船舶设计外文翻译---船舶最大下沉量

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 increase in draft. Sinkage increases with ship speed, until it reaches a maximum at just below the critical speed
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 sinkage, 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 φ(x,y)satisfies the linearized equation of shallow-water theory(SWT)
yy 0
xx βφ + φ= (1)
Where 2F h β=1-, with F =U /h 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 F h replacing the Mach number. For 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
'US ()(x,0)=2y x h ±Φ± (2)
where S(x) is the ship’s submerged cross -section area at station x . The boundary condition (2) indicates that the ship 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 ship’s cross -section thickness over the water depth. There are also boundary
conditions at infinity, essentially that the disturbance velocity ∇Φ vanishes in subcritical flow
(0β<).
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 0β→, or F 1h →.In particular the subcritical (positive upward) force is
given by Tuck (1966) as
2
F =B'(x)S'()log dxd x ξξξ- (3)
with B(x) the local beam at station x . Here and subsequently the integrations are over the wetted length of the ship, i.e.,
22L L X -<<where L is the ship’s waterline length. This force F is usually negative, i.e., downward, and for a fore-aft symmetric ship, the
resulting midship sinkage is given hydrostatically by
22S V s C L ⎛⎫= ⎪⎝⎭ (4) where ()V S x dx =⎰is the ship’s displaced volume, and
2
'()'()log 2s W L C dxd B x S x A V
ξξξπ=-⎰⎰ (5) where ()w A B x dx =⎰is the ship’s waterplane area. The nondimensional coefficient 1.4s C ≈ has been shown by Tuck &Taylor(1970) to be almost a universal constant, depending only weakly on the ship’s hull shape.
Hence the sinkage appears according to this linear dispersionless theory to tend to infinity as 1h F →.However, in practice, there are dispersive effects near 1h F = which limit the sinkage, and which cause 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 later). 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 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 hulls. 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, 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 will 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 ships traveling in channels, the width of the channel becomes increasingly important
around 1h F =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 water. 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 1h F =, as we shall discuss.
Transcritical shallow-water theory (TSWT)
It is not possible simply to set ‚0β= in (1) in order to gain useful information about the flow near 1h F =. 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 (1976) (see also Mei & Choi,1987) is to derive an evolution equation of Korteweg-de Varies (KdV) type for the flow near 1h F =. 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 derivation, resulting in a two-dimensional KdV equation, which generalizes (1) by adding two terms to give
231h 03xx yy X XX xxxx U βΦ+Φ-
ΦΦ+Φ= (6) The nonlinear term in X XX ΦΦbut not the dispersive term in
xxxx Φwas included by Lea & Feldman (1972). Further solutions of this nonlinear but nondispersive equation were 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 domains, 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 provided an analytic solution for the linear case where the term X XX ΦΦis 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 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 Φ.In
open water, however, there is already an adequate balance with the two-dimensional term in yy
Φ.
This is in contrast to finite-width domains, which tend to amplify transcritical effects and cause the flow to be more nearly unidirectional. Hence nonlinearity becomes important in finite-width channels to such an extent that steady flow becomes impossible in a narrow range of speeds close to critical (see e.g., Constantine 1961, Wu & Wu 1982).
Therefore, for slender ships in shallow water of large or in finite width, we can solve for maximum squat using the simple transcritical shallow-water (TSWT) equation
0xx yy xxxx βγΦ+Φ+Φ= (7) (Writing
23h γ=), subject to the same boundary condition (2). The term in ƒ provides dispersion that was absent in the SWT,and limits the maximum sinkage.
Conclusions
We have used two slender-body methods to solve for the sinkage and trim of a ship traveling at arbitrary Froude number, including the transcritical region.
The transcritical shallow water theory (TSWT) developed by Mei (1976) has been extended and exploited numerically, using numerical Fourier transform methods to give sinkage and trim via a double numerical integration. This theory has also been extended to the case of a ship moving in a channel of finite width; however, the numerical difficulty in computing the resulting force integral, and its limited validity, mean that the open-water theory is more practically useful.
The finite-depth theory (FDT) developed by Tuck & Taylor (1970) has also been improved and used for general hull shapes. This theory gives a sinkage force and trim moment that are slightly oscillatory in h F . Since the theory involves summing infinite-depth and finite-depth
contributions, both of which vary with 2U at high Froude numbers, any error will grow
approximately quadratically with U . Therefore we cannot use this theory at large supercritical Froude numbers. Also, the difficulty in finding the infinite-depth contributions numerically, as well as the extra numerical integration needed to compute the force and moment, make the FDT slightly more dif. cult to implement than TSWT.
In practice, scenarios in which ships are at risk of grounding will normally have h/L <0.125. Since the TSWT is a shallow water theory and it works well at h/L = 0.125, we expect that it will give even better results at smaller, practically useful values of h=L . Also, since the TSWT and FDT give almost identical results for h/L <0.125, and the TSWT is a much simpler theory, we recommend it as a simple and accurate method for predicting transcritical squat in open water.
备注：T.P.Gourlay and E.O.Tuck ．The Maximum Sinkage of a ship[J]．Jourmal of Ship Research ，2001．50~58
<文献翻译二：译文>
船舶最大下沉量
T. P. Gourlay and E. O. Tuck
澳大利亚阿德莱德大学
一艘在等深为h 的浅水中平稳前行的船舶通常趋向于受到向下的合力并产生船体下沉,
处我们同时利用”线性跨临界浅水方程”和”完全分散限深理论”研究典型船体在接近临界速度时的水流和计算这些船体的最大下沉量、纵倾角和船尾位移。

