Ship wakes: Kelvin or Mach angle?

Love The Way You Lie 2022-07-13 01:27 71阅读 0赞

# Ship wakes: Kelvin or Mach angle? #

[M. Rabaud][], [F. Moisy][]

The V-shaped wakes behind objects moving on calm water is a fascinating wave phenomenon with important practical implications for the drag force on ships and for bank erosion along navigable waterways.

The wake pattern was first explained by Lord Kelvin, who by recognizing the dependence of the phase speed *c*φ of surface gravity waves on their wavelength (dispersion) predicted that the wake half-angle α = arcsin(1/3) = 19.47o should be independent of the object's velocity *U*.

However, Kelvin's analysis is called into question by numerous observations of wakes significantly narrower than he predicted, which have been rationalized by invoking finite-depth effects, nonlinear resonances or solitons, unsteady forcing, and visualisation biases.

Analysing a set of airborne images taken from the Google Earth database, we show that ship wakes undergo a transition from the classical Kelvin regime at low speeds to a previously unnoticed high-speed regime α = 1/ *U* that resembles the Mach cone prediction α = arcsin( *c* φ/ *U*) for supersonic airplanes.

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   <td><img src="http://www.fast.u-psud.fr/~moisy/wake/fig1.jpg" alt=""> <p style="font-family:arial,helvetica; font-size:11px; margin-top:5px; margin-bottom:5px"> Airborne images of ship wakes taken from the Google Earth database. (a), Cargo ship near Antwerpen, with α = 20<sup style="font-size:11px">o</sup>&nbsp;and Froude number&nbsp;<em>Fr</em>&nbsp;= 0.15. (b), Speed boat near Toronto, with α = 9<sup style="font-size:11px">o</sup>&nbsp;and&nbsp;<em>Fr</em>&nbsp;= 1.03. For each image, the wake angle α is defined from the slope of the line going through the brightest points resulting from sun glitter or whitecaps (yellow dotted line), which trace the maximum amplitude of the wake.</p> </td> 
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   <td><img src="http://www.fast.u-psud.fr/~moisy/wake/fig2.jpg" alt=""> <p style="font-family:arial,helvetica; font-size:11px; margin-top:5px; margin-bottom:5px"> Log-log plot of the wake angle α as a function of the hull Froude number&nbsp;<em>Fr</em>&nbsp;=&nbsp;<em>U</em>&nbsp;/ (<em>gL</em>)<sup style="font-size:11px">1/2</sup>. Red circles: angles measured from the 37 airborne images of the dataset. Blue line: model. Blue dotted line: asymptotic Mach law. Yellow squares: numerical simulations.</p> </td> 
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## Capillary effects ##

For objects of small size, surface tension effects become important. We have performed experiments in the swimming-pool of Orsay and in a small-scale towing tank at laboratory FAST using surface-piercing vertical cylinders of various diameters *D* translated at constant velocity. We find that the angle of maximum wave amplitude still decreases as 1/*U*, with an additional dependence with respect to the Bond number Bo = *D*/λc, where λc = 16 mm is the capillary length.

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   <td> <p style="font-family:arial,helvetica; font-size:11px; margin-top:5px; margin-bottom:5px"> Two experiments showing the wake behind a vertical cylinder translated at constant velocity in deep water. Download movie:&nbsp;<a href="http://www.fast.u-psud.fr/~moisy/wake/capillary-gravity-wakes-lowres" style="color:rgb(38,48,51)" rel="nofollow">Low Res</a>&nbsp;|&nbsp;<a href="http://www.fast.u-psud.fr/~moisy/wake/capillary-gravity-wakes-highres" style="color:rgb(38,48,51)" rel="nofollow">High Res</a>.</p> </td> 
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## Publications ##

