Proceedings of the Symposium Aspects of Navigability of Constraint Waterways, Including Harbour Entrances, Volume 3, Delft, The Netherlands

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Scheepshydromechanica Archief

Mekelweg 2, 2628 CD Delft

Tel.: 015- 786873- Fax: 015-781836

delft 1978

delft, the netherlands,april 24-27,1978

SYMPOSIUM

P1978-5

aspects of V°1 3

navigability of

constraint waterways,

including

harbour entrances

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lume

aspects of

navigability of

constraint waterways,

including

harbour entrances

pmcaed

delft ,the netherlands,april 24-27,1978

volume 3

papers 16-28

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SYMPOSIUM

aspects of

navigability of

constraint waterways,

including

harbour entrances

effect of

squat on the

resistance of

ships in canals. This

effect

appears

to

be most promiment for

ships with a blunt stern shape where boundary-layer separation is likely to occur. A formula is derived to determine the resistance of such ships in canals of restricted width. The formula still contains empirical coefficients that can

be determined from experiments. For push-tows one coefficient may be sufficient.

Values of this coefficient are given based on experiments in the Delft Hydraulics Laboratory and on published data of other investigators. Some restrictions with

respect to the applicability are made. Remarks are also made on the relation between the resistance of towed and self-propelled ships, especially in

view of

the ship's

speed

limit. In a general way the thrust-to-power relation is

discuss-ed, and some effects on the speed-power relation both for ships sailing outside

the canal axis and for ships sailing on flowing water are mentioned. Finally, the most interesting conclusions are summed up.

1 INTRODUCTION'

On water of infinite width and depth the return flow of a ship and the

water-level depression and squat associated with it are relatively small. If the ship's

fairway is restricted both in width and depth the return flow, water-level

de-pression and squat may be considerable. By a simple one-dimensional calculation,

first applied to this problem by Thiele [1], the return flow, water-level

depres-sion and squat can be approximately calculated. Therefore, the relationships for

the conservation of energy (Bernoulli's theorem) and mass are applied on the

water motion relative to the ship, which yields the following equations:

2gZ = (V + u)2 - V2 (Bernoulli) _(I)

VAc = (V + u) (Ac - Am - AA) (conservation of mass) (2)

where g (m/s2) is the acceleration due to gravity; Z(m) is the depression of

the water level, assumed to be equal to the ship's squat; V(m/s) is the ship's

speed; u(m/s) is the velocity of the return flow; A (m2) is the wetted

cross-sectional area of the undisturbed canal;-Am(m2) is the midship cross-sectional _area; and AA,(m2) is the decrease of the canal's wetted cross-sectional area due to water-level depression and squat. For most canals, _{a good approximation of AAc} is given by

AAc = Bc Z, (3)

[I] THIELE, A.,

"Schiffswiderstand auf Canalen",

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where Bc(m) is the surface width of the undisturbed canal. Using this approxi-mation, Schijf et al [2] presented diagrams for the determination of return flow and water-level depression (or squat), valid for the _{so-called lower critical} speed-range according to Kreitner [3]. Figure 1 shows these results in a

com-bined form. 04 030 020 010 0 o 0.30v V = ship's speed

2 AN1 mdship section (ma)

vL =speed limit relative to the canal (m/s) Ac canal cross - sectional area (ma) u _{=return flow} _Bc _{surface width of canal (m)}

h' = Ac/Bc (m)

010 020 0.40 0.50

Figure 1 Ship's speed, return flow and squat, according to Schijf

Schijf also stated that a self-propelled vessel could not exceed the lower critical speed, which was for that reason denoted by him as limiting speed. When applying Eqs. 1 - 3 or Schijf's diagram for calculating return flow,

water-level depression and squat, there should be kept in mind the assumptions made

at the derivation of the relations it is based upon, see, e.g., [2.1.

An assumption that is often neglected is that ships should have a sufficiently long parallel midship section. According to Schuster [4], a ship's length of

SCHIJF, J.B., and JANSEN, P.P.,

Rome, 18th International Navigation Congress, Section I, Communication 1, 1953, pp. 175 - 197

KREITNER, J.,

"Uber den Schiffswiderstand auf beschranktem Wasser",

Germany, Werft, Reederei, Hafen, 15 (1934) 7, April, pp. 77 - 82

1

SCHUSTER, S.,

"Untersuchungen iiber StrOmungs- und Widerstandsverhaltnisse bei der Fahrt von Schiffen in beschranktem Wasser".

Berlin, J.S.T.G., 1952, pp. 244 - 279 -2-060 0.70 Z 4

bk

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(7)

at least half the channel width is required. Some measurements supporting this assumption are reported by Bouwmeester et al [51.

In general, experiments show higher values of return flow and water-level de-pression than the calculated values. The experimental results agree better with the theoretical values if the blockage factor (Am/Ac) increases. Ships travel-ling outside the canal axis cause a stronger mean water-level depression than ships moving along it. Finally, the differences between measured and calculated

water-level depressions increase when the (self-propelled) vessel approaches the speed limit according to Schijf. These trends are illustrated in Figure 2.