引言
对于在水深为h 且无宽度限制的浅水流中的一艘瘦长型的从船底至顶均为垂直舷侧的物体,Michell (1898) 证明了小扰动速率的电位 φ(x,y)满足浅水理论(SWT)线性方程
yy 0
xx βφ + φ= 其中2F h β=1-, 且
F =U /h ，傅汝德数建立在x -wise 流速U 和水深h 的基础上。

此方程与描述通过瘦长型翼型的线性空气动力学的流体的方程 (见Newman 1977 p. 375)是大致相同的, 不同的是用F h r 代替了马赫数。

对于一艘常见横截面形状的瘦长型船舶来说, Tuck (1966) 指出解决方程（1）受到如下形式的船体边界条件的限制
'US ()(x,0)=2y x h ±Φ±
其中S(x)是在x 处水下横截面区域.边界条件(2) 指示船舶在(x ,y)水平面处的表现如一个瘦长形的对称翼型，其厚度S(x)/h 是通过对水深求全船横截面厚度的平均数获得。

在无限宽水流中同样也有边界条件，主要是扰动速率∇Φ 在缓流 (0β<)中消失了.
同空气动力学中一样, 方程(1)的解答仅仅针对充分缓流(椭圆形)或充分缓流(双曲线
形). 对以上任何一种缓流, 方程的解答中都存在奇点如0β→, 或F 1h →。

特别地，亚
临界力(正向上)由Tuck (1966)给出
2
F =B'(x)S'()log dxd x ξξξ-
B(x)是x 位置处的横梁. 此处和之后的积分下限是在船舶浸湿长度之内，即
22L L X -<<这里L 是船舶水线面的长度。

这个F 力通常是负的，即方向向下，并且对于一艘首尾对称的船舶，静力学中给出最终的船中下沉量
22S V s C L ⎛⎫= ⎪⎝⎭其中()V S x dx =⎰是船舶排水容量，且
2
'()'()log 2s W L C dxd B x S x A V
ξξξπ=-⎰⎰ 其中()w A B x dx =⎰是船舶水线面区域。

