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

    Could you expand on the relationship between Reynold's Number and induced drag?
    Any reference? I haven't encountered that as long as the Reynold's number is reasonably high. Could this be a difference in the definition of induced drag between naval architects and aerodynamicists?
     
  2. DCockey
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    DCockey Senior Member

    I don't see any disagreement between the observation that induced drag is better thought of in terms of lift and span/draft and therefore chord shouldn't be selected based on induced drag considerations, and your observations that frequently a larger chord relative to span is better. They seem to be in alignment.
     
  3. Jenny Giles
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    Jenny Giles Perpetual Student

    I understand that. Thanks.

    We are looking at some simple calculations of induced drag on thin wings at small angles where the force perpendicular to the wing must be balanced by the drag and the suction, so we use
    suction = lift X sin(aoa) - induced drag
    and for small aoa,
    CL/aoa = CS/aoa^2 + Cdi/aoa^2
    so the drag and suction are related but i suppose we use a bit of a trick to avoid calculating the induced drag directly.

    Great discussion so far.
     
  4. DCockey
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    DCockey Senior Member

    Usually it's the other way around. Induced drag can be calculated directly by a Treffitz plane analysis, and then the leading edge suction is arrived at by the equation you give.
     
  5. Jenny Giles
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    Jenny Giles Perpetual Student

    Yes, that is the standard way but LE suction is an interesting line of attack for splash problems and planing which is why i was asked to look into it.

    thanks for all your help and advice!
     
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  6. daiquiri
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    daiquiri Engineering and Design

    I presume you didn't want to use the word "belief". It is not a belief. It is a fact. High aspect ratio wings DO produce less total drag than the low aspect-ratio ones, for a fixed total lift.

    At the end, that's what you have said in your opening post - keeping a constant area and increasing the span equals to increasing the aspect ratio.
    I will add that if you keep a constant span (again, for a fixed lift) and vary the area, you again end up with the inverse proportionality between the total drag and the AR. So, having as high AR as compatible with other constraints is beneficial in terms of total drag.

    However, that's just the aero/hydrodynamics part of the problem. Then you have other structural and operational constrains you have to consider (you've mentioned some), usually so important that they become main players when determining span, thickness, planform etc.

    It is similar to the process which leads to the choice of the optimum hullform. If you limit yourself to just the minimum drag in flat water, you end up with a very slender, canoe-like shape (as has been discussed countless times here). But then you have to consider other things - like seakeeping, maneuverability, stability, interior layout, rules requirements, passenger comfort, volume for cargo holding, port loading/unloading operations etc. - which at the end produce a practical optimum hull very different from a theoretical one based on just speed and displacement.

    So, I really fail to see where's a myth here. It's just one of many design aspects to be considered.

    Cheers!
     
  7. daiquiri
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    daiquiri Engineering and Design

    I just saw that you were talking about the induced drag only. I was refering to the total drag, which is of major interest in design. My mistake.
     
  8. DCockey
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    DCockey Senior Member

    Only if the increase in AR is accompanied by an increase in span and/or decrease in lift. This is where the myth comes from.

    If you keep a constant span and fixed lift coefficient, and vary the area and therefore change lift and AR, the induced drag changes.

    However:

    If you keep a constant span and fixed lift, and vary the area and therefore AR, the induced drag stays the same irrespective of the AR.

    (See first post for derivation of these.)

    The "myth" I was talking about is confusion between these two facts. Dicsussions of induced drag almost always use lift and drag coefficients and the non-dimensional AR instead of a dimensional wing size, and reach the conclusion is that a higher AR reduces induced drag. Then some folks assume that reducing area while keeping span constant will reduce induced drag, which is only true if the lift is also reduced.

    Perhaps my first paragraph caused some confusion. The myth I was refering to was about induced drag, not total drag. I added a clarification to my original post.

    There is some AR and therefore area where total drag (induced drag + viscous drag) will be a minumum for a given lift and span. Since the induced drag is independent of AR and area for a given lift and span (assuming similar planview shapes) the optimum drag will depend on how the viscous drag changes with AR and area. As a first approximation it will be with a AR and area which results in the airfoil sections being at the lift coefficient which corresponds to the optimum 2-D CL/CD for the section. As lift of the wing/keel/rudder increases the optimum AR and area will also increase. However other considerations may keep this AR for minimum total drag from being desirable or even feasible.

    I agree. AR and area should not be selected on the basis of induced drag alone, and if the span is at the limit then induced drag is not a factor unless planview shape is changed.
     
  9. DCockey
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    DCockey Senior Member

    I was busy with my reply and didn't see your note. But I appreciated your earlier note because I realized that I was not clear enough that I was talking about induced drag only.
     
  10. daiquiri
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    daiquiri Engineering and Design

    We agree then. The attached xls file displays graphically what happens to both the induced and the total drag when AR is increased. For those who prefer graphs and numbers to equations.
    Fixed values:
    Lift = 1000 N
    Span = 1 m
    Speed = 5 m/s

    Cheers
     

    Attached Files:

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  11. DCockey
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    DCockey Senior Member

    Thanks for the plot!
     