The same conclusions as for the mean water-level depression can be drawn with 1.00 0.80 0.60 0 'N 0.40 1 0.20 0 1.0 ,T3 0.74 IN 7

ta

IN 1.0 -3-0.78 0.82 0.86 0.90 \i/ V L 0.94 0.98

BOUWMEESTER, J., KAA, E.J. van de, NUHOFF, H.A., and ORDEN, R.G.J. van,

Leningrad, 24th International Navigation Congress, Section I, Communication 3, 1977, pp. 139 - 158

McNOWN, J.S.,

"Sinkage and Resistance for Ships in Channels",

Proc. ASCE, J. of the Waterways, Harbor and Coastal Engineering Division, 102 (1976) WW3, August, pp. 287 - 300 . /4c31. AC/AC A'c /Ac 0.5 . 0.3

_'

:

:.0.1 a

'

A:

...

A II

xxt,,,-x motor research push-tow i vessel vessel 0 _{motor vessel}1 x research vessel. push-tow 4a.

,

I

x

xt .

I' :

... ,, _...,. '48,,, I

.. _

0 0.20 0.40 0.60 0.80 0 0.02 0.04 0.06 0.08 0.10 0.12

2 ( calculated)i) Cm) --11. AM/AC

calculated with method- Thiele (Eqs 1-3)

?) ratio of the canal cross-section between ship and bank (Ac)and the total canal cross-section (Ac)

Figure 2 Measured and calculated water-level depressions

regard to the squat of the vessels. Figure 3 shows a comparison of the measured mean squat of vessels in relatively wide canals with the calculated values

according to Eqs. 1 - 3.

Several authors have paid attention to the important effect of the squat on the added resistance of ships in restricted water. Schijf et al [fireported measure-ments where the added resistance in a canal ap eared to be approximately

propor-tional to the squat. More recently, Mc Nown [6] suggested that the added power

required to cope with the effect of channel size on ship resistance could be determined from the squat of the ship with

PA = pgZ Am V (4)

where PA(W) is the required power increment, p(kg/m3) is the specific density

of the water, Z(m) is the ship's squat calculated according to Eqs. (1) and (2), Am(m2) is the midship sectional area, and V(m/s) is the ship's speed.

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0 0 0.10 0.20 0.15 0.00

In a discussion on McNown's paper, the author 1_7] assumed that such

an approach would only be feasible

for ships where boundary-layer separation plays an important role, as is the case for several kinds of push-tows. In the next Chapter

the effect of squat on the resis-tance of ships is treated, based on simple momentum considerations.

2.1 General

If the air resistance is disregarded, the resistance of any vessel, travelling

at constant speed in a prismatic fairway is the force resulting from the normal pressures on and the shear stresses along the hull of the under-water vessel

(with appendages, if any). If the ship is sailing at zero drift angle at the

axis of the fairway, the components of the normal and shear stresses

perpen-dicular to the ship's axis neutralize each other for reasons of symmetry. Thus, only the components of the stresses acting in or parallel to a vertical plane

through the ship's longitudinal axis will contribute to the resistance.

2.2 Frictional resistance

The contribution of the shear stresses to the resistance can be determined

approximately from:

R =

CF IpV2S,2 (5)

where RF(N) is the frictional resistance of the ship, CF(dimensionless) is the

frictional resistance coefficient, V(m/s) the ship's speed relative to the water,

and 5(m2) the wetted hull area of the under-water vessel. The frictional

resis-tance coefficient of a flat plate can be determined approximately according to the ITTC - 1957 method from:

lo

CF - 0.075 (

lg

yL,

- 2)-2 +

ACF' (6)

where L(m) is the plate's length, v(m2/s) is the kinematic viscosity of the water

[7] KAA, E.J. van de,

Discussion of "Sinkage and Resistance for Ships in Channels", by J.S. McNown, Proc. ASCE, J. of the Waterways, Port, Coastal and Ocean Division,

103(1977) WW3, August, pp. 389 - 392 -4-x o lb 0 x A, X41, Q X AcS . Y-L o 0 .

.

x motor vessel 0.03< Am/Ac<004 0.75 V/ VL .-5. 0.95 0 push-tows 0.09< Am/Ac< 012 0.75..e.V/VL,e0.99

ships sailing close to the

channel axis

X 0 ships sailing close to U.'

banks

X

r

0.00 0.05 0.10 0.15

calculated squat/ h'

Figure 3 Measured against calculated squat

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and ACF(dimensionless) is a roughness allowance. For ships in restricted water the sum of ship's speed V and return flow u should be substituted for V in Eqs. (5) and (6).

2.3 _{Pressure resistance on unrestricted water, squat and trim disregarded}

In Figure 4 sketches are given of the distribution of the horizontal and vertical components of the normal pressures on a ship travellingunderdifferent conditions

with respect to channel dimensions and squat.

