Asymetrical centreboards

There is considerable confusion about induced drag and aspect ratio. The effect of aspect ratio on induced drag depends on what is kept constant, and on how induced drag is defined.

Textbooks generally discuss lift, induced drag and aspect ratio in terms of drag coefficients, ie lift/area and drag/area. But in an application generally the actual lift and drag are what matter, ie lift coefficient*are and drag coefficient*area. This is one of the primary sources of confusion about the effect of aspect ratio on induced drag. Confusion also arises from several different uses of "induced drag". For this discussion I'm using the standard definition of induced drag as the drag due to trailing vorticity which does not include drag due to viscous effects. Note that the total drag of a wind with infinite aspect ratio (2D) increases with increasing angle of attack and lift due to viscous effects. Also, span-wise area, twist (if any) and corresponding lift distributions are assumed to remain the same as aspect ratio is changed.

Let's start by considering the situation where lift is constant and and area is constant. Since area is constant the lift coefficient also remains constant. To increase aspect ratio in this situation the span is increased and the chord is decreased. This is the textbook situation. The result is the induced drag decreases. Since area and lift coefficient remain constant the viscous drag generally does not change significantly.

However in many situations the lift is constant and the span is constant. Increasing aspect ratio is accomplished by decreases in chord and area. The lift coefficient (lift/area) increases as the area is decreased. In this situation the induced drag remains essentially constant. While the decrease in area may result in a decrease in viscous drag the viscous drag coefficient will increase as the lift coefficient increases. The net result may be a decrease or in increase in viscous drag.

If lift is constant and span is constant then induced drag is independent (to first order) of area, chord and aspect ratio. In this situation aspect ratio, chord and area should be selected on the basis of other effects such as viscous drag, not on induced drag.

Induced drag, lift aspect ratio, span, and area were discussed at length in the http://www.boatdesign.net/forums/hydrodynamics-aerodynamics/myth-aspect-ratio-36836.html thread.

 

I think it's most convenient to think of aspect ratio as the nondimensional version of span (squared). AR = span^2/Area instead of AR = span/mean_chord.

In most design problems, the quantity that has to be kept constant is the lift, in metric terms. The lift typically has to equal the weight, or the side force from a sail rig. When compared on the basis of equal lift, induced drag is inversely proportional to span^2 and also inversely proportional to velocity^2. Skin friction is the biggest contributor to parasite drag in a well-designed craft, and it is proportional to wetted area and proportional to speed^2. The designer then endeavors to cut the parasite drag by reducing the area, and cut the induced drag by increasing the span, until other design considerations come into play. The result is high aspect ratio.

The reduction in induced drag with speed is something that isn't at all apparent when looking at the nondimensional coefficients. At low speed, it's most important to attack the induced drag. At high speed, it's most important to attack the parasite drag.

In aircraft, one of the best predictors of good lift/drag ratio is the wetted aspect ratio: ARwet = span^2/total_wetted_area. This is a figure of merit that takes into account the biggest factors associated with two different sources of drag. The induced drag is inversely proportional to span^2, so 1/induced drag is proportional to span^2. The skin friction is proportional to wetted area, so 1/parasite_drag goes as 1/ wetted_area. Multiply these together to make a composite metric, and you get span^2/wetted_area. Increase the metric and you're probably reducing the drag. The beauty of wetted aspect ratio is it includes things like fuselages, engine nacelles and pylons, and tail surfaces that aren't included in the wing planform area. On a boat, it would include the wetted area of the hull, bulb and rudder as well as the keel itself.

Dan Raymer illustrates this by comparing the Avro Vulcan with the Boeing B-47: http://www.homebuiltairplanes.com/f...rry-blumenthals-concepts-raymer_adaca_p24.jpg. The B-47 had three times the aspect ratio of the Vulcan, but the maximum lift/drag ratio of the Vulcan is just as good. The reason is the B-47 had a lot of extraneous wetted area compared to the Vulcan, which was almost a flying wing. Their wetted aspect ratios are very similar, and so their L/D is similar.

 

I'm sure this is all highly confusing for the average person. Is anybody aware of a "How to design and optimise a centreboard / keel" article? If not I'll put it onto my to do list.

 

For global drag "calculation" ie: A-Cat hydrofoil, I use to consider minimum global drag is achieved for 1 design speed when Induced drag = friction drag.

Around this design speed it's no more the optimum but not that far.

Is it a correct approach?

Regards

EK

 

Incorrect.

Minimum total drag is achieved for a design speed when the total drag does not change with small changes in the design parameters. This may be a local or a global minimum.

