Hull Asymmetry and Minimum Wave Drag

Discussion in 'Hydrodynamics and Aerodynamics' started by DCockey, May 28, 2011.

  1. Leo Lazauskas
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    Leo Lazauskas Senior Member

    Richard,
    To help your ruminations, here's a paper where the assumption of linearity
    seems quite well confirmed by experiments.
     
    Last edited: Aug 12, 2015
  2. Richard Pitblad
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    Richard Pitblad Richard Pitbladdo

    Thanks for the references, it's always nice to gather more material, but they don't quite address the point I am trying to make (maybe if those catamaran hulls were spaced closer together rather than outside the Kelvin angle).

    First, while a slight bit off topic, let me give some easy counterexamples on superposition. First, assume a "trimaran" with slender hull, plus full length amas each spaced, say, 15% of hull length athwarship of the center hull. Further assume the amas are very deep and infinitely thin, so they have no sources or sinks. Clearly, the way to model this situation is to assume perfect reflection of the waves off the amas back to the center hull. Superposition is violated and wave resistance is affected, not by vortices or nonlinearities or any mistakes in the mathematical derivations, but by appropriate modeling of boundary conditions (relecting barrier). If superposition is violated with very deep and very narrow amas, there is no reason to think it is not also violated with less extreme ama shapes.

    One could take this example further and form a horizontal thin plate beneath the hull connecting the side fences, making a box. If appropriately shallow, supercritical flow could be induced, and zero wave drag could be created along the lines of a work I'm sure you are familiar with [XUE-NONG CHEN and SOM DEO SHARMA (1997). "Zero wave resistance for ships moving in shallow channels at supercritical speeds." Journal of Fluid Mechanics 335].

    Getting back to topic, let me consider a design problem of optimizing the deep-water shape of the hull from midships to stern, taking the shape of the hull from bow to midships as a given. To keep things simple, assume the front half is slender, with high prismatic cx, and the design speed corresponds to one wavelength bow to stern. We can safely ignore any feedback of the shape of the stern half on the wave resistance exerted on the bow half.

    Now consider the wake pattern of the bow in isolation, which would be a hump-speed pattern since the bow is a half wavelength. In order to minimize wave resistance, the stern half needs to capture as much energy of the bow's wake pattern as possible, by "surfing" down the bow wave. Using an energy analysis, all that matters for the design of the stern is the shape of the bow wave during that half wavelength. It is clear that the stern half ought to reach out as far as the bow wave hollow has spread at midships, in order to "catch" the entire breadth of the bow wave.

    This is what made me very suspicious of the symmetry argument, although I realize I have not disproved that resistance is the same going forward and backward. It could simply be that wider is better in general for wave resistance (and we know what that does for viscous resistance). The other reason that makes me suspicious is to observe the trend toward asymmetry of bows and sterns in those displacement designs where the greatest degree of design and experimental resources have been applied, namely cruise ships, cargo ships, and large military vessels.

    One more thing for now. Assume for the sake of argument that wave resistance is the same going forward and backward. That does not mean in any way that optimal hull designs must be fore-aft symmetric, but would only imply that a nonsymmetric hull can only be optimal among multiple optima which would necessary include its mirror image.
     
  3. DCockey
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    DCockey Senior Member

    There appears to be some fundamental confusion about the use of "superpostion" and it's relationship to the solution of flow around arbitray bodies using the linear approximations.

    Superpostion of the individual solutions for flow around three bodies is not the same as the solution for the flow around the three bodies together. The zero-normal velocity boundary condition will not be satisfied on each body by the combined solution.

    Simple example for flow without a free surface. The potential flow around a sphere in a uniform stream is given by a doublet plus the free stream located at the center of the sphere with the strength of the doublet determined by the radius of the sphere and the free stream velocity. The zero normal boundary condition will be satisfied at all points on the surface of the sphere. However the flow around three adjacent spheres is not given by three doublets plus the uniform flow. The zero normal velocity boundary condition will not be satisfied on the surface of the spheres (other than perhaps at a few coincidental locations). In principal, depending on the strength of the doublets and their locations, one, two or three surfaces could be found with zero normal velocity. However these surfaces will not be spherical.

    Another way to model the flow past a sphere or other body is to distribute singularities (sources, doublets, vortices) of unknown strength over the surface of the body, and then determine the strength of the singularities which satisfy the zero normal velocity boundary condition on the surface of the body. In analytical form this involves the solution of an integral equation. Panel methods are examples of numerical methods used to solve the integral equations.

    For the flow past three adjacent spheres or other bodies singularities of unknown strength are distributed over the surface of the bodies, and then the strength of the singularities of the three bodies are determined which satisfy the zero normal velocity boundary conditons on the surfaces of the three bodies.
     
