Lift & Drag - Airfoil shape or Aspect Ratio?

Hello everyone

I am working on a project (control oriented) which is centered around a model of a sailboat. In order to perform certain analyses, I need to know the lift and induced drag coefficients of the keel and rudder.

Here's what I've collected so far: there are sources that mention that drag,lift/aoa relations can be found for keels of different aspect ratios (which is a geometric property of the lengthwise section of the keel/rudder/wing). On the other hand, many sources, including simple software tools that I found have those relations w.r.t. the airfoil shape (i.e the cross section).

I had found the first approach first, so I took the measurements required to find the aspect ratio. However, later on, I found that more sources (specially aerodynamic/wind design related) give that data for the airfoil shape.

I do not know which approach to follow or which is correct. If it was the latter, I would probably just look at my keel and rudder and roughly match them with one of the standard airfoils shapes for which the data is available.

Any tips?



P.S.
- I do not have access to a wind tunnel, or any advanced aerodynamics software, nor is that really within the scope or focus of my project. However, I do want to get an accurate answer.

- All my boat design/sailing/aero-hydrodynamics knowledge is what I've gathered in just the past couple of weeks, so excuse my novice-ness.

- I would like to apologize for seeming like a leech, and joining the forums just to ask a question. I hope you don't mind sharing any useful knowledge, and I'd love to contribute with anything if I get the chance.

 

Those that are for the 2d airfoil shape do not take the 3d effects into account. That is the induced drag term. The wiki on induced drag gives you the bare bones of how to do this. The induced drag is basically purely a function of the lift being generated by a foil, and varies as lift squared. The constant that relates the lift to the drag squared is dependent on span, and on how well the load distribution across the span matches the ideal case. In the wiki, they sort of hide this. S is surface area. AR is span squared over S. So replace S*AR with span^2.

And no need to apologize. Welcome to the forum.

One usually is interested in the relative size of the different drag components. At a constant lift and varying velocity, induced drag decreases with velocity squared and viscous drag increases at a rate of about velocity squared.

 

To put this in a bigger perspective, the three-dimensional flow around a lifting surface like a keel or wing difficult to solve. A widely used approximation is to break the 3D problem into two separate two-dimensional problems.

The first is the flow taken in the plane normal to the axis of a wing of infinite span. This gives the 2D section characteristics from airfoil theory. This is a good approximation to the local flow about a wing of moderate to high aspect ratio. Keels typically have an aspect ratio that is really too low to meet this criterion, but 2D airfoil theory is still used because there's not a cheap alternative.

The second problem is solved in a plane that is perpendicular to the flow direction, called the Trefftz plane. Ideally, this plane is taken to be at infinity, but the wake is also assumed to trail straight back without rolling up. The flow in the wake determines the lift-induced drag, which must be added to the profile drag (from 2D airfoil theory) to get the total drag for the lifting surface. The induced drag depends on the square of the span. When you divide the square of the span by the area, in order to get a nondimensional coefficient, this results in the aspect ratio. That's why you see aspect ratio coming into the picture. It's also convenient that all section shapes have a very similar slope to their lift vs angle of attack curves, so the actual section shape doesn't need to be known to solve this part of the problem.

The two solutions are tied together to get an approximation to the 3D flow. The induced velocities from the wake modify the local angle of attack at each section. This is generally a reduction from the freestream angle of attack, due to the "downwash" from the wake. The section shape, through its zero lift angle of attack lift curve slope, along with the local chord length, determines the lift at the local angle of attack. The profile drag, which is due to viscous effects in the boundary layer, is added to the induced drag due to the wake, to get the total drag.

 

Thanks for the answers. I just borrowed a couple of yacht design references which confirm what you're saying.

Basically, wings of the same airfoil shape have approximately the same maximum lift coefficient, but the angle of attack at which they reach it varies with the aspect ratio. This is crucial to my application, since I'd like to study the effect of the boat's leeway (the angle of attack in this case).

The main problem is: Where do I find data that takes into account both airfoil and aspect ratio? The books only have quantitative plots. The software tools only consider airfoil shape. So far, the best thing I've found are old NACA reports, but those either neglect aspect ratio, or when it's considered, it's for very high aspect ratios (for wings); much higher than the aspect ratios of my keel and rudder. 0.45, and 2.5 respectively.

