Downwash equation for Bell-Shaped spanload distribution

Well, I have been hesitating befoe to post the following question, as it is a NASA workpaper I am a bit shy regarding the issue:
in the workpaper attached:
1-The equation of the Bell shaped spanload distribution is y=(1-x^2)^1.5
2-And the derivative (the downwash equation) is y'=1.5*(x^2-0.5)

(y' is the derivative of y and u' the derivative of u)

while candidly using the generic derivative formula for the function (u^n)'=n*u'*u^(n-1)
So with u=(1-x^2), u'=-2x and n=1.5 I found y'=-3x*(1-x^2)^0.5

I don't know where I am messing, any good Samaritan to open my eyes ?

Thanks in advance

 

The downwash is not the simple derivative of the spanwise lift. The formula for downwash in lifting line theory is an integral equation which involves the derivative of the circulation.

 

Thank you DCockey,

I realize it is not as straightforward as I previously thought,
I need to revisit seriously Kutta-Joukowsky & Co in order to understand in depth.

Cheers

Erwan

 

Sigh....all mathematical models are wrong...some mathematical models are useful.
First you need to read (and understand the imposed conditions of) Prandtl's 1918-1922 papers to get a background of where modern "lifting line" theory comes from (note that the Kutta-Joukowsky condition is circa 1910 so is included in Prandtl's and Munk's work). Both circulation (gamma) and the Kutta-Joukowsky condition are mathematical contrivances that make the solution discrete. Neither is based in reality. Nor is "lifting surface" theory espoused by Kerwin in 1964+.
Collectively, in this forum, we are engineers. We need to address what is achievable with the materials and capabilities at hand. For the most part we are limited in span, bi-directionality, twist, camber, dihedral, and feedback. This is not the case in nature. While the paper you cite, and Prandtl's work, addresses the desired condition of no tip downwash; that condition is not achievable in most modern naval architecture situations. It is both a function of our material and control limitations, as well as our environmental requirements.

 

Thanks for your advices,
Unlike most of attendents on this forum, I am not engineer, just a little mathematics background, so mostly self-educated regarding Fluid Dynamics.
I candidly assume that exists an apparent wind velocity, which allows you to put your righting moment at full use, (Then it should be an Elliptical lift distribution)
and for any apparent winds above this one, the best trade off is the Bell-Shaped one, with the center of effort @ 34% instead of 42%.
That is why I am interested to understand downwash/ Effective AoA in deepth.

Cheers

 

Bell shape lifing distribution results in the minimum induced drag with lift and bending/heeling moment held constant and span allowed to vary.

Elliptical lift distribution results in the minimum induced drag with lift and span held constant and bending/heeling moment allowed to vary. Larger span for constant lift (not lift coefficient) results in lower induced drag and higher bending/heeling moment.

If you compare two wings with the same span and same lift, one with elliptical lift distribution and the other with bell shaped lift distribution, you will find the wing with bell shaped lift distribution has higher induced drag but lower bending/heeling moment.

If you compare two wings with the same bending/heeling moment and same lift, one with elliptical lift distribution and the other with bell shaped lift distribution, you will find the wing with bell shaped lift distribution has lower induced drag but larger span.

Lift distribution for minimum induced drag depends on the constraints. Span (mast height for a boat), bending heeling moment, or some combination of the two. If the constraint is a combined one it is possible the lift distribution for minimum induced drag will be neither ellipticial or the "bell shaped" given in the references but rather somewhere between those two.

Important - The discussion above is for absolute lift, bending/heeling moment and span.

Note that "induced drag" includes only the drag associated with trailing vorticity and does not include any drag due to viscosity. Also heeling moment does not include the effect of gravity on the sail and rig. Including viscous effects and gravity effects will change the optimum but the bell shape and elliptical shape are indicative of what the optimum will be with those effects included.

Erwan - are you interested in how to trim the sails on an existing rig in different wind conditions, or in how to design a new rig?

 

No, you miss the difference between the two situations. Elliptical loading gives the best L/Di for a given Wing Span. Bell-Shaped loading gives the best L/Di for a given Wing Weight. Those are two different things, especially applied to powered aircraft and the aero-hydrodynamics of a sailing vessel.

<X-post with DCockey above, his is much better>

 

An alternate view of optimum lift distribution:

 

Well, there goes my doctorial paper. Good thing I can show prior art.
Where can I get a copy of the 2nd presentation? ESA?

 

Google Al Bowers NASA

 

Thanks you All for your lights

DCockey, You read in my mind:

My "Case Study" is an A-Cat rig which means constant span, constant area, constant area distribution, constant weight.
The benchmark rig is supposed to have a sailplan with an elliptical leading edge or quarter chord, and straight trailing edge as this kind of sailplan is supposed to achieve Oswald coefficient=1 with Elliptical lift distribution.

The maximum available 2D Cl (Lift coef) will provide the maximum Apparent Wind Speed which would match the max righting moment for an Elliptical Lift Distribution.

Then Assuming you can achieve a Bell Shaped distribution with a lower Center of Effort, with the same max 2D Cl, I should be able to identify the minimum Apparent Wind Speed consistent for Bell Shaped Distribution.