V PV2/2g pnv2/2g I

A=ON1111

I I stern ,low,eag lpgBT2 + C ' IpV2BI ₍₇₎ P .-pgBT2 + C " IpV2BT (8) P (9) (10) (11)

I. V

/AV, VIA4 WAW/ AVC., AVCA, V ANV/A

unrestricted water restricted water

zero squat ; zero angle of trim squat Z ; zero angle of trim

Figure 4 Normal pressures acting on a ship with no trim

The midship section is supposed to be rectangular and prismatic _{over a relatively}

large distance. For such a ship on unrestricted _{water the effect of return flow}

and squat on the pressure resistance can for _{a first approximation often be}

dis-regarded. The forces due to the normal pressures can then be written as:

FB . 1_2pgBT2 ₊

lp ff

_{Fi(y,z)V2 dy dz} = AB Fs

+ IP II

F"(y,z)V2 dy dz = S A S FV = pgAwT +

lp If

_Aw F"(x,y)V2 dx dy +B/2 V2 AB = BT + I p'(y) -5- dy B/2 +B/2 V2 As = BT +

I

p"(y)

dy B/2

where lpgBT2 and pgAwT are the resultants _{of the hydrostatic pressures; FB,Fs and} Fv (all in N) are the resultants of the horizontal _{resp. vertical components of}

the normal pressures acting over the bow, stern and bottom area of the ship; B(m) is the width, T(m) the draught and V(m/s) _{the speed of the ship; AB, As and}

A (all in m2) are the projections of _{the wetted areas of bow, stern and bottom}

on the midship section and the waterplane _{area respectively; F'(y,z), F"(y,z),}

F'"(x,y), p'(y) and p"(y) are dimensionless functions, dependent on the res-pective coordinates; x,y,z(m) _{are cartesian coordinates (the longitudinal,}

trans-verse and vertical distances to the ship's _{centre of gravity); and Cp' and Cp"} are dimensionless coefficients dependent _{on the speed and draught of the ship}

and the bow- and the stern shape respectively. bow

WV2/2g

V

(10)

The vertical force Fv is in equilibrium with the ship's weight, so for ships

with rectangular cross-sections of equal draught over the whole ship's length

the integral in Eq. (9) is zero. As Fv does not influence the resistance of the ship, it will be disregarded in this paper from now on.

The pressure resistance due to the normal pressures is found by subtracting Fs according to Eq. (8) from Fs according to Eq. (7):

R = (C' - C ")1pV2BT = C IpV2BT, (12)

where C is a dimensionless constant, dependent on the form, draught and speed

P 2

of the ship.

2.4 Pressure resistance on restricted water, with trim disregarded

In the same way as in the preceding paragraph the forces acting on the bow and

the stern of a ship sailing in restricted water with a speed V and a squat Z, causing a return flow velocity u, can be described (see also Figure 4):

lpgB(T + Z)2

lp If

F'(y,z)V2dy dz lpgB(T + Z)2 + C ' IpV2B(T +Z) (13)

AB

1PgBT2

lp If F"(y,z)

(V + u)2dy dz = lpgBT2+C "

Ip(V +

u)2BT

FS _As

= pgBTZ + lpg8Z2 + Cp' ipV2B(T + Z)- Cp"

Ip(V +

u)2BT.

The values of C' and C" will be different from the corresponding values on

P P

unrestricted water (as the bow and stern areas are larger on restricted water due to the squat). As a first approximation, however, these constants are

as-sumed to be the same for both conditions.

2.5 Pressure resistance on restricted water, taking trim into account

If the squat varies along the ship's length, the horizontal component of the

pressures acting on the ship's bottom has to be taken into account. Sketches

of such situations are given in Figure 5. If the pressure resistance is derived

in the same manner as in the preceding paragraphs, both situations yield the same expression for the pressure resistance:

R = pgBTZ +

PgBZ2

+ C 'IpV2E1(- + ZB)- C

"Ip(V

+ u )2BT (16)

B

where Zs(m) and Zs(m) are the squat of the ship respectively at the bow and at the stern, and us(m/s) is the return flow at the stern.

--10.V stern w(,,,,$)2t2g bow P. V2/2 g

4vAL

-6-Figure 5 Normal pressures acting on a ship with trim

stern

(VUs)2/2 g

(11)

The values of C' and

CP" will be different from the corresponding values on

unrestricted water. As a first approximation, however, these constants are

assumed to be the same for both conditions.

2.6 _{Total resistance of ships on restricted water}

A general formula for the total resistance of vessels with rectangular cross-sections is found if the frictional resistance according _{to Eqs.} _{(5) and (6)} is added to the pressure resistance according to Eq. (16):

RT = CF

Ip(V +

u)2S + pgBTZs + lpgBZB2 + C 'IpV2B(T + ZB)-

_{Cp"Ip(V +}

us)2BT,(17)

where RT(N) is the total resistance (disregarding air resistance) _{and for u the} mean value of the return flow along the ship's hull may be taken as a first

ap-proximation. From this Equation it will be clear that for a proper determination of the resistance of ships on restricted water both values of Cp' _{and Cp" have} to be known separately. A simplification of this formula is possible under spe-cific circumstances:

On unrestricted water return flow and squat are approximately zero and thus

the conventional expressionsforfrictional and pressure resistance are found. Corrections are required to cope with the effects of the small return current

velocities and squat, e.g., the application of form factors _{to the resistance}

coefficients.