*  **Mach-like capillary-gravity wakes**  
    F. Moisy and M. Rabaud, *Phys. Rev. E* **90**, 023009 (2014). **Editor's suggestion**  
    \[[Abstract][] | [PDF][]\]
 *  **Scaling of far-field wake angle of non-axisymmetric pressure disturbance**  
    F. Moisy and M. Rabaud, *Phys. Rev. E* **89**, 063004 (2014).  
    \[[Abstract][Abstract 1] | [PDF][PDF 1]\]
 *  **Narrow ship wakes and wave drag for planing hulls**  
    M. Rabaud and F. Moisy, Ocean Eng. **90**, 34 (2014).  
    \[[Abstract][Abstract 2] | [PDF][PDF 2]\]
 *  **Du neuf dans les sillages**  
    M. Rabaud, F. Moisy, *Reflets de la Physique* **39** 10-13 (2014).  
    \[[Abstract][Abstract 3] | [PDF][PDF 3]\]
 *  **Ship wakes: Kelvin or Mach angle?**  
    M. Rabaud and F. Moisy, *Phys. Rev. Lett.* **110**, 214503 (2013). **Editor's suggestion**  
    \[[Abstract][Abstract 4] | [PDF][PDF 4] | [Suppl. Material][]\]

## Research highlights ##

*  [Wake up][], Nature Physics News and Views **9**, 390 (july 2013).
 *  [Vague à lame][Vague _ lame], Paris-Sud Magazine **84**, 35 (juin 2013).
 *  [Blick auf Bugwellen mit Google Earth][], Pro-Physik (30 may 2013).
 *  [Physicists rethink celebrated Kelvin wake pattern for ships][], Physics World (30 may 2013).
 *  Le sillage fait des vagues, Pour la Science **429**, 7 (2013) - [communiqué en ligne 29 mai 2013][communiqu_ en ligne 29 mai 2013].
 *  [Lord Kelvin Wipes Out on Speed Boat Wakes?][Lord Kelvin Wipes Out on Speed Boat Wakes], Science Now (9 may 2013).

Last modification: January 06 2015, 02:00:59.

[M. Rabaud]: http://www.fast.u-psud.fr/~rabaud
[F. Moisy]: http://www.fast.u-psud.fr/~moisy
[Abstract]: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.023009
[PDF]: http://www.fast.u-psud.fr/~moisy/papers/2014_mr2_pre.pdf
[Abstract 1]: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.063004
[PDF 1]: http://www.fast.u-psud.fr/~moisy/papers/2014_moisy_pre.pdf
[Abstract 2]: http://www.sciencedirect.com/science/article/pii/S0029801814002522
[PDF 2]: http://www.fast.u-psud.fr/~moisy/papers/2014_rabaud_oe.pdf
[Abstract 3]: http://dx.doi.org/10.1051/refdp/201439010
[PDF 3]: http://www.fast.u-psud.fr/~moisy/papers/2014_rabaud_reflets.pdf
[Abstract 4]: http://prl.aps.org/abstract/PRL/v110/i21/e214503
[PDF 4]: http://www.fast.u-psud.fr/~moisy/papers/2013_rabaud_prl.pdf
[Suppl. Material]: http://www.fast.u-psud.fr/~moisy/papers/2013_rabaud_prl_sm.pdf
[Wake up]: http://www.fast.u-psud.fr/~moisy/papers/2013_nphys2687.pdf
[Vague _ lame]: http://www.fast.u-psud.fr/~moisy/papers/2013_PSMag-84_p35.pdf
[Blick auf Bugwellen mit Google Earth]: http://www.pro-physik.de/details/news/4826871/Blick_auf_Bugwellen_mit_Google_Earth.html
[Physicists rethink celebrated Kelvin wake pattern for ships]: http://physicsworld.com/cws/article/news/2013/may/30/physicists-rethink-celebrated-kelvin-wake-pattern-for-ships
[communiqu_ en ligne 29 mai 2013]: http://www.pourlascience.fr/ewb_pages/a/actualite-le-sillage-fait-des-vagues-31486.php
[Lord Kelvin Wipes Out on Speed Boat Wakes]: http://news.sciencemag.org/sciencenow/2013/05/lord-kelvin-wipes-out-on-speed-b.html