Design of dagger rudder for trimaran

[tspeer]

Originally Posted by jmckerrow

...I've done some research and found it is quite straight forward to design and build a rectangular planform NACA 00XX board. The problem I have is how to get other planforms such as elliptical or a tapered trailing edge.
...

The rectangular planform may be a lot more efficient than you think, especially compared to a single-taper planform with the greatest chord at the surface.

I take it your daggerboard rudder is transom-hung, and therefore will act a lot like a surface-piercing foil. Although at low Froude numbers (based on the chord of the rudder) the water surface acts like a solid surface (because the pressure around the foil is small compared to the pressure due to gravity as the surface is deformed), at these speeds the Reynolds number is so low that you're likely to be more concerned with other issues than the planform shape.

At high chord-Froude numbers, the surface acts differently, and the rudder behaves like there's another rudder extending out of the water in line with it, but deflected the opposite way so as to cancel out the lift on the rudder right at the surface. This also doubles the contribution to induced drag of the part of the rudder that is near the surface. As a result, while the optimum planform for an inboard rudder that is up against the rigid wall of the hull might be best shaped with a straight taper, the optimum planform for a surface-piercing foil at high speed has its maximum chord farther down.

The resulting optimum planforms are not even elliptical, they are somewhat egg-shaped, with the maximum chord at more like 56% of the span.


A rectangular planform is a much better approximation to this optimum planform than is a tapered planform. You can use this spreadsheet to calculate the difference in induced drag of the straight planform compared to the tapered planform, or the optimum planform - with or without the approximate surface effects. The lifting line theory it uses is the same theory that says an elliptical planform is best for an airplane. The infinite-Froude number approximation used by the spreadsheet is also compared with more sophisticated methods in:

Kuhn, John C. and Scragg, Carl A, "Analysis of Lift and Drag on a Surface-Piercing Foil. Kuhn," The Eleventh Chesapeake Sailing Yacht Symposium, January, 1993.

They find the infinite Froude number approximation to be conservative (estimates less lift and more drag) than the more sophisticated finite Froude number calculations. The spreadsheet actually allows you do put in a surface effect value that is somewhere in between fully free and no surface, so you can probably find a "calibrated" value that matches the Kuhn and Scragg results for the optimum case, and use that for making reasonable engineering estimates of your candidate designs.

Kuhn and Scragg also present experimental results for a rectangular planform hydrofoil with comparison to all three theoretical methods. It's hard to precisely read the data for the rectangular foil from their plots, but lift and drag coefficients appear to be indistinguishable from the optimum planform at the same speed and angle of attack.

So there's little justification, from a hydrodynamic point of view, for going with the tapered planform. And when you factor in the practical considerations of the dagger-board rudder, it looks like a slam-dunk decision to me.

At most, I'd consider tapering, say, the bottom 25% of the span by sweeping the leading edge back to something like half the chord at the tip, while maintaining the trailing edge straight. This will do a couple of things. First of all, it makes it easier to thread the board into the sleeve! It will improve the span-loading a bit and marginally reduce the drag - if you take a look at modern sailplane wings, they are all going with this approach.

[jmckerrow]

Originally Posted by tspeer

The rectangular planform may be a lot more efficient than you think...I'd consider tapering, say, the bottom 25% of the span by sweeping the leading edge back to something like half the chord at the tip.

Thanks, Tom, for your detailed reply. I have appreciated the large amount of valuable insight you have provided to us amateurs, via the web.

Does it matter what form the taper takes - linear, elliptical, parabolic?

Also, do you have any opinion on why Ian Farrier (and others) use a tapered trailing edge rather than leading edge? Most of his rudders are kick-up types - would that have anything to do with it?

Cheers,
Jamie

[tspeer]

I'd go with linear taper. The difference in induced drag between that and curved is not great, but the ability to fair to an accurate section shape with the straight taper is significantly different. All you need are two templates to get it right.

I suggest you go to the Yahoo F-boat forum or Farrier Marine and ask Ian yourself. Actually, the rudder on my F-24 is a semi-ellipse, so he doesn't always put the taper just on the trailing edge!

