The Myth of Aspect Ratio

Hello Tom!

That's not too difficult to implement roughly in Michlet to get some ideas of the effect on wave resistance. It will be difficult to position the appendage very accurately hull unless many stations are used for the hull. I am pretty sure the extra wave resistance will be very small unless the keel is fairly thick and voluminous, and/or the Froude number is small.

At high Froude numbers, "diverging wave" resistance is the major component of the total wave resistance and these waves are damped very strongly as submergence depth increases.

There is an early version of Xu's paper I cited earlier available in the Proc. of the 18th Symposium on Naval Hydrodynamics at:
http://www.nap.edu/openbook.php?record_id=1841&page=273

My guess is that it would have to have a fairly high aspect ratio (AR > 5) for lifting line theory to be accurate within about 10%, but maybe that's good enough.

Leo.

 

Two keels, same depth/span, same side force. One is twice the aspect ratio of the other, thus half the area so it will be operating at twice the overall lift coefficient and the average pressure difference between the "upper" and "lower" sides will be twice that of the lower aspect ratio keel.

What is the difference on wavemaking resistance? Does twice the pressure difference but half the chord result in the same wavemaking resistance? Perhaps the Froude Number based on chord comes into play. If the Froude Number based on chord is sufficiently high the chord is small compared to the wavelength. Then perhaps the chordwise pressure distribution doesn't matter, and only the spanwise load distribution is important for the wavemaking. This would line up with the comment that Froude Number relative to span/depth is the appropriate one to consider. Then the question is how high is sufficiently high for the Froude Number based on chord?

 

This is the publication that made me bring up the topic about chord based Froude number... unfortunately I cannot find it anymore. I know I've had the booklet, but I've probably loaned it during the years and the lender never returned (like all my Marchajs & Bethwaites ).

As I recall, there was a very graphic illustration (from SPLASH-code?) about the Fn effect on spanwise loading of a surface piercing, straight foil, losing nearly half of its effective depth as the chord Fn got higher.

 

I may need to order a copy of the Eleventh Chesapeake Sailing Yacht Symposium proceedings. I'm curious about the design tradeoffs vs graphs of non-dimensional coefficients.
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One of the reasons for starting this thread was that issues involving span and chord, ie aspect ratio, are usually discussed in terms of non-dimensional coefficients. Basing design decisions directly on the non-dimensional graphs can sometimes lead to erroneous design decisions because changing one physical parameter can lead to the simultaneous changing of multiple non-dimensional paramters. Changing chord while keeping lift constant changes both the aspect ratio and the lift coefficient. For a "wing" without free surface effects the change in drag can be very different than what would be expected based on a chart of the changes of drag with aspect ratio with the lift coefficient remaining constant.

 

If you have the time, maybe you could post a graph or two showing a simple example of how things can go wrong if non-dimensional values are used to assess some simple keel shapes.

 

Just to clarify there is no problem with using non-dimensional values to assess a keel shape. Potential problems can arise when using certain curves of non-dimensional parameters in making design decisions without taking into account that what is constant and what is varying in the non-dimensional coefficient plots may be different than what happens given the requirements and constraints of the design. An example is (trailing vorticity) induced drag which started this thread.

Good though about an example graph or two, but it may be a while before I'm able to do so.

 

In that way we are getting back at the page 3, post #36 and following ones, of this discussion. A sole induced drag is not an issue for a designer.
The total drag is an issue, and there's no myth in saying that the total drag diminishes with AR, any way you look at it (as you also have emphasized that fact in the edit of your first post).

So, this is indeed an interesting and educative theoretical discussion (particularily in it's digressions), but from a practical (designer's) point of view it will hardly change anything.

Cheers!

 

Would you agree that plots of CD due to induced drag vs CL of various aspect ratio wings should not be used as the rationale that "higher AR" is lower drag when the depth/span is constrained? Higher AR may be lower drag but not because of lower induced drag with that common constraint. Different physics involved. Also the angle of attack/leeway angle becomes larger for the same lift if the area is reduced and span/depth held constant to increase AR.

Dicussions of drag reductions due to higer aspect ratio keels and rudders should emphasize the role that viscous drag plays and not concentrate on induced drag.