    Last edited: Feb 18, 2012
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  4. groper
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    groper Senior Member

    This is what i was getting at earlier Leo ^^^. Like i said, even Michlet predicts the least resistance for an assymetric shape - better than anything symmetrical godzilla could come up with... The reason is described above ^^^, the stern half of the hull needs to form an effective sink for the bows source...
     
  5. DCockey
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    DCockey Senior Member

    I need to add that depending on the size and arrangement of the bodies the superposition solution may be "close enough" for some purposes.
     
  6. DCockey
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    DCockey Senior Member

    Leo, could you expand on the last statement above. In particular what would superpostion be allowable for?
     
  7. DCockey
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    DCockey Senior Member

    Let's look at some of the assumptions associated with the statements that in a linearized, free surface potential flow analysis the resistance is the same going forward or backwards:

    1) Resistance consists of wave resistance only.

    2) Viscous effects including boundary layes are not included, irregardless of direction of motion.

    3) The immersed portion of the hull is fixed relative to the nominal free surface, and does not change when the direction of motion is reversed.

    4) Linear free surface approximation.

    ----------

    Now let's look at what happens to a vessel in real water:

    1) Resistance includes wave resistance and changes in resistance due to non-zero viscosity.

    2) Viscous effects are included, and dependent on the shape of the hull relative to the direction of motion. The change of trim angle will depend in part on the shape of the hull. The difference in viscous effects will affect the wave resistance as well as other forms of resistance. Unless the hull is symmetric fore/aft the viscous effects and resulting changes in resistance will be different depending on the direction of motion.

    3) Unless the vessel is externally constrained, there will be a change in draft and trim angle. These changes will affect the viscous effects, wave resistance and other forms of resistance. The immersed portion of a symmetric hull will be assymetric due to the change in trim angle. A hull with an immersed portion which is symmetric when underway in one direction will be assymetric when at rest or underway in the opposite direction.

    4) Free surface is non-linear.

    ---------

    So why might the "best" hull shape for resistance be assymetric fore/aft even if the lowest wave resistance?

    - Viscous effects depend on direction of motion

    - Change in draft and trim angle will depend on the shape of the hull. An assymetric hull may have a more diesirable change in draft and trim angle.

    - Free surface is non-linear. However the magnitude of the differences due to non-linear vs linear free surface compared to the differences due to viscous effects and draft and trim will depend on hull shape.
     
    Last edited: Feb 19, 2012
  8. DCockey
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    DCockey Senior Member


    Richard, any suggestions of where I could learn more about the concept of "filling out the Kelvin angle" and drag reduction?
     
  9. philSweet
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    philSweet Senior Member

    If you take a fixed bow section and optimize the stern for an arbitrary velocity, you don't end up with a symetrical hull. But if you optimize both the bow and the stern, you do. This idea of chasing the radiating wave pattern with the stern is not as effective as modifying the bow to produce a pattern more amenable to cancellation by the stern. Am I missing something here?
     
  10. DCockey
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    DCockey Senior Member

    Supposition or based on ? Inviscid theory or physical?
     
  11. Leo Lazauskas
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    Leo Lazauskas Senior Member

    Richard wrote...

    Thanks for the references, it's always nice to gather more material,
    but they don't quite address the point I am trying to make (maybe
    if those catamaran hulls were spaced closer together rather than
    outside the Kelvin angle).


    MY REPLY
    Remember that for catamarans there might not be an optimal spacing
    in thin-ship theory for some Froude numbers.
    For those cases, the minimum wave resistance occurs for infinitely-spaced
    demihulls, i.e. two separate ships.


    First, while a slight bit off topic, let me give some easy
    counterexamples on superposition. First, assume a "trimaran" with
    slender hull, plus full length amas each spaced, say, 15% of hull
    length athwarship of the center hull. Further assume the amas are
    very deep and infinitely thin, so they have no sources or sinks.
    Clearly, the way to model this situation is to assume perfect
    reflection of the waves off the amas back to the center hull.
    Superposition is violated and wave resistance is affected, not by
    vortices or nonlinearities or any mistakes in the mathematical
    derivations, but by appropriate modeling of boundary conditions
    (relecting barrier). If superposition is violated with very deep
    and very narrow amas, there is no reason to think it is not also
    violated with less extreme ama shapes.


    MY REPLY
    Super-position isn't violated. The boundary conditions are just
    not valid.

    I take your point that the theory does not include reflections,
    and they might be important for closely-spaced demihulls.
    If you are going to include those sort of (probably 2nd order) effects,
    maybe you should also include other effects, such as the welling-up of
    water between the demihulls, bores, and, possibly, solitons.