 

http://ntrs.nasa.gov/search.jsp

Might find something here.

 

The standard old time reference is Fluid-dynamic Drag by Dr. Sighard F. Hoerner. It's expensive, but worth it for the straight forward methods of analyzing drag problems based on years of practical engineering. It's from the days of slide rules so don't expect modern computer methods.

Lohring Miller

 

Get "Theory of Wing Sections" by Abott & VonDoenhoff. Or it's online predecessor, NACA-TR-824, "Summary of Airfoil Data". It explains how section data relate to the 3D aerodynamics.

You can also use software like XFLR5 to calculate the 3D lift and drag. However, it is still best for medium to high aspect ratio surfaces.

A panel code, such as CMARC, is a more capable program that can calculate the 3D flow around the hull and keel. However, even for a panel code, a low aspect ratio surface is still a challenge when the flow separates from the leading edge and rolls up into a vortex that trails back along the surface. The influence of the free surface is also an issue that isn't really captured very well at the kind of Froude numbers that are probably relevant to your boat.

The unfortunate fact is that to accurately calculate the flow around a low-performance configuration, you need a very sophisticated program. The alternative is to make do with simpler methods to get an approximate estimate, and then use empirical data to improve the estimate.

 

Amjams, is it a university project you're working on? If yes, what university - if I may ask?
Cheers

 

Thanks for all the suggestions. I think I've come close to an answer. There appears to be a formula that computes the lift coefficient as a function of the planform's AR and the lift coefficient of the 2D airfoil section.

http://en.wikipedia.org/wiki/Lifting-line_theory#Useful_approximations
http://en.wikipedia.org/wiki/Thin_Airfoil_Theory#Thin_airfoil_theory

Now I need to determine somehow the airfoil shapes of the the rudder and keel of the model boat.

Yes, more or less a university project.

 

If you have experimental Cl-Alpha, Cm-Alpha and Cl-Cd curves for airfoils you intend to use, for example from Abbott&Doenhoff (see TSpeer's post above), then you can forget about the thin airfoil theory and can calculate the 3-D wing characteristics by using these formulae: http://www.mh-aerotools.de/airfoils/hdi_aoawing.htm
a0 in these formulae is the slope of the linear part of the Cl-Alpha curve, as seen in this figure (a0 = dCl/dAlpha):



I don't know where the approximated formula in the Wikipedia ( AR / AR+2 ) comes from, should make some math to get the meaning of it. But I also don't see the usefulness of it, since the original formula is so simple to calculate that it doesn't need further approximations.

Of course, the above formulae are all based on some assumptions and approximations (linearizations and small angles of attack) which are not always appropriate for design of keels, rudders or (worse yet) sails, but they will get you sufficiently close to the ballpark.

Cheers

 

Go look up Harrington's paper "Rudder Torque Predictions" in 1981 SNAME Transactions. Covers low aspect 3D foils against a hull as well as planform and sweep effects, both of which are important in low aspect (~< AR=3) foils.

Edit: Now that I look at my copy, you really should go back to the source; DTMB report 933 "Free Stream Characteristics of a Family of Low Aspect Ratio Control Surfaces", 1958 and the DTMB translation 321 of Thime, "Design of Ship Rudders", 1965. I pulled alot for my cross faired data out of Harrington, but he got the data from the reports. FWIW, most of that data is presented in the 1966 edition of Principles of Naval Architecture, you just have to interpert the individual effects yourself.

 

Thanks. I"ll try both formulas and see which gets a better fit for the measurements I have.

edit: I forgot to mention that I saw that approximation used here: https://www.academia.edu/5132922/In..._Different_Keel_Geometries_on_A_Sailing_Yacht page 3

SNAME articles are inaccessible. Let me know if there's a way to get access somehow, because I need a particular paper from there.

 

Attached a simple spreadsheet regarding Martin Hepperle's document

http://www.mh-aerotools.de/airfoils/hdi_aoawing.htm

(It's intended to help on understanding the behaviour of AC72 wings, foils and rudders. Works well for me)

Hope it helps!

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