An example of the confusion is the discussion of Hydrodynamic Side Force, in the "Hydrodynamics of Hull, Keel and Rudder", Chapter 5 by J A Keuning of Sailing Yacht Design: Theory 1998, University of Southhampton, pp 71-73. He starts with a discussion of induced drag and shows the classic circa 1921 plot from Prandtl of lift coefficient vs angle of attack for rectangular wings of different AR, and then concludes that paragraph with:
So an effective wing in respect of high lift production at small leeway angles and at the cost of minimal induced drag is found in the high aspect ratio wing. These high aspect ratio foils enable the yacht to sail at relative low leeway angles at least when the speed is sufficiently high. The keel becomes effective also because a reduction in the wetted area of the keel leads to lower frictional resistance. Obviously enviormental conditions such as water depth as well as constructional considerations may lead to retraints on keel design.

This sounds great, lower leeway angles, lower reduced drag AND lower frictional resistance! Let's consider two scenarios where the lift stays the same and AR is varied:

Scenario A: AR is increased by increasing the span/depth and decreasing the chord. Area stays constant. CL of the wing/keel/rudder stays the same. Induced drag will be less. Leeway angle will be less. Frictional resistance will stay about the same, not be significantly reduced. Two out of three.

Scenario B: AR is increased by decreasing chord and area. Span/depth is fixed. CL increases proportional to the AR. Induced drag will be the same. Frictional resistance will be reduced. Leeway angle will be GREATER. One out of three, and the promise that leeway angle will be reduced is the opposite to what happens.

CL proportional to the AR for a given lift if the span/depth is held constant seems to be what is frequently missed. It's easy enough to look at a plot and assume Lift and CL are the same. Only true if the area is fixed.

Keuning goes on to discuss two drawbacks to high aspect ratio keels; possible stalling of the keel coming out of a tack, and possible increased drag when sailing upwind in waves due to periodic stalling.

 

What it mean is a practical designer shouldn't be using considerations of induced drag when making decisions about keel and rudder chord and area. Once the span/depth is selected induced drag is close to a given. That may be a change for some designers.

Primary hydrodynamic considerations for selecting keel and rudder chord and area should be based on viscous drag and how large a stall margin is needed. Of course section profile selection also affects these. Leeway angle may also be a consideration. And then there are structural and ballast considerations.

 

I agree with you on this - it's too easy to be dazzled by narrow, sleak shapes, forgetting that in the end it's span that counts. Certainly has happened to me, and seems even to happen to authorities such as Keuning.

I could not resist the temptation (and waste my time ) to run my VPP for a base boat + 2 variations: 1 with a double chord keel and another with 1/2 the chord keel, all with same draft & approximately same weight. The difference in performance (according to my 20 years old VPP) was surprisingly small. For the narrow keel, I adjusted the rudder to take more load, else leeway was excessive and performace poorer. The sailboat is quite a self-regulating system.

 

Sorry, the first variation was with a 1,5 times chord, not double.

 

Interesting. If someone just looked at the plots of lift coefficient vs angle of attack for different aspect ratios they could reasonably assumed (as Keuning apparently did) leeway would be less with the narrower chord, higher aspect ratio keel. What is overlooked is the keel will operate at a different lift coefficient if the area is changed when the aspect ratio is changed.

 

Sailors want to minimize leeway because they are working with a fixed configuration.

Designers want to maximize leeway (up to a point) by reducing chord so the boat has less wetted area and the parasite drag is reduced.

The optimum chord length is the one that results in the keel operating at the best section L/D, which can be determined by the non-dimensional section data.

But your point is well taken that people confuse aspect ratio with skinny-ness, and think that a skinny keel will have less induced drag regardless of its span. Aspect ratio is really the nondimensional form of span-squared. Area is used to nondimensionalize all the length-squared parameters. A keel with a different chord will have pretty much the same induced drag as a baseline keel of the same depth, despite the difference in aspect ratio. The span-squared is the same, but the longer chord keel has more area in the calculation of the nondimensional aspect ratio.

And it's easy to forget that it is the dimensional lift that has to be kept constant, not the lift coefficient, because the dimensional lift is what is determined by the sailplan. This is easy to see when comparing the dimensional and nondimensional formulas for induced drag:

Cdi = CL^2/(pi*AR*e)

Di = L^2/(pi*qbar*depth^2*e)

The nondimensional form also fails to show that induced drag decreases with speed-squared when lift is held constant, while most other forms of drag increase with speed-squared.

 

Tom, thanks for reinforcing several of the points from earlier in the thread.

 

Another way to look at it: The relevant "skinniness" parameter is not AR, but AR/CL. This directly gives the induced-drag/lift ratio:
Di/L = 1 / [ pi * (AR/CL) * e ]
So even a low-AR surface will be efficient if the CL is low enough.

This is a somewhat trivial re-write of the basic relations, but it's a good way to look at a case where the CL is unusually low because of non-aero factors, such as size rules or structural requirements.