  12. Ad Hoc
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    Ad Hoc Naval Architect

    I think what Mike is referring to, correct me if i am wrong Mike, is that the induced drag is usually (from most text books) calculated as:

    D (induced) = 2/(rho.pi.b^2).[L/V]^2

    Where V is the velocity. Rn is proportional to V, ergo induced drag is influenced by Rn.

    But if you non-dimensionalise into a coeff, it becomes Cl^2/(pi.AR)

    AR being aspect ratio.

    But this all assume constant sections anyway and constant circulation around the hydrofoil.
     
  13. DCockey
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    DCockey Senior Member

    If L is lift and b is span then that's the same equation I had, just written slightly differently.

    I wouldn't agree that anything which is a function of V can be regarded as function of Rn even though it's not function of viscosity. Change the chord while keeping V and L constant; Rn changes but induced drag doesn't. Change the kinematic viscosity but keep the density constant; Rn changes but induced drag doesn't.

    The circulation is proportional to the lift, it's only constant if lift is constant.
    These equations hold independent of section as long as the spanwise lift distribution is similar. For un-cambered, symmetric sections and no twist (true for all most all keels and rudders) the section profile has no effect on induced lift.

    I have a guess at where the idea in naval architecture that induced drag is a function of Reynold's Number comes from, but I'll put it in another post.
     
  14. Ad Hoc
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    Ad Hoc Naval Architect

    Well, I'm not sure how you square the circle of scaling then?...assuming you are not using a scale of 1:1 for your testing.

    If you change the chord, one assumes the tip chord is also changed. Since at the tips the +P (under the foil) and -P (above) pressure short circuits so to speak and spills over causing the tip vortex. It is well documented that the longer the tip the greater the loss of lift owing to the reduction in +P. Hence the shortest tip foil with greatest span, for a given area is more efficient than one with a longer tip, or chord tip. Or do you not agree with this theorem?

    I'm a naval architect not a aerodynamicist. It has been years since i delved into hydrofoil theory, i rare design with foils like this, except for rudders. My memory is not so good and requires "updates". But that is much more simple. I'd be interested in what you have to say..and Daiquiri too, a real heavy weight in Aerodynamics.
     

  15. DCockey
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    DCockey Senior Member

    Not sure what circle needs squaring. Induced drag scales with dynamic pressure and linear size squared.

    The induced drag I'm refering to is the drag due trailing longitudinal vorticity, and is an invicid phenomena. There is much more to drag of a wing/keel/rudder than just induced drag.

    What I've presented assumes the spanwise lift distribution remains constant as AR is varied. Differences in spanwise lift distribution are accounted for by Sigma which will be discussed in a moment. The strength and distribution of the trailing vorticity including the tip vortex is directly tied to the spanwise lift distribution. The theory accounts for it. (Simplistic presentations of induced drag mention only the concentrated tip vortex but any finite span wing producing lift will have induced drag, whether a concentrated tip vortex is present or not.)

    As long as changes in AR occur such that the chord length distribution along the span remains the same and the quarter chord sweep angle stays the same, the spanwise lift distribution stays reasonably the same for AR > 2 or so. Chord length distribution remaining the same means if the chord distribution is elliptical it remains elliptical, if it is a staight 4:3 taper then it remains a straight 4:3 taper, etc. Remember that a 1 meter wide keel projecting 1 meter below the hull has an effective AR or 2 or more.

    Differences in spanwise load distribution (including resulting tip vortex strength) are accounted for by the factor Sigma in the equation relating induced drag to lift:
    Induced Drag = 1 / [Pi * (0.5 * Density of Water * Speed ^ 2)] *[ (Lift / Span) ^ 2] * (1 + Sigma)
    Sigma depends on the shape of the spanwise lift distribution but is independent of aspect ratio and lift magnitude. Change the planview shape and Sigma changes. Sigma = 0 for an elliptical load distribution, and greater than 0 for other load distribution shapes. Principals of Yacht Design, 3rd Edition has a graph, Fig 6.9 on p 110, which shows the increase in induced drag (Sigma) for taper ratios from triangular (0.0) to rectangular (1.0). For AR of 1 it barely gets over 1%, for AR of 6 it gets up to 7%. The minimum Sigma occurs for taper ratios slightly less than 0.5 and is less than 2%. (There is some confusion in PYD about the definition of AR. The definition is stated as AR = effective keel depth / chord. This would result in an AR which is 1/2 of the a corresponding wing given the mirror effect of the hull. But the equations shown use the wing definition of AR. I don't know which is applicable for the chart mentioned.) Kuethe and Chow, 1976, show a Sigma value of 4.6% for a rectangular wing at an AR of 6. So the effects of planform shape are not major as long as it's a reasonable planform and may not be significant.

    It's been over three decades since I studied wing theory in any detail, so this has been a good (re-) learning experience for me.
     
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