The Myth of Aspect Ratio

Discussion in 'Hydrodynamics and Aerodynamics' started by DCockey, Feb 20, 2011.

  1. DCockey
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    DCockey Senior Member

    When the design of keels and rudders is discussed statements are sometimes made which imply that aspect ratio is a critical parameter when consider lift and drag, and that higher aspect ratios are better.
    “The aspect ratio is the most important ratio for the lift and drag of a wing.”
    “the higher aspect ratio rudder is more efficient (i.e. less drag for the same amount of turning input)”
    But is this generally the case for keels and rudders on boats?

    The very short answer: Draft of the keel or rudder is critical for induced drag, not the aspect ratio. (Added to avoid confusion) Area and therefore aspect ratio are important when total drag including viscous drag, not only induced drag, is considered. Now for the longer answer:

    For the purposes of this discussion I’ll be talking about wings but remember that a keel or rudder can be considered as a half-wing moving through the water, with the other half of the wing “mirrored” by the hull. First, some definitions:
    Induced Drag: Drag increment of a wing due to the longitudinal vorticity shed into the wake as a result of a finite wing producing lift.
    S : Span, for a wing this is the distance from wing tip to wing tip. For a keel or rudder it is approximately twice the distance the keel or rudder extends under the hull.
    A : Planform Area
    CL = Lift /[ (0.5 * Density of Water * Speed ^ 2) * A] : Lift Coefficient
    CD = Drag / (0.5 * Density of Water * Speed ^ 2) * A] : Drag Coefficient
    CDI = Induced Drag / (0.5 * Density of Water * Speed ^ 2) * A] : Induced Drag Coefficient
    AR = S ^ 2 / A : Aspect ratio, the ratio of the span to the average chord.

    So where does the common belief that high aspect ratio wings produce less induced drag come from? It probably starts with plots of CL vs CD or CDI for wings of various AR’s which show larger CL and smaller CD/CDI as AR increases. Or perhaps from this equation from wing theory agrees with the plots:
    CDI = CL ^ 2 / (Pi * AR) * (1 + Sigma)
    Sigma = 0 for an elliptical load distribution, and greater than 0 for other load distribution shapes. It is not a function of aspect ratio.

    It’s obvious based on the plots mentioned above or this equation that the higher the AR the lower CDI is for a given CL. But what does this mean for a boat?

    How fast a boat sails depends on the physical lift and drag and the size of the boat, not on non-dimensional coefficients. The amount of lift which a keel or rudder needs to provide is determined by what is required to offset the force of the sails and/or steer the boat. Using the definitions of CL, CDI and AR in the equation above and simplifying results in:

    Induced Drag = 1 / [Pi * (0.5 * Density of Water * Speed ^ 2)] *[ (Lift / Span) ^ 2] * (1 + Sigma)

    AR, Aspect Ratio, vanishes from the equation! For a given Lift and load distribution the Induced Drag is proportional to the Span squared only!

    So are the statements that Induced Drag decreases as Aspect Ratio increases fundamentally wrong? No, as long as the Planform Area, A, is held constant and the Aspect Ratio is increased by increasing the span and decreasing the chord. But usually the depth of a keel or rudder is restricted for a boat for a variety of reasons:
    - Draft limitations due to the waters then boat will sail in, storage limitations, etc.
    - Required height of rudder above the bottom of the keel
    - Bending moment at the root of the rudder or keel. Bending moment for a given lift increases as depth increased.
    - Measurement rules.

    To decrease Induced Drag increase the depth of the rudder and/or keel to the maximum possible. Then select the chord (and implicitly AR), planform shape, based on other considerations: maximum lift required, drag due to area, section lift/drag characteristics, planform and tip shape effect on induced drag, maximum lift required, thickness needed to meet structural considerations and space for ballast, etc.

    --------------------------------------------

    I’m not the first on this forum to make the point about span, not aspect ratio being the critical parameter for induced drag. Tom Speer discussed it in 2002; see post #10 of http://www.boatdesign.net/forums/sailboats/tandem-keel-1058.html Note that his notation differs from what I used above. In particular S is used for planview area, not span.
     
    Last edited: Feb 23, 2011
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  2. gonzo
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    gonzo Senior Member

    They usually compare a keel to an airplane wing. Fast airplanes have delta wings. They are similar to early 20th century keels.
     
  3. DCockey
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    DCockey Senior Member

    Not sure who you are refering to?
     
  4. gonzo
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    gonzo Senior Member

    Fast planes have low aspect wings.
     