If ZB << T and products and squares of small quantities _{are disregarded, the} total resistance can be calculated approximately from:

RT = CF110(V + u)2S + pgBTZ + Cp'IpV2BT -

Cp"Ip(V +

u )2BT. (18)

This equation clearly shows that the squat of the ship at the stern deter-mines to a large extent the resistance of ships _{on restricted water.}

If also the trim is negligible, then:

RT = CF-110(V + u)2S + pgBTZ + C 'IpV2BT

-P

d) _{If boundary layer separation occurs along}

coefficient CJ' may be approximately zero.

could be determined with one experimental

3 RESISTANCE OF PUSH-TOWS

3.1 Boundary layer separation

Push-barges in Western Europe have in general _{a wedge-shaped bow, but the stern}

actually looks like a vertical plane. A body plan of the _{most common standard}

barge, the Europe-II type, is given in Figure 6. It

will be

_{clear that if such}

MIS

Figure 6 Body plan of a Europe-II barge

-7-main dimensions LoA r76.5 m B 11.4m T _40m C

".1p(V +

u)2BT. (19)

the whole midship section, the The whole pressure resistance coefficient C only.

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a push-barge is towed, boundary layer separation will play an important role in

the resistance of the barge. Figure 7 shows the axial components ux of the

cur-rent velocities behind the stern of such a barge as a fraction of the velocity (V + u). The return flow was calculated according to Thiele (Eqs. 1-3). The con-voy consisted of 2 Europe-II barges (wide formation), with a push-boat behind them. The current velocities were measured in the vertical plane through the x-axis of one of the barges. The Reynolds length-number (related to the barge

length)

Rn = (V + u) L/v (20)

during these model tests (scale 1 : 25) was between 1.5 *106 and 3 * 106. Both

for loaded and empty barges the boundary layer seperates close to the ship's bottom, as could be expected in view of the blunt stern shape. For that reason

it may be assumed that Cr" (see Eq. 19) is very small. This is in agreement with

the measured water levels behind the stern of the barges (see Figure 7). These levels appear to be approximately equal to the original levels minus the mean

water-level depressions as calculated from Eqs. 1-3.

ship : 1x2 Europe -rr barges with .a push-boat, scale 125, draught 0.16 m / 0.074 m (towed)

canal bottom width 472 m, depth 04m

original level

original level minus mean water-evel depression

original levels

original levels minus mean water-level depression

draught 016 m

canal bottom

0-0 V=1.0 m/s X-X V.0.5 m/s

measured levels behind

barges

00 10

ux/(V6)

( ux/(V.G),with ux measured behind the centre of one of the barges and 5 calculated from Eqs 1-3

-8-draught 0.074 m

measured level behind bar. 55

0.0 1.0

ux/(V+6) V=1.0 m/s, canal bottom and

water-levels relative to the barges 0-0 V. 1.0 m/s , ux /(V 6), with us

measured behind the centre of one of the

barges and calculated

from Eqs 1-3

Figure 7 Velocities behind the stern of a Europe-II barge in a push-tow with

1 x 2 barges

V= 1.0 m/s

(13)

Though push-tows with a length of one barge only are not unusual, the most com-mon convoys in Western Europe consist of 2 x 2 or 2 x 1 barges. The vertical

sterns of the barges are then connected in the middle of the push-tow. The shape

of the stern is the same as that of the bow of a Europe-II barge, thus much better

now. Again current velocities were measured behind a towed model scale 1 : 25

of such a formation. The Reynold's length numbers during the tests were about

3.5 * 106 and about 6 * 106. The results of the measurements are presented in

Figure 8.

2x2 Europa barges without push-boat, scale 1:25, draught 0.12 m (towed)

canal : bottom width 4.72 m, depth 0.24 m, side slopes 1:4

original levels

canal bottom

V= 0.9 m/s

--41- original levels minus mean water-ievel depression

_L_

00 0.5 1.0 00 0.5 1.0 00 0.5 1.0

Ux/(V*U) canal bottom and water-levels relative to the barge

V= 0.5 m/s 0-0 V= 0.9 m/s

ux/(V.5), with ux measured behind the centre of one of the barges and 0

calculated from Ecis. 1-3

X--X V= 0.5 m/s

Figure 8 Velocities behind the stern of a Europe-II barge in a push-tow with

2 x 2 barges

The boundary layer separation point is situated at about one-third of the draught from the bottom of the barges, so still close to the bottom. The measured _water levels behind the barges are higher than the original levels minus the (calcu-lated)mean water-level depressionsbut still closer to these levels than to the

original levels (see Figure 8). So it may be assumed that C " (see _{Eq. 19) is}

rather small.

