Sinkage and trim in shallow water of finite width

In t 10 (lue tion

In a recent paper [1], a theory was presented for ship dyna-mics in shallow water of infinite width. Only a relatively simple extension of this theory is necessary to treat the case where vertical side-walls are present, and the results of such an analysis are described in the present note.

For simplicity, attention will be concentrated on tile case of subcritical motion along the centre of a canal of rectangular cross-section, the width of which is comparable with a ship-length but large compared with the canal depth and the beam and draft of the ship. However, the formulae obtained give the same limiting behaviour when the width tends to zero as well-known (e.g. [2], and the Appendix to this paper) hydrau-lic approximations for sinkage in narrow canals. Since the finite-width results also tend to the results of [1) as the width tends to infinity, they may be considered to be uniformly valid for any canal width.

Analysis for Finite Width

The formulation of the problem and the method of solution are for tile most part identical with that described in [1], and only the differences due to finite width will be discussed here. The analysis to follow relies heavily on that given in [11, which should be read first by mathematically inclined readers. How-ever, it is suggested that those interested less in mathematical analysis than in practical conclusions may skip the present section. except to observe that the final results are contained in the formulae (14) - (21).

A result of [1] is that in the "outer" region far to the side of the ship, the effect of the ship is formally analogous to that of a thin wing in a compressibic fluid; hence the only dif-ference for the present case is that this thin wing is situated between two parallel walls, the sides of the canal. On the other hand in the "inner" region close to the ship, the side-walls do not appear (because they are required to be at a distance large compared with the beam of the ship) and hence the "inner" solution is identical with that found in [1), and tells us that the "effective thickness" distribution of the thin wing in the aredodynamic analogy is S (x) ¡h, where S (x) is the cross-sec-tional area of the ship at station x and h is the canal depth.

Thus we can set up an "outer" boundary value problem as follows: In the (x, y) plane, W (x, y) satisfies

2qc _&'I'

(1Fa")

₊

=0

(1)

x2 !y

where Fh = U / Vgh is the Froude number based on water depth. 0n the limiting plane surface of the wing' y = O ±

± (2)

2 h

while oit tile side walls y ± l w

=0

(3)

E. O. Tuck David Taylor Model Basin

Washington, D.C. A (k) = Çdke A (k) (-1k! v7 Fh2) sinh ( w ¡/1 -Fh2 k!)

2tJ

i S'(x) 2 h

Using the Fourier integral theorem, we have by inversion that

A (k) (

k VI - F11) sinh (f w j/l - Fh21k!)

Jdx

C S'(x) L ik S* (k) 2h where S (k) _{= 5}dx S (x) e11 L

is the fourier transform of the area curve. Hence

i sgnk S* (k)

2h JI1Fh2

Thus the longitudinal disturbance velocity near the ship is alf'

u(x)'c-U

(x,o)

ax

00

fdke

k! S* cotli (I w Vi

_{- Fh}

_k!) froni which the first approximation to the pressure (and hence the forces on the ship) follows immediately as in [lj. As w - co the hyperbolic co.tangent in equation (8) tends to unity, and the result of [I], namely

U

f

S'()

u(x) = -- - --

i dj,

(9) -

2rhViFi12J

X-L .00

cosech (f w JIl F

k!) Schittsteehntk Bd. 14 - 1967 - Heft 73 - 92

Sinkage and Trim in Shallow Water of FinhIIJE

UIIVERSITET Laboratorium voor Scheepshydromechanica Archief Mekelweg 2, 2628 CD Deift Tel.: 015- 76873 - Fax: 015.78823

where w is the canal width. 'Fhe solution to this problem is

obtained most easily by taking the Fourier transform with

respect to x of the potential ljJ, writing

00 'P (x, y) = f dke lff* (k;_y),

2z J

whence 2lJc*

_kl(1_F112)ll%*=0.

ay-

-In the subcritical range Fh < 1, the solution of (5) which

satis-fies (3) is

lfD = A (k) cosh (1k! Vi Fh2 (!yI w)) (6) for some A (k) . But then

'1'.. (x, O +) = c-c (4) (S) U 4.t_{h JI! - Fh2}

follows after sorne manipulation. At the other end of the scale, as w - O we have

u (x)*

2 n h w (1 F12)

_Jdke

S* (k)

=U.