由Tuck & Taylor(1970)给出的非色散系数 1.4s C ≈已被证明接近恒定不变，只是很微弱的受船壳形状影响。

此处下沉量根据线性非色散理论将趋向于无穷大。

然而，实际情况中，在1h F =附近存在的色散效应限制了下沉量并导致其在临界速度处达到最大值。

精确的最大下沉量的全船实验数据也非常的有限。

然而，根据线性无粘理论，最大下沉量对于给定的船型跟船长成正比（见下文）。

这意味着最大下沉量的模拟实验，能够给出全面的结果，提供的深吃水船舶也保持不变。

这个最大下沉量的幅度相当大。

例如，1964年Graff er al 研究的Taylor 系列A3模型在水深吃水比为4时具有船长的0.89%的下沉量。

这相当于一艘200米的船舶，船中部下沉1.88米。

在1991年Du & Millward 利用NPL 系列船壳进行了船舶坐底量实验。

他们获得了在水深吃水比为2.3时150B 型的船舶中部最大下沉量为船长的1.4%。

这相当于200米长的船舶船中下沉
2.8米。

考虑到这个因素，当最大下沉发生时通常会有一个显著地船首纵向上扬角度，船尾处的下沉更加大，对于一艘200米船舶来讲可能更多。

值得注意的是只有的傅汝得数相应的船舶才能达到这个最大下沉量。

因此，对于油轮或者散货船起最大下沉量的预测将会减少。

由于航行的傅汝德系数内的船舶大都比较瘦长，我们希望细长体理论能够提供一个关于最大下沉量的好结果。

对于航道中行驶的船舶，当在1h F =附近且流布稳定时航道的宽度变得更加重要（如见，Wu & Wu 1982）。

Hence 在航道中的实验不能很准确的用来预测船舶在开敞水域的最大下沉量。

Graff et al 在大水箱中的实验，相当于实验宽度的36倍，跟开敞水域理论相比已经是个不错的结果。

然而，尽管有如此大的试验箱，当
1h F =时岸壁效应依旧产生影响，因此仍
需要讨论。

浅水理论 这是不可能简单的在（1）中设置0β=，为了增加在1h F =时的有用信息。

根据空气
动力学，还需要包括其他在SWT 中忽略的方面。

在1976年Mei 建议的方法是一个1h F =时KdV 方程。

通常的一维形式既包括此类方程非线性和色散条款。

由于不难推到纳入第二空间的维数，通过添加两个方面给出二维KdV 方程，从而给出了
231h 03xx yy X XX xxxx U βΦ+Φ-
ΦΦ+Φ= X XX ΦΦ中的非线性项被 Lea & Feldman(1972)加入其中，但没有包括xxxx Φ中的色散项。

这个非线性但非色散方程的对水道中的船舶的进一步求解被 Ang (1993) 获得。

Chen &Sharma (1995) 考虑到了水道中船舶产生的孤波的不稳定问题，利用 Kadomtsev-Petviashvili 方程, 即方程（6）一种不规则形式。

尽管主要针对的是宽度有限的水域，他们的方法仍适用于开放水域，虽然这样的计算量较庞大。

非线性和色散方面的内
容进一步被陈（1999）列入，从而允许有限宽度的结果计算覆盖傅汝德数的较大范围。

Mei (1976) 研究了方程（6）在开放水域中的完全形式并对忽略了X XX ΦΦ项的线性情况提供了解析解法。

他阐明了对于船体足够细长的船舶，方程（6）中的非线性项不如色散项重要并可以被忽略；同样，反过来说对于船体肥大的船舶非线性项则是主要的。

这在 Gourlay (2000)中也有涉及。

如前所述，大多数可以跨临界速度航行的船舶相对而言都是较为细长的。

对于这些船舶，色散限制了其在开放水域中的下沉量，而不是非线性。

非线性项因其必要性通常包含在一维KdV 方程中，作为steepening 中介以平衡xxxx Φ中的色散项的宽化效应。

但是在开放水域，在yy Φ中的二维项对该效应已有足够的平衡。

这是相对于宽度有限水域来说的，在限宽水域有扩大跨临界效应的趋势，并引起水流更加接近于单一方向。

此时非线性特性在限宽水道变得如此重要，以至于在接近临界速度的狭小速度范围内稳定的水流可能性极小(见 Constantine 1961, Wu & Wu 1982).
因此，对大宽度或无限宽度的浅水域中的瘦长型船舶来说，我们可以利用简化跨临界浅水域(TSWT) 方程解决最大下沉
0xx yy xxxx βγΦ+Φ+Φ=
(注：
2h γ=)，服从边界条件(2). 公式中的项提供了SWT 中缺少的色散性并限制了最大下沉量。

结论
我们已经用两个细长体理论来解决客船在任意傅汝德系数的下沉和纵倾，包括跨区域的。

有Mei 在1976年提出的TSWT 理论利用数值模拟进行了扩展和利用，通过数值傅里叶变换给出一种双重数值积分方法来计算下沉和纵倾。

这个理论同样被用于宽度受限的航道中运动的船舶。

然而，其在数值计算上的困难意味着在开敞水域理论更加的实用。

由Tuck & Taylor(1970)开发的FDT 技术同样被发展和用于一般的船型。

这个理论给出的下沉力和纵倾有略微的研究价值。

由于涉及到受限和无限宽度的条件限制，在傅汝的系数较大时都发生变化。

任何错误都将增加二次。

因此，在傅汝得数较大时不能用此理论。

此外，在寻找无限宽度条件时也存在困难，以及需要额外的数值积分来计算力和力矩，使FDT 的偏差变小，从而比TSWT 更实用些。

在实践中，在h/l 小于0.125时通常存在触底的危险。

由于TSWT 理论适合于浅水区和h/l 为0.125时，我们希望它能够给出在h=l 时较小的，实际有用过的值。

此外，由于TSWT 和FDT 在h/l<0.125时给出的结果几乎相同，而TSWT 是一个更加简单的理论。

我们推荐它作为开敞水域中船体坐底的一个简单实用的预测方法。

γ。