For self-propelled vessels the boundary layer separation point will be liable to shift upward as a result of the suction developed by the propeller. For push-barges, the distance to the propeller near the stern of the push-boat is such

that hardly, if any, shifting of the separation point due _{to suction has to be} expected. Also, the place of the separation is to _{some extent dependent on the}

Reynolds' length number.

measured levels behind barges

(14)

-9-Figure 9 Pressure coefficients of push-tows consisting of Europe-II barges

LUTHRA, G.,

"Untersuchung der Nachstromverteilung eines im Verband schiebendem

Schubboots in Pontonform"

Hansa, 111(1974) 18, September, pp. 1515 - 1521

MULLER, E., und BINEK, H.,

"Systematische Modellversuche mit Schubleichterverbanden" Schiff und Hafen, 28(1976) 11, November, pp. 1126 - 1128

°

-1/

est

No

202 E.roo-it

Ref DO) ,A./...c

Ref. 110) , AM/Ac . DHL, Au/Ac winos,no push <01 >01 <0.1 boat o

.

,.

..

I .l e

lEuro, -1/ barge. no push-boat

Ref 110] ,A./Ac <01

\--r=

o 0 o o a x x o oo . 0 o G 1o2 Eur,e-11 x OHL. Am fAo 0 OHL. o 0/IL. Am/Ac

barges. with push-boat <01. pooh- boat draught <al ? bar, >0.1 than that and barges draught greater of the push c

with the same

boat

025 050 0.75 0.25 0.50 0.75 1.00

V/VL

3.2 Resistance of towed barges

A general expression for the determination of the resistance of vessels was given in Eq. (17). If the effect of trim is disregarded and products and

squares of small quantities are neglected, the simplified Eq. (19) results.

From the preceding sub-section it might be concluded that for the resistance of push-barges Cp" is much smaller than Cp'. If for that matter the term

containing Cp" in Eq. (19) is disregarded, then:

R = C

Ip(V

+ u)2S + pgBTZ + C ipV2BT, (21)

with

0

CF = 0.075 [1 lg

(V + u)L

21+

ACF' (22)

where C is a coefficient to be determined by experiments.

To test the applicability of Eqs. (21) and (22) for determining the resistance of push-tows with Europe-II barges, some measurements were executed in the Delft Hydraulics Laboratory, from which values of Cp were determined. For the rough-ness allowance ACF (for prototype-model correlation) a value of 2.5

*10-4

was

taken. Return flow and squat were determined according to the Thiele-method

(Eqs. 1-3). As for ship speeds higher than the speed limit, these equations

cannot be solved, so only for ship speeds below the speed limit Cp-values were determined. The length and the wetted surface of the push-boat, if present

during the measurements, were disregarded when calculating the Cp- values.

The Cp- values obtained in this way are plotted in Figure 9 against the ratio of the ship's speed V and the speed limit VL. Some published results from other

sources [8,91are also plotted in Figure 9. The scatter in the results presented

025 050 0. 1

---. WA

0.5 CF o 0.5 cp

10

05 cp I o -0.5 0.5 s°,

2 x 2 Europe-2 barges. with push-boat

Ref 191. Am/Ac <01

o OHL. Au/Ac <0.1

025 0.50 075 100

(15)

is rather high. This is not surprising, as in the Cp- value all experimental error is accumulating and the whole range of possible draughts of Europe-II barges is represented, thus the shape of the formation (e.g., the

draught-to-width ratio) is varying strongly.

A large number of resistance data, obtained from hydraulic model investigations, of American barges was reported by the U.S. Army Engineer Division [10]. Stern and bow of each barge generally had the same shape. Sixteen different barge types were investigated, moving in six different formations in a large number of canal profiles. Push-boats were not attached behind the barges. Just a few

results -for one barge type towed alone but in a large number of canal profiles-are plotted in Figure 10. Also from published data for some other barge-types

and convoys [11,12,13,14] Cp- values were calculated.

0.5 0 Cp -1.0 0 0.5 0 Cp -1.0

Figure 10 Pressure coefficients of an American barge

BAIER, L.A.,

"Resistance of Barge Tows, Supplement"

Cincinnati, U.S. Army Engineer Division, Ohio River, Corps of Engineers, June, 1963

Centre National de Recherches des Constructions Civiles, "Recherches sur la Navigation en sections limit-6es" Belgium, Bull. PIANC, 250 (1976) 24, pp. 49 - 72

GUtVEL, R., BELVAL, A., et VRAIN, B.,

"Essais de traction, sur modele reduit, de navires _{on d'engins dont la} carene est mal profil6e"

France, Nouveautes Techn. Mar., 1974, pp. 51 - 63

HEISE, G., und SCHNEIDER, M.,

"Widerstandsuntersuchungen an Modell-Schubverbanden"

Germany, Schiffbautechnik, 10 (1960) 11, pp. 560 - 564

HELM, K.,

"Einflusz der verschiedenen Flachwasserprofile _{auf Widerstand und Vortrieb}

von Binnenschiffen mit Rechnungsbeispiel fiir die Binnenwasserstrasze _der Klasse IV" Hansa, 102(1965) 11, pp. 1093 - 1105, 102(1965) 12, pp. 1178 - 1184

.

s...0.1 .