  5. michael pierzga
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    michael pierzga Senior Member

    " But usually the depth of a keel or rudder is restricted for a boat for a variety of reasons:
    - Draft limitations due to the waters then boat will sail in, storage limitations, etc.
    - Required height of rudder above the bottom of the keel
    - Bending moment at the root of the rudder or keel. Bending moment for a given lift increases as depth increased.
    - Measurement rules. "


    I dont understand ? What happened to righting moment and displacement ?

    A deep fin capped with a lead bulb generates the required righting moment with the least lead...hence lighter overall displacement

    You cant just talk about generating lateral resistance when considering keel profiles.

    This holds true with rudders...they have to be in the water to work..a single rudder will be deep, high aspect, twin rudders will be shallow, to generate the same bite.
     
  6. DCockey
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    DCockey Senior Member

    Certainly more to keel profiles than lateral resistance. Didn't claim otherwise. Space for ballast is a consideration.

    I don't think I said anything which excludes a fin with a lead bulb. But if a design uses a fin with a lead bulb the assumption should not be made that for a given draft a fin with a smaller chord and therefore higher aspect ratio will have lower induced drag.

    Yes, rudders need to be in the water to work. Didn't say anything about them not being.

    I am not advocating shallower draft for keels or rudders. What I am saying is that for a given draft, the induced drag is the same irregardless of the width of the rudder or keel.
     
  7. gggGuest
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    gggGuest ...

    Well yes, but the various other forms of drag vary of course.

    Its a true enough statement, but I'm curious what the point of making it is. Its hard to think anyone who takes a significant interest in design won't know.
     
  8. DCockey
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    DCockey Senior Member

    Discussions of induced drag, from keels and rudders, both in books and on this forum, frequently focus on aspect ratio and coefficients, not the actual lift and drag and draft of the keels or rudders. This can easily lead to the mistaken assumption that reducing chord will decrease induced drag. Chord should be selected based on other considerations. That's the reason. It's great to hear that it is well known.
     
  9. Jenny Giles
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    Jenny Giles Perpetual Student

    Many also have sharpish leading edges.

    Speaking of which...
    Does leading edge suction on thin keels play much part in the total drag or do they usually have a very rounded leading edge so it doesn't come into play?

    TIA
     
  10. DCockey
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    DCockey Senior Member

    A delta wing is generally selected because the aircraft flies at supersonic speeds. Sharp leading edges reduce drag at supersonic speeds due to shock waves. Not a consideration for boat keels.
    Depends on the profile and angle of attack the keel sees. If the flow separates from the leading edge then the drag increases and lift decreases. If the flow doesn't reattach the keel is stalled, drag increases a lot and lift drops. If performance is a consideration when designing the boat the keel section should be selected so that separation doesn't normally occur.
     
  11. Jenny Giles
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    Jenny Giles Perpetual Student

    Thanks for answering.

    I was given some code to calculate induced drag and leading-edge suction by a member of this forum (thanks, Leo!!!!). He said it only works for thin flat wings.

    Do you know if there is a way to interpolate the suction between a thin wing where there is a direct relationship between induced drag and suction, and a wing with a rounded leading edge?

    sorry to keep bothering.
     
  12. gonzo
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    gonzo Senior Member

    The reason I posted about Delta wings, is that airplane wings keep on being used as examples to justify keel designs.
     
  13. Perm Stress
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    Perm Stress Senior Member

    My applause!

    Simple and straight explanation.
     
  14. Perm Stress
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    Perm Stress Senior Member

    As usual in discussions on single design parameter, "all the rest is assumed to be the same".
     

  15. DCockey
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    DCockey Senior Member

    Physical world discussion:

    "Leading-edge suction" generally refers to the the low pressure peak at the leading edge of an airfoil when the airfoil is at an angle of attack. Induced drag is caused by the finite span of a wing which causes a cross flow and the leading edge and trailing vorticity in the wake. They are fundamentally different and not directly related. Consider two wings with different symmetric sections, one with a "sharp" leading edge and the other with a very rounded leading edge. The wings have the same planform shape and are at the same angle of attack. The overall lift generated, the span-wise distribution of the lift, and the resulting induced drag will be the same for both wings. But the pressure difference distributions on the wing surfaces will be different. The wing with the sharper leading edge will have a larger negative pressure peak near the leading edge than a the one with a well rounded leading edge. This is due to the higher speed of the flow around the sharper edge.

    So the relationship between "leading-edge suction" and induced drag is indirect. Both are connected with the production of lift, but arise due to different mechanisms.
     
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