Then, putting together the True Wind Gradient& Shear, the boat speed and True Wind Angle, the Downwash/ Upwash along the span, I should have enought elements to calculate the local lift coefficients along the span, and from this, using a well known wing section similar to a teardrop mast + full batten sail (ie:Eppler 376), changing its camber with XFOIL; It should be possible to provide the required change in (local section camber, local Angle of Attack ) which achieves the local required lift coefficient at minimun 2D Drag.

So I candidly believe it could be interesting for both How to trim sail, May be for How to cut sail, and could be a guide line for new concept rig too.

For the moment I deliberatly forget the Apparent Wind Speed range just beyond the Elliptical Optimum and just before the Bell-Shaped Optimum. I will explore the powers between 0.5 and 1.5 with a 0.10 iteration in a second phase.

In addition Excel Sheets save you the job of calculating primitive functions to perform Integral calculations.

Thanks for reading, Hope you didn't waste your time.

Cheers

 

I should add:

It could be interesting for How to cut sail,

probably only for third league players in the sail industry, as big names like North are probably well beyond this stage.

 

Here's a spreadsheet that will calculate the downwash from a given planform. And here is how I used it to look at the issue of sail design. Unfortunately, it won't handle swept planforms. For that, you'll need a vortex lattice program like AVL. The relationship between downwash and spanwise lift distribution in the Treffz plane is the same for swept and unswept planforms, however. It's just that the planform itself will have a different shape to produce the same lift and downwash distributions if it is swept.

I think the most interesting part, though, is it will also allow you to go backwards and calculate the planform (or twist) from a given downwash distribution. It turns out the downwash distribution, rather than the spanwise lift distribution, is really what is important to minimizing the induced drag. The classic elliptical lift distribution results from a uniform downwash, and a linear variation of the downwash along the span can produce a bell-like lift distribution. The linear downwash distribution minimizes the drag when there is a constraint to the moment due to the lift. This was originally presented by R. T. Jones (Jones, Robert T., "The Spanwise Distribution Of Lift For Minimum Induced Drag Of Wings Having A Given Lift And A Given Bending Moment", NACA-TN-2249, 1950.) and then extended to sail rigs in the presence of a nonuniform wind by Peter Lissaman (Lissaman, Peter B. S., "Lift In A Sheared Flow", AIAA Ancient Interface Symposium (year unknown, probably late '70's).

For a swept flying wing, which is Bowers' application of the bell-shaped lift distribution, there is a coupling between the pitching moment and the bending moment. The further outboard the center of lift, the more nose-down pitching moment. A flying wing needs to have a nose-up pitching moment about the aerodynamic center in order to be speed stable in trimmed flight, and this puts a constraint on the moment from the spanwise lift distribution. The way to get the necessary pitching moment with minimum induced drag is to use a linear downwash distribution. So he is actually stumbling onto the right answer by specifying a lift distribution that happens to result in a near linear downwash distribution.

For a sailboat, the righting moment of the hull puts a constraint on the heeling moment from the sail rig. The answer to this is again the linear wake wash distribution. Although the spreadsheet only considers a uniform wind, you can account for some of the effects of a wind gradient by changing the lift curve slope. And the spreadsheet allows you to account for the gap in the lift distribution between the foot of the sail and the water surface, whether that's due to the wind gradient going to zero at the surface or a physical gap between the foot and the water. It turns out the gap has a significant effect on the spanwise lift distribution, but the minimum drag wake wash is still linear.

 

Erwan,

I think the A-cat hardly ever sails in the "elliptical wind range". You are fully trapezing in 6-7 knots? And thus heeling moment constrained. So I think you should be interested in the Bell-range rather than the elliptical.

Tom's note on the sweep back and pitching moment has an interesting connection to the sailboat mast rake: when depowering a raked aft rig, lowering the center of effort also moves it forward, decreasing weather helm.

 

Thank you very much Mr Speer and sorry for late answer, this corona has triggered unexpected issues and I wanted to use Vortex a little bit before to post, but unfortunatly I didn't do it yet.

In fact I discovered VORTEX more than 10 years ago, but as I was unable to understand it at this time, I just forget it. Today I fell a bit more equiped to try to use it. An interesting homework during home lock-in period.
Never used AVL yet, but as it seems to be the 3D tool complementary for XFOIL I think I will not escape the User Manual reading (Another interesting homework)

Thanks Mikko, you are perfectly right, with the righting moment assumption I usedfloater mode),
The max AWS for Elliptical lift spanload is around 7 m/s
The mini AWS for Bell-Shaped spanload is around 9 m/s

In the 80's I was interested in R/C airplanes & Flying Wings, that is when I discovered the Horten wings and what I understood for the swept of the wing+big twist was that it makes the tips of the wing to act like an horizontal tail.

Today's some windsurf sails for foiling board seems to approach this Bell Shaped distribution:
The mast has a lot of bend, probably more parabolic than elliptic, the leech is straight from the square head to the deck, (so, below the wishbone), and when watching the camber distribution of the sail, it seems to be significant in the first 60% and very small above.
According to their limited righting moment, I would not be surprised that the windsurf sailmakers try to achieve this Bell-Shaped distribution above-mentionned.

Thanks again for your kind consideration

Best regards

Erwan