,

4,

'

- "

a. ..

'

...

.0 _0% 0

one barge only,

L.76.2m; B.16.5 Tz0.9 ; 1.6 , 4.3 blockage factor o Am/A,>0.2 0.1< Am /A, bow stern 008 m; L.76.2 m m 23 mi : . 4.5m i1.0mo R-12.2 m 5.6m < 0.2 _13.5m,

40e.

...%.'

_.e...

v.,

_°

tAraiti4.8.48:4.1%*

_ii-v.rie

.1! -

.0

one barge only

I_ ..76.2m ; B.16.5 T =09 ; 1.6 ; 4.3 blockage factor x Am /A, <0.002 0.002<Am/A, m , bow m 2.3 : 4.5m

1

L=76.2m stern i1.0m R .12.2 rn 5.6m < 0.1 6.5m 10.5m 0.75 1.00 0.25 0.50 V/VL 0.75 1.00 025 050

--p V/ VL

(16)

All the measurements that were analysed showed the following general trends:

Empty barges (without push-boat), on unrestricted water (Am/Ac < 0.002)

show an increase in C -value if the (absolute) ship speed becomes very high. Under mostly all other conditions, Cp tends to decrease with increasing

values of V/VI,' This effect could be the result of putting Cp" equal to zero

wich is, of course, never exactly correct. Also the assumption that the trim may be disregarded and that the squat is small compared with the ship's

draught becomes less correct if the speed approaches the speed limit.

With increasing blockage factors, Cp tends to decrease. The same reasons that were supposed to cause the preceding trend can be due to this effect. For push-tows with Europe-II barges, Cp is slightly higher for one-barge-long

convoys than for two-barge-long formations. The American convoys, where al-ways bow and stern had the same shape, show the inverse trend. The

differen-ces in stern shape and its effect on boundary layer separation are supposed

to be the reason for these effects. Energy losses at the connection between the barges could possibly be the reason for the larger resistance of the longer convoys.

Neglect of the push-boat in the calculations has only a minor effect on the C -values of loaded push-tows (see Figure 9). The strong wake behind the loaded barges is probably the reason for the apparently small resistance of the push-boat. Only for loaded push-tows measurements with and without push-boat were analysed, and for empty barges this trend will probably not occur. If the draughts of push-barges and push-boat are the same, some effect of

the push-boat on the resistance of a 2-barge convoy can be found in Figure 9.

The Cp-values found actually are remarkably similar, taking into account the

large differences in barge form, push-tow formation (from one single barge to 2 x 3 and 4 x 2 barges, with and without push-boat) and draught-to-width ratios (from empty three-barge wide convoys to loaded single barges). For Am/Ac < 0.1, by far the most measurements yielded Cp-values between 0.0 and 0.2. For 0.2 >

Am/Ac > 0.1 and V/VL < 0.95, most Cp-values were between -0.1 and 0.1. Finally,

if V/VL > 0.95, Cp varied strongly, even attaining extreme values of -1.3

(Ame-rican barge) and + 0.8 (empty Europe-II barges, one barge long with push-boat). However, for such speeds the resistance determined from tests with towed ships

are not anymore representative for the resistance of self-propelled vessels. This topic will be dealt with in the next paragraph.

3.3 Speed limit and resistance augment fraction

The resistance data analysed in the preceding paragraph all concerned convoys

towed in canal models. When approaching and passing the speed limit no spec-tacular rise in the resistance of the towed vessels was observed. Thus, the

existance of a speed limit requires either a sharp decline in the ratio between

propeller thrust and engine power, resulting in a limiting value of the thrust as supposed by Schofield [151, or an increasing difference between the resis-tance of the self-propelled and the towed vessel. That difference can be ex-pressed in the (dimensionless) resistance augment fraction a, defined by:

RT (1 + a) = R = T,

where RT(N) is the resistance of the towed vessel and R(N) the resistance of

the self-propelled vessel wich has to be equal to the total thrust T(N) for ships with constant speed. The resistance augment fraction a can be expressed in the (dimensionless) thrust deduction fraction t (which is an allowance for the same difference between towed resistance and total thrust) by:

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[151 SCHOFIELD, R.B.,

"Ship speeds in restricted waterways"

(17)

a = t/(1 - t). (24)

To investigate the resistance of a self-propelled push-tow when approaching the speed limit, the self-propulsion points of a 4-barges tow were determined

in a model scale 1 : 25. The width of the tow was 0.912 m, the draught 0.12 m,

and the over-all length 7.54 m. The trnsh-boat was equipped with three ducted

propeller systems in tunnels, and could simulate prototype-engine powers up to about 10,000 kW. The model canal was trapezoidal with a width of 4.72 m at

the bottom, a depth of 0.239 m, and side slopes 1 : 4.

The speed limit under such conditions is 0.935 m/s. However, the highest self-propulsion point that could be attained was a speed of 0.932 m/s at 30.3 revo-lutions per second of the propellers. At a speed of 0.930 m/s two self-propul-sion points were measured, at 28.9 s-1 and 33.8 s-1 of the propellers. The highest number of revolutions that could be attained was 34 s-1, due to the restricted power of the model push-boat. Thus, at lower speeds only one

self-propulsion point could be expected, which actually happened.