If total drag = induced drag + frictional drag (wave drag, etc is neglected or ignored)

change in total drag = 0 for small changes in design parameters

change in induced drag + change in frictional drag = 0 for small changes in design parameters

change in induced drag = - change in frictional drag for small changes in design parameters

This is true assuming the assumption that all forms drag other than induced drag and frictional drag can be neglected.

 

Put another way, the minimum drag occurs when the SLOPE of the induced drag is equal and opposite to the slope of the friction drag on a "drag component" vs "speed" plot. At a constant load, induced drag decreases about as speed^2 and viscous drag increases about as speed^2. Given a big design domain, what you said may also be true. But more often than not, rule constraints or structural constraints or other practical matters force you into a corner where you can't get at the optimal answer so you have to compromise.

The other issue is gnarly. I basically consider the daggerboard's induced drag to be an aero force, not a hydro force. When fiddling with rigs and boards, I try to optimize the overall aero performance which includes hydro induce drag. So I sum the entire hull's viscous drag and balance that against the total aero force. Your ratio of the board's viscous drag to its induced drag never directly enters into my calculations. I think the actual coupling between the two is fairly weak.

 

OK I have put together a quick spreadsheet that calculates required lift and resulting drag if the righting moment, boat speed and a few geometric and foil parameters are known.

The only thing to optimise is Cl / Cd and the span of the foil. Cl / Cd will depend on the chosen profile and operating point. The foil span will be limited by practical and structural considerations.

I also included some data for 9% thickness foils. An ordinary NACA 0009 a "mild" laminar profile and an "extreme" laminar profile. As can be seen the maximum Cl / Cd is actually not very close to the laminar bump at low lift coefficients. The profile will only operate in the "laminar area" at low lifts and high speeds (downwind reaching). Therefore a laminar profile only has advantages under these conditions. Upwind, where speeds are low and lift has to be high, the ordinary NACA 0009 has the best performance with Cl / Cd of 92.7. The extreme laminar profile only has a Cl / Cd of 62.5.

The only difference between a symmetrical foil and an asymmetrical foil is the Cl over Cd curve.

The leeway angle can be read from the Cl over angle of attack curve.

Hope I didn't make too many mistakes. Have fun playing with the numbers.

 

Nicely put.

 

I have mentioned my use of the NACA 0009 on my custom windsurfer fin in other threads. If it is sized correctly the 0009 allows quicker acceleration from tacks and turns.

 

Thanks for the precision DCockey,

But is it antinomic ?
Induced drag= f(1/V^2) and friction drag= g(V^2)
When curves cross (Induced=friction), their respectives tangents are around +/-45° and little change in one drag is offseted by the opposite change in the other drag.

Of course under assumption of constant lift and neglecting other form of drag

Cheers

EK

 

Thanks PhilSweet for sharing your methodology,

For optimization, I was wondering wether considering the foil in isolation was a good approach compared to a more global approach I suspect could be more relevant, but more difficult to define smartly (at least for me)

Thanks

EK

 

If lift is constant, Induced drag is proportional to 1/V^2 and friction drag is proportional to V^2, then the V at which the sum of induced drag and friction drag will be minimum will be the speed at which induced drag = friction drag. This is the simplified version of determining the minimum drag cruise speed for an aircraft where lift = weight at all speeds or the aircraft falls out of the sky.

The assumption that friction drag is proportional to V^2 is valid for airfoil sections if the angle of attack and lift coefficient are constant. However if lift is kept constant and speed varies then a reasonable approximation for frictional drag coefficient of an airfoil is:
Cdfric = c0 + c1 * Cl^2 where c0 and c1 are constants which depend on the airfoil section and Cl is the lift coefficient.
If lift is constant then Cl will be proportional to 1/V^2. Note that the Cl^2 term in the expression above is not related to induced drag, and occurs airfoils with two-dimensional flow, ie effective infinite span.

With this improved model for friction drag the speed for minimum total drag with constant lift no longer occurs when friction drag = induced drag.

 

Depending on what you are trying to optimize, the two types of drag may be different by very large amounts.

A different problem, relevant to this thread, would be determining how far to raise or lower a daggerboard under given conditions. If we simplify the problem by assuming a board with constant chord and use the approximations (given the lift & speed) that friction drag = f(span) and induced drag = f(1/span^2), then it works out that total drag is minimized when the friction drag is twice the induced drag.

This result is also relevant to the questions of how high to fly in a surface-piercing foiler and how large to make the foils in a foiler with submerged foils.

 

Thank you David & Doug,

in the proxy equation of friction drag is c0 the section drag coef at 0 lift ?

Cheers

EK

 

Yes, c0 is the section Cd at zero lift.

Note that for asymmetric sections zero lift may not occur at the defined geometric zero angle of attack.