-12/airplane/cyl.html

Joukowski Transformation & Kutta Condition (Java Simulator)
http://www.grc.nasa.gov/WWW/K-12/airplane/map.html

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### philSweetSenior Member

yup. I get that. I was referring to the inviscid 2d model daiquiri posted.

I have one piece of advise for any newbie to this thread. Turn your ceiling fan off. Sitting under a fan when reading this thread is not helpful.
It's downright distracting, in fact.

Jehardiman.- in answer to your question- circulation is as real as the uniform flow it is being superimposed upon, no more and no less.:idea:

(edit) I guess that should read potential flow, not uniform flow. The circulation being added after the transform that inflates the foil

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