the_maximum_sinkage_of_a_ship

1、Journal of Ship Research, Vol. 45, No. 1, March 2001, pp. 50–58The Maximum Sinkage of a ShipT. P. Gourlay and E. O. TuckDepartment of Applied Mathematics, The University of Adelaide, AustraliaA ship moving steadily forwa

2、rd in shallow water of constant depth h is usually subject to downward forces and hence squat, which is a potentially dangerous sinkage or increase in draft. Sinkage increases with ship speed, until it reaches a maximum

3、at just below the critical speed pgh. Here we use both a linear transcritical shallow-water equation and a fully dispersive ? nite-depth theory to discuss the ? ow near that critical speed and to compute the maximum sink

4、age, trim angle, and stern displacement for some example hulls.IntroductionFor a thin vertical-sided obstruction extending from bottom to top of a shallow stream of depth h and in? nite width, Michell (1898) showed that

5、the small disturbance velocity poten- tial ê4x1y5 satis? es the linearized equation of shallow-water the- ory (SWT)?êxx Cêyy D 0 (1)where ? D 1 ? F 2 h, with Fh D U= pgh the Froude number based on x-wise s

6、tream velocity U and water depth h. This is the same equation that describes linearized aerodynamic ? ow past a thin airfoil (see e.g., Newman 1977 p. 375), with Fh replacing the Mach number. For a slender ship of a gene

7、ral cross-sectional shape, Tuck (1966) showed that equation (1) is to be solved sub- ject to a body boundary condition of the formêy4x10 5 D US04x52h (2)where S4x5 is the ship’s submerged cross-section area at stati

8、on x. The boundary condition (2) indicates that the ship behaves in the 4x1 y5 horizontal plane as if it were a symmetric thin airfoil whose thickness S4x5=h is obtained by averaging the ship’s cross- section thickness o

9、ver the water depth. There are also boundary conditions at in? nity, essentially that the disturbance velocity ïê vanishes in subcritical ? ow (? > 05, or else behaves like an out- going wave in supercritica

10、l ? ow (? p?=? and k < ?p?=?, where ? is imaginary, similarly cancel each other, so (11) reduces toF D ? ?U 22? hZ p?=?0k p???k2 S S4k5S B4k5dk (16)Thus the transcritical sinkage force is a predominantly subcritical-

11、like or elliptic phenomenon, depending on the long- wave part —k— < p?=? of the wave number range where all distur- bances due to the ship tend to zero at in? nity. This is consistent with the result of Tuck (1966) th

12、at the dispersionless supercritical sinkage vanishes for ships with fore-aft symmetry. Conversely, the subcritical trim moment for fore-aft symmetric vessels can be writtenM D ?U 22? hZ ? p?=?ik p?k2 ?? S S4k5xB4k5dk (17

13、)since xB4k5 is pure imaginary and odd in k. There is zero contri- bution to the moment from the range —k— < p?=?. (For supercrit- ical ? ow the lower terminal p?=? in (17) is replaced by zero.) Hence transcritical tr

14、im is a predominantly supercritical-like or hyperbolic phenomenon, depending on large wave numbers where the ship produces an outgoing short wave at in? nity. Again, this is consistent with the trim vanishing in subcriti

15、cal dispersionless ? ow for fore-aft symmetric ships (Tuck 1966), and taking a large bow-up value in supercritical ? ow, which is anticipated in the transcritical range. For fore-aft symmetry, the hydrostatic coupling be

16、tween sink- age and trim vanishes, and (15) becomesF D??gAWsM D?gIW? (18)where AW D R B4x5dx and IW D R x2B4x5dx. Therefore the sink- age and trim displacements are given bys D F 2 h2?AWZ p?=?0k p? ??k2 S S4k5 S B4k5 dk?

17、 D F 2 h2? IWZ ? p?=?ik p?k2 ?? S S4k5 xB4k5 dk(19)Finite-width channelIn the case of a ship moving along the center of a channel of width 2w, assuming steady ? ow we can still solve (7) using Fourier-transform technique