The reason for tapering the leading edge and maintaining the trailing edge straight is to minimize the spanwise velocity in the boundary layer at the trailing edge. The boundary layer at the leading edge is thin and robust, so it can stand the spanwise flow better. At the trailing edge, the boundary layer is being stressed by an adverse pressure gradient that makes it want to separate. So you don't want to make things worse there.

Since shallow draft is one of your concerns, have you considered using a pair of rudders rather than a single rudder. Obviously mounting 2 rudders is going to be more work but there would be several advantages. The two rudders can have less draft than the single rudder but they could still have a greater aspect ratio. I believe there should also be some reduction in induced drag by splitting the lift across two surfaces rather than one but I'll let Tom tell us if that is correct. The load on the tiller will be less since the distance from the center of pressure will be closer to the hinge axis. Best of all, since the rudders will be much smaller you can probably find a dingy centerboard that will be work. Something like a pair of Laser centerboards would probably work great and you won't have to cut that clump of epoxy out of your hair. I don't know how it gets there but there is always epoxy in my hair before the project is complete.

[tspeer]

Originally Posted by des

...The two rudders can have less draft than the single rudder but they could still have a greater aspect ratio. I believe there should also be some reduction in induced drag by splitting the lift across two surfaces rather than one but I'll let Tom tell us if that is correct. ...

It's a common misconception that induced drag depends on aspect ratio. It doesn't. Induced drag depends on span (squared). Different rudders of the same span but different aspect ratio will have similar induced drag when they produce the same lift (in lb or N). The lower aspect rudder will have greater profile drag because it has more area, but not more induced drag.

If the rudders are widely separated, then you can get back to the same drag as the single rudder of similar planform. So you've achieved your goal of reducing the draft without a serious penalty. But if the rudders are close enough to interfere with each other, the drag will be higher. If the rudders are very close, the induced drag will be doubled because the span is 30% less (assuming the single and double rudder planforms are similarly shaped).

On a trimaran main hull, it won't be possible to get the rudders separated by very much. But rudders in all three hulls would be a different story. I think I'd go with 1 rudder for a trimaran and raise it for the shallows.

[SailDesign]

Originally Posted by Tspeer

On a trimaran main hull, it won't be possible to get the rudders separated by very much. But rudders in all three hulls would be a different story. I think I'd go with 1 rudder for a trimaran and raise it for the shallows.

Twin rudders, as Tom says, are great for wide hulls, especially where you can get one out of the water on each tack. In those cxases, I've found 55% of the area of a single does just fine. On a trimaran? One on each ama, with all the associated kit to control them? Not for me, but YMMV.

Steve

Tom

I remember a lecture from college about biplanes and surface loading. The point of the lecture had to do with splitting the load across multible surfaces to avoid long spans. As I recall, induced drag is a function of load per unit of span. If the load is spread across two surfaces, the loading will be 1/2 as much per unit of span assuming that the two surfaces have the same span as the single surface it replaces. And since the coefficient of induced drag = coefficient of lift^2/(Pi * aspect ratio * e) and the induced drag = the coefficient of induced drag * density * area * velocity^2. This along with the aspect ratio= span^2/area can be rearranged to
Drag induced = lift^2/(Pi * span^2 * density * velocity^2 *e)
By spreading the lift across two surfaces, the lift per rudder is 1/2 as much.
Induced drag for 2 rudders = 2 * (lift/2)^2/(Pi * s^2 * q * V^2 *e)
= 1/2 * L^2/ ( Pi * s^2 * q * V^2 *e)
Of course, this assumes that the two surfaces do not interact.

[tspeer]

Yes, that's true. You're essentially moving half of the trailing vortices away from each surface. This is essentially where the benefit of winglets comes from - the winglet displaces some of the vortex vertically where it has less effect on the wing.

You kept the span constant in your comparison, but that's almost never done in practice. Instead, most designers keep the aspect ratio of each demiwing equal to the aspect ratio of the baseline monoplane. As a result, the span of the biplane is 70% of the span of the monoplane. When you substitute span/sqrt(2) into your formula, it cancels out the 1/2 in the final result. That's why I said, "If the rudders are widely separated, then you can get back to the same drag as the single rudder of similar planform."

Ilan Kroo presents the classic biplane theory in his digital textbook.