    One could take this example further and form a horizontal thin
    plate beneath the hull connecting the side fences, making a box.
    If appropriately shallow, supercritical flow could be induced, and
    zero wave drag could be created along the lines of a work I'm sure
    you are familiar with [XUE-NONG CHEN and SOM DEO SHARMA (1997).
    "Zero wave resistance for ships moving in shallow channels at
    supercritical speeds." Journal of Fluid Mechanics 335].


    MY REPLY
    Yes, those "super-conducting" vessels with S-shaped centreplanes
    travelling in a finite width, finite-depth channel are fascinating.



    Getting back to topic, let me consider a design problem of
    optimizing the deep-water shape of the hull from midships to stern,
    making the shape of the hull from bow to midships as a given.
    To keep things simple, assume the front half is slender, with
    high prismatic cx, and the design speed corresponds to one
    wavelength bow to stern. We can safely ignore any feedback of the
    shape of the stern half on the wave resistance exerted on the bow
    half.

    Now consider the wake pattern of the bow in isolation, which would
    be a hump-speed pattern since the bow is a half wavelength.


    MY REPLY
    Plus a huge wave created by the transom stern you have created, if
    it is considered to be fully-wetted.

    Or, plus a large component of "drag" due to the missing hydrostatic
    pressure on the transom stern if it is considered to be running
    fully dry.

    Or, you could model it with an infinite hollow trailing behind the
    stern.



    In order to minimize wave resistance, the stern half needs to
    capture as much energy of the bow's wake pattern as possible, by
    "surfing" down the bow wave.


    MY REPLY
    Is that the same as the definition of wave resistance, which is
    the energy left in the far-field wave pattern?

    There is a good derivation of wave resistance using energy radiation
    arguments in Newman's book, "Marine Hydrodynamics".
    Essentially, the wave resistance is an integral from -pi/2 < theta < pi/2
    of a function which is the square of the complex amplitude multiplied
    by a [cos(theta)]^3 factor.


    It is clear that the stern half
    ought to reach out as far as the bow wave hollow has spread at
    midships, in order to "catch" the entire breadth of the bow wave.


    MY REPLY
    I don't understand what this means.



    This is what made me very suspicious of the symmetry argument,
    although I realize I have not disproved that resistance is the
    same going forward and backward.


    MY REPLY
    But fore-aft symmetry is necessary, not sufficient.



    It could simply be that wider
    is better in general for wave resistance (and we know what that
    does for viscous resistance). The other reason that makes me
    suspicious is to observe the trend toward asymmetry of bows and
    sterns in those displacement designs where the greatest degree
    of design and experimental resources have been applied, namely
    cruise ships, cargo ships, and large military vessels.


    MY REPLY
    But those ships nearly always have transom sterns.



    One more thing for now. Assume for the sake of argument that wave
    resistance is the same going forward and backward. That does not
    mean in any way that optimal hull designs must be fore-aft symmetric...


    My REPLY
    It does in thin-ship theory: fore-aft symmetry is necessary.


    Whew!
    Leo.
     
  12. Leo Lazauskas
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    Leo Lazauskas Senior Member

    Sorry, I still don't understand what you are getting at.
    Could you answer my earlier question. Or did I miss it in
    this tangled thread we have woven :)
     
  13. Leo Lazauskas
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    Leo Lazauskas Senior Member

    It is worth pointing out that Michell's thin-ship theory simplifies this by
    putting all the sources and sinks on the centreplane.
    The strength of the sources is
    sigma = 2U dY(x,z)/dx
    where U is ship speed and dY(x,z)/dx is the longitudinal slope of the hull at
    the point (x,z).
     
  14. Leo Lazauskas
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    Leo Lazauskas Senior Member

    The last statement just repeats the earlier assertion that if waves are small,
    you can linearise the free surface. It doesn't matter whether the waves are
    created by sources, vortices or dipoles.
     

  15. Leo Lazauskas
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    Leo Lazauskas Senior Member

    There are, of course, many other complications.

    The "sinkage" of the hull is mainly due to the mean lowering of the surface
    around the hull, and if the hull also trims in sympathy with the surrounding
    water surface, it will create waves very similar to what it would in its
    static attitude with an initially flat surface.

    When a hull squats, the water around it also changes shape.
    It can be argued that in equilibrium, the hull in the "dish" shape it has
    created has roughly the same wetted area it had at rest.

    In thin-ship theory, there also issues to do with "consistency as Tuck pointed
    out in a discussion of a paper by Doctors.
    Essentially, squat is 2nd-order in thinness, and if you start including 2nd-order
    effects it might be wise to start including other 2nd-order effects. Cherry-picking
    effects in order to improve agreement with experiments will, no doubt, lead to
    improvements for that set of hulls, but it will lead to worse predictions for other
    hulls.
     
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