1、 The Maximum Sinkage of a Ship T. P. Gourlay and E. O. Tuck Department of Applied Mathematics, TheUniversity of Adelaide, Australia A ship moving steadily forward in shallow water of constant depth h is usually subject to downward forces and hence squat, which is a potentially dangerous sinkage or i
2、ncrease in draft. Sinkage increases with ship speed, until it reaches a maximum at just below the critical speed gh . Here we use both a linear transcritical shallow-water equation and a fully dispersive finite-depth theory to discuss the flow near that critical speed and to compute the maximum sink
3、age, trim angle, and stern displacement for some example hulls. Introduction For a thin vertical-sided obstruction extending from bottom to top of a shallow stream of depth h and infinite width, Michell (1898) showed that the small disturbance velocity potential (x,y)satisfies the linearized equatio
4、n of shallow-water theory(SWT) yy 0xx (1) Where 2Fh , with F = U / ghh the Froude number based on x-wise stream velocity U and water depth h. This is the same equation that describes linearized aerodynamic flow past a thin airfoil (see e.g., Newman 1977 p. 375), with Fh replacing the Mach number. Fo
5、r a slender ship of a general cross-sectional shape, Tuck (1966) showed that equation (1) is to be solved subject to a body boundary condition of the form U S ( )( x , 0 ) =2y xh (2) whereS(x) is the ships submerged cross-section area at station x. The boundary condition (2) indicates that the ship
6、behaves in the (x ,y) horizontal plane as if it were a symmetric thin airfoil whose thickness S(x)/h is obtained by averaging the ships cross-section thickness over the water depth. There are also boundary conditions at infinity, essentially that the disturbance velocity vanishes in subcritical flow
7、 ( 0 ). As in aerodynamics, the solution of (1) is straightforward for either fully subcritical flow (where it is elliptic) or fully supercritical flow (where it is hyperbolic). In either case, the solution has a singularity as 0 , or F1h .In particular the subcritical (positive upward) force is giv
8、en by Tuck (1966) as 22UF = B ( x ) S ( ) l o g2 1 F h d x d xh (3) withB(x) the local beam at station x. Here and subsequently the integrations are over the wetted length of the ship, i.e., 22LLX where L is the ships waterline length. This force F is usually negative, i.e., downward, and for a fore
9、-aft symmetric ship, the resulting midship sinkage is given hydrostatically by 22 21 hShFVsCL F (4) where ()V S x dx is the ships displaced volume, and 2 ( ) ( ) l o g2s WLC d x d B x S xAV (5) where ()wA B x dx is the ships waterplane area. The nondimensional coefficient 1.4sC has been shown by Tuc
10、k &Taylor(1970) to be almost a universal constant, depending only weakly on the ships hull shape. Hence the sinkage appears according to this linear dispersionless theory to tend to infinity as 1hF .However, in practice, there are dispersive effects near 1hF which limit the sinkage, and which cause
11、it to reach a maximum value at just below the critical speed. Accurate full-scale experimental data for maximum sinkage are scarce. However, according to linear inviscid theory, the maximum sinkage is directly proportional to the ship length for a given shape of ship and depth-to-draft ratio (see la
12、ter). This means that model experiments for maximum sinkage (e.g., Graff et al 1964) can be scaled proportionally to length to yield full-scale results, provided the depth-to-draft ratio remains the same. The magnitude of this maximum sinkage is considerable. For example, the Taylor Series A3 model
13、studied by Graff et al (1964) had a maximum sinkage of 0.89% of the ship length for the depth-to-draft ratio h/T = 4.0. This corresponds to a midship sinkage of 1.88 meters for a 200 meter ship. Experiments on maximum squat were also performed by Du & Millward (1991) using NPL round bilge series hul
14、ls. They obtained a maximum midship sinkage of 1.4% of the ship length for model 150B with h/T =2.3. This corresponds to 2.8 meters midship sinkage for a 200 meter ship. Taking into account the fact that there is usually a significant bow-up trim angle at the speed where the maximum sinkage occurs,
15、the downward displacement of the stern can be even greater, of the order of 4 meters or more for a 200-meter long ship. It is important to note that only ships that are capable of traveling at transcritical Froude numbers will ever reach this maximum sinkage. Therefore, maximum sinkage predictions w
16、ill be less relevant for slower ships such as tankers or bulk carriers. Since the ships or catamarans that frequently travel at transcritical Froude numbers are usually comparatively slender, we expect that slender-body theory will provide good results for the maximum sinkage of these ships. For shi
17、ps traveling in channels, the width of the channel becomes increasingly important around 1hF when the flow is unsteady and solitons are emitted forward of the ship (see e.g.,Wu& Wu 1982). Hence experiments performed in channels cannot be used to accurately predict maximum sinkage for ships in open w
18、ater. The experiments of Graff et al were done in a wide tank, approximately 36 times the model beam, and are the best results available with which to compare an open-water theory. However, even with this large tank width, sidewalls still affect the flow near 1hF , as we shall discuss. Transcritical
19、 shallow-water theory (TSWT) It is not possible simply to set 0 in (1) in order to gain useful information about the flow near 1hF . As with transonic aerodynamics, it is necessary to include other terms that have been neglected in the linearized derivation of SWT (1). An approach suggested by Mei (
20、1976) (see also Mei & Choi,1987) is to derive an evolution equation of Korteweg-de Varies (KdV) type for the flow near 1hF . The usual one-dimensional forms of such equations contain both nonlinear and dispersive terms. It is not difficult to incorporate the second space dimension y into the derivat
21、ion, resulting in a two-dimensional KdV equation, which generalizes (1) by adding two terms to give 231 h03x x y y X X X x x x xU (6) The nonlinear term in X XX but not the dispersive term in xxxx was included by Lea & Feldman (1972). Further solutions of this nonlinear but nondispersive equation we
22、re obtained by Ang (1993) for a ship in a channel. Chen & Sharma (1995) considered the unsteady problem of soliton generation by a ship in a channel, using the Kadomtsev-Petviashvili equation, which is essentially an unsteady version of equation (6). Although they concentrated on finite-width domain
23、s, their method is also applicable to open water, albeit computationally intensive. Further nonlinear and dispersive terms were included by Chen (1999), allowing finite-width results to be computed over a larger range of Froude numbers. Mei (1976) considered the full equation (6) in open water and p
24、rovided an analytic solution for the linear case where the term X XX is omitted. He showed that for sufficiently slender ships the nonlinear term in equation (6) is of less importance than the dispersive term and can be neglected； also that the reverse is true for full-form ships where the nonlinear
25、 term is dominant. This is also discussed in Gourlay (2000). As stated earlier, most ships that are capable of traveling at transcritical speeds are comparatively slender. For these ships it is dispersion, not nonlinearity, that limits the sinkage in open water. Nonlinearity is usually included in one-dimensional KdV equations by necessity, as a steepening agent to provide a balance to the broadening effect of the dispersive term in xxxx .In open water, however, there is already an adequate balance with the two-dimensional term in yy .