S (x) IS0 (10)

I F2

where S0 = wh is the cross-sectional area of the canal. This agrees with the result of a crude hydraulic theory described in the Appendix.

The remainder of the analysis proceeds as in Il]; thus the vertical (upward) force on the ship is

F U 5 dx u (x) B (x)

L

4'rh i/i F112

Jdk k! S

(k) B6 (k) coth ( w ¡/1 - Fh2 Ik

(li)

where ß* (k) = J dx B (x) e1, and B (x) is the waterplane breadth (beam) curve of the ship. The trim moment (bow-up) is obtained from the saine formula but with

__xB* (k)

_{= -J dx xB (x) e}

L

insted of B6 (k)

Again, as w -* the hyperbolic co-tangent tends to unity, and after some manipulation') we recover the result given in

[1], namely

F=

2 th Vi - Fh2

Jdx J d B' (x) S' () log ix

As w - O

F--

QU2

JdkS*(k)

B»(k)

2twh (1F112)

eU2 ('_{dx S (x) B (x)} (13) S0 (1

_{- 1h2) J}

L whidi is the hydraulic approximation.

The sinkage and trini displacements may now be obtained as in Ill by use of Archimedes principle, which yields a pair of simultaneous linear equations; the final results are repeated here for completeness. If we define non-dimensional sinkage and trim co-efficients C, CT by the equations

s _{= C} Fh2

, (14)

L

ViFh2

t==CT , (15)

Vi

where s is the sinkage of the centre of gravityx = 0, and t the bow-up trim angle in radians, then we bave

CF (LC1

Cs=

- , (16)

1-a 13

Cr

CMI3CF

(17)

1) Actually the Fourier-transformed integral (11) is more

con-venient for purposes of numerical computation even when w =

and the numerical results given in (1] were calculated using (11)

rather than (12). 4nL

_JdxB()

L CM = 0012 0.010 OO8

r

0.006 0.004 0.002 o

Fig. I Sinkage at Various Widths

WIDTH _{- 0.5} .0 LENGTH HYDRAULIC APPROXI MATIOi-_/

awl PI

- 93 - Schiftstechnjk Bd. 14 - 1967 - Heft 73 where CF- and C1 are corresponding non-dimensional force and moment coefficients, and where

J dx x B (x) L f dx B (x) L , (18) L f dx x B (x) (19) J dx x2 B (x) L

In the present case of finite width, the force and moment

coef-ficients are

Jdk kl S* coth ( w Vi F112 1k:)

4g J dx x B (x)

L

Discussion of Results

It may be recalled from [1] that in the case of infinite width the nondimensional co-efficients Cs, C-1-, C, C1 were de-pendent only on the relative geometry of the s h i p (speci-fically only oli the shape of its waterplane and cross-sectional area curve), and were not functions of water depth or of speed. Now, however, each of them depends on the width of the chan-nel and the Froude number F11, und we can write (say) C as a function of a single variable

C5 = C5 (W) (22)

where

W = --- Vi - Fh2

(23)

is an "effective" width/ship]ength ratio. Since the width w

occurs only in combination with the expression V1-Ft, it is immediately apparent that if Fh is close to unity, an y canal becomes in effect a very narrow canal, and the hydrau-lic approximation (13) to equation (11) applies. Thus the mathematical singularitY of the theory as Fh- i is always more serious (like a pole, rather than an inverse square root) in any finite width then it is in infinite width of water; of course the actual physical flow has no singularity, but no

presently available theory can adequately describe the non-linear phenomena which occur near critical speed. The above arguments supports the frequent observation that the effect of side-walls, however wide, should not be neglected near criti-cal speed.