Under bollard pull conditions the total thrust of the ducted propeller system could be determined by measuring the total force required to arrest the ship. Also the thrust of the propeller alone could be determined. The ratio between total thrust and propeller thrust appeared to be about 1.4. When the ship was moving, only the thrust of the propeller could be measured. A rough estimate, assuming reasonable values of the wake fraction and following the trends indi-cated in screw diagrams of ducted propeller systems, yields a difference between

total thrust and propeller thrust of 20 to 30 percent.

The measured propeller thrusts at the self-propulsion points are plotted in Figure 11 against the ship's speed, along with the measured resistance of the towed convoy with and without push-boat. Until 90 percent of the speed limit the propeller thrust follows the resistance of the towed convoy without push-boat. At about 96% of the speed limit the thrust equals the resistance of the towed convoy with push-boat. At still higher ship's speeds, a very sharp rise in thrust is required to propel the ship. When the speed limit is approached

80 60 ,7,- 40

1

20 -13-push-boat bottom o - 0---0 X---X with width 0 towed towed propeller 2x2 Europa 472 m, depth resistance resistance thrust - IL 0.24 of push of push-of self -propelled barges, draught m, slopes -tow without tow with 1:4 push - tow 0.12 push-boat push-boat m, in trapezoidal canal, 0 ' g

00

01

02

03

04

05

06

07

08

09

10 11 .ship's speed (m/s)

Figure 11 _{Resistance and thrust of a push-tow in} _{a canal}

very closely (difference less than 0.5%) a further increase in propeller _thrust causes a decline in the ship's speed. Thus, Schofields' supposition of _{a limiting}

value of the propeller thrust when the ship attains the speed limit obviously does not hold. On the contrary, _{an improving propeller thrust coefficient could}

be expected in view of the decreasing advance coefficient of the propeller when approaching the speed limit. This was what actually happened _{during the tests}

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For ship speeds below 90 percent of the speed limit a resistance augment frac-tion of 20 to 30 percent may be assumed for the resistance determined without a push-boat. That is at least what can be expected if the rough estimate of the effect of the duct is correct. From data on the thrust deduction fraction pub-lished by Luthra [8] an approximate value of a , 0.25 was calculated for a

similar loaded 4-barge push-tow with a Push-boat. This fraction was rather constant between 55 and 85 percent of the speed limit.

Summarizing, the resistance augment fraction may be assumed to be in the order

of magnitude of 0.25 for loaded push-tows travelling at speeds below 95% of the speed limit. At higher speeds a very strong increase in this factor occurs,

resulting in values far beyond unity when approaching the speed limit very

clo-sely.

4 POWER AND SPEED OF PUSH-TOWS

4.1 Resistance and thrust

If a push-tow is travelling at constant speed, the total thrust is equal to the

resistance of the self-propelled vessel. For a given convoy speed the towed resistance can be determined as shown in Sub-section 3.2. If the effect of the resistance augment fraction on the resistance of the self-propelled vessel is taken into account (see Sub-section 3.3), the total resistance in any canal, and thus the total thrust required, can be calculated approximately. If, more-over, the thrust as a function of the engine power is known, a relation between

power and speed is found. The thrust-to-power relation will first be discussed now.

4.2 Thrust and power

The thrust T(N) is related to the engine power according to:

T =

noR ns Ps /VA' (25)

where no is the (dimensionless) propeller efficiency, to be determined from tests with free running propulsion systems or to be found from systematic screw

series; 11R is the (dimensionless) relative rotative efficiency (an allowance for the difference in efficiency between the free-running propulsion system and the

same system working behind the ship); ns is the (dimensionless) shafting efficiency,

including losses in the reduction gear (if any); Ps(W) is the shaft power of the engine; and VA(m/s) is the speed of advance of the propeller behind the ship, in

principal given by:

VA = (V + u ) (1 - w), (26)

if the (dimensionless) wake-fraction w is determined on unrestricted water.

For a first approximation, the product of the relative rotative efficiency

(flR 1.00 to 1.05, see a.o. Luthra [81) and the shafting efficiency

(ns ,

0.95

to 0.99, see a.o. Henschke [161) may be taken equal to unity. For loaded 4-barge push-tows, Luthra [8]reports some values for the wake fraction (w 0.3). For

low keel clearances higher values (order of magnitude: w 0.5) are possible, as was found during some tests in the Delft Hydraulics Laboratory. Empty push-tows could have lower wake fractions than the values reported by Luthra for

loaded push-tows.