As examples of computation of sinkage and trim by equa. lions (16) - (21), we show in Figure 1 curves of sinkage against Fh, for values of width/length of infinity, 3.4, 1.0 and 0.5. The ship form used is a Taylor Standard Series with a prismatic coefficient of 0.64 and the dimensions of Model "A3' described in 131; in fact the value 3.4 for w/L cor-responds to the dimensions of the towing tank in which the measurements of 11 were performed. Clearly at w/L = 3.4 and for moderate values of F11, the departure from the result for infinite width is small2), but it is already as much as 10 o/ at F1 = 0.8 even for such a wide channel. Also shown is the result of the hydraulic approximation at w/L = 0.5 which is, as expected, an under-estimate of the sinkage at low Froude number, but hecomes as F11 - 1 a better approximation (in

terms of relative error) to the finite width theory.

o

05

FIg. 2 Sinkage and Trim at Finite Widths Relative to Infinite Widths

It is instructive to plot the r a t i o of tile finite-width sinkage or trim to the corresponding values for infinite width, against

the parameter W defined by equation (23). This yields a uniqae curve for any given ship which tells us (for instance) the per-centage extent to which a measurement in a towing tank of finite width gives an over-estimate of the infinite-width sinkage. In fact it appears from calculations for a variety of ships that this curve is very nearly a u n iv e r s a I curve for all ships. Thus in Figure 2 we present computed points for five different ship-shapes covering a wide (but by no means exhaustive) range of practical forms; tite results appear re-2) Too small, in fact to give any significant improvement in agreement between the theory of (1) and experiments of ¡3], although the difference is in the right direction. The discrepancies between theory and experiment (f1J. Figure 2) are to be explained

by finite-depth and non-linear near-critical effects rather than by

finite-width effects.

Schjftstethnjk Bd. 14 - 1967. Heft 73 ₉₄

-mlrkably close to a universal curve for sinkage, and reason-ably so for trim. There would scent to be no physic.al or mathe-matical reason for this to happen; that it does so is fortunate, and il is therefore suggested that Figure 2 may be a useful tool for extrapolating sinkage and trim from one width to another - regardless of the form of tite ship.

Ii is notable that the effect of finite width is far more serious for sinkage titen it is for trim. At moderately low values of Fh tile abscissa of Figure 2 is very nearly equal to the width! length ratio, and it can be seen that the effect of finite width on t r i m is negligible for any width greater than about one cltip-lctgih. On the other hand, at this width the sinkage is already increased by over 30 0/o; even in a very wide chan-nel with w/L = 2.5 there is a 6 0/ increase in sinkage at low F11, and a proportionatleiy greater increase as F11 increases towards unity (which reduces the value of W).

APPENDIX

An Empirical Result for Narrow Canals

Suppose we have a canal of arbitrary constant cross-section of area S and width w, and suppose the cross-sectional area S (x) of the ship is small compared with that of the canal. Let there be a uniform stream U far upstream, and make tite approximation that only tite longitudinal disturbance u (x) to

titis stream due to the (fixed) ship need be considered. Then continuity requires

US) = [U + u(x)] [S0

+

_w'(x)S(x)1,

(24) where (x) is the elevation of the water surface due to tue presence of the ship. But from Bernoulli's equation applied at

the free surface

U2=[U±u(x)j2+g(x).

(25)

Diopping terms of second order of smallness we obtain front

(24)

OUwt(x)US(x)+S0u(x),

and from (25) O = U u (x) + g (x) Eliminating (x) gives u(x) U S (x)

So -

U2 w g U S (x) / S0 (26) U2

1 -

_gh

where Fi = S0/w is a mean depth. In the case of a rectangular canal li h and (26) reduces to the result (10) of the text.

(Received 26uh1 June 1966)

References

[1] _{E. O. Tuck, "Shallow-Water Flows past Slender Bodies",} Journal of Fluid Mechanics, 1966.

[21 R. S. Garthune e t a 1, "The Performance of Model Ships

in Restricted Channels in Relation to the Design of a Ship

Canal", TMB Report No. 601, August 1948.

[31 W. Graff e t a 1, "Some Extensions of D. W. Taylor's Standard Series", Trans. S.N.A.M.E. 72 (1964) 374.

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