4.3 Power and speed

Combination of Eqs. (17), (23), (25) and (26) yields a general relationship be-tween the power and speed of ships:

[16] HENSCHKE, W.,

"Schiffbautechnisches Handbuch, Band I" Berlin, VEB-Verlag Technik, 1966

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5.1 Ships with a small keel clearance

It has already been stated that the formulae derived in this paper are based on the assumption that the midship section of the ship is rectangular and prismatic over a sufficiently large distance. Moreover, it is assumed that the return flow is present all around the vessel. This is only the case if the boundary layers

along ship and channel bottom are not large compared with the keel clearance. According to Schlichting [17], the boundary layer thickness of a flat plate equals:

6 = 0.37 X0'8 V-0'2 v0.2 (29)

where 6 (m) is the distance perpendicular to the plate with a mean current velo-city of less than 99% of the undisturbed current velovelo-city (relative to the plate);

X(m) is the distance to the leading edge of the plate; V(m/s) is the plate's speed;

and (m2/s) the kinematic viscosity of the water. Measurements in the Delft

Hydraulics Laboratory of the current velocities along a 1 : 25 model of a

push-tow yielded a boundary layer thickness of the same order of magnitude as

calcu-lated from Eq.(29) by taking the ship's length as X and the sum of the ship's speed

and return flow as V. From this it may be inferred that a return current between ship and channel bottom may only develop if the keel clearance of the ship is not much smaller than the boundary layer thickness of the ship, calculated from Eq.(29) by substituting the ship's length for X and the sum of the ship's speed and return flow for V. In actual practice, for inland vessels of the types cur-rently using the fairways of Europe, caution is required for keel clearances

less than 0.5 m.

5.2 Push-tows travelling outside the canal axis

If ships are travelling close to one of the banks of a canal, the squat of the ships will in general be larger than if they move along the axis (see Figure 3). Then, the resistance will also be larger, and consequently the ship's speed will be lower if the engine power is the same. This effect may be considerable on restricted fairwaysbutis not taken into account in this paper.

5.3 Push-tows travelling up- and downstream

Up till now only push-towing in still water has been dealt with. When travelling on flowing water it is generally expected that ships sailing downstream will have a higher ship's speed relative to the water than ships sailing upstream,

and ships on still water will be in-between. However, several model tests with free running models in the Delft Hydraulics Laboratory showed the opposite ef-fect. The differences observed varied with the water depth, the ship's speed

[0]

SCHLICHTING, H.,

"Boundary Layer Theory"

Karlsruhe, Verlag G. Braun, 1955

-15-(1 - w)-15-(1 + a)

Ps (V + u5) LCF2 Ip(V + u)2S + pgBTZS + lpgBZ2B+Cp'12-pV2B(T +

ZB)-no nRs

2

-

Cp"Ip(V + us)2BT] (27)

which, after some simplifications, for push-tows can be written as:

Ps

(1 - w)(1 + a)

(V + u)

LCF1p(V + u)2S + pgBTZ + C IpV2B11. (28)

no 2

In the Sub-sections 3.2, 3.3 and 4.2 order-of-magnitude values of most coefficients are given. The value of Cp may only be applied if u and Z are calculated according

to Thiele (Eqs. 1-3).

no has to be found from tests with the free-running

propul-sion system.

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and, of course, the _{current velocity. The order of magnitude}

of this effect may be illustrated by the following _{example. A four-barge push-tow,}

scale 1 : 25,

with a draught of 0.132 _{m and a width of 0.912 m was put in the}

model canal des-cribed in Sub-section 3.3. On still water the speed of the _{push-tow varied from}

0,48 to 0.56 m/s (depending _{on the number of revolutions of the}

propeller). Under the same conditions, but _{now going upstream (against a}

mean undisturbed current

velocity of 0.30 m/s over the

midship section), relative speeds _{from 0.50 to 0.58} m/s were attained. Going downstream the speed attained varied _{from 0.44 to 0.54} m/s. These data concerned ships _{travelling along or close to the canal}

axis;

close to one of the banks the _{differences were even much}

more pronounced.

Further analysis of these phenomena _{needs still to be done.}

6 CONCLUSIONS

A general expression for the resistance of _{a towed vessel is given by:}

R = C Ip(V+ u)2S+pgBTZ5 +1pgBZ2

+c,Ipv2B(r+z )_ c

)2BT.

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2 B p2

From this equation it will be clear that for a proper description of the

resis-tance of ships on restricted water the effect of the shape of _{the bow and the}

stern has to be known separately. Moreover, return flow and _{squat have a great}

impact on the resistance of ships. If the squat of the ship at the bow is lar-ger than at the stern, less power will be required to attain _{a certain speed.} The squat of the ship at the stern largely determines the resistance.

Eq. _{(17) should only be applied if the keel clearance is} _{sufficiently large}

compared with the boundary layer thickness of the ship's hull (see Sub-section 5.1).

- _{For push-tows, boundary layer separation plays an important role in the} resis-tance, from which it may be concluded that Cr" is small compared with Cr'. Dis-regarding the dynamic pressure at the stern of the push-tow, neglecting products

and squares of small quantities and assuming _{zero trim, Eq. (17) becomes (see}

Sub-section 3.2):

RT = CF2Ip(V + u)2S + pgBTZ + C IpV2BT

The return flow and squat in Eq. (21) can be calculated according to Thiele (Eqs. 1-3). Then CD-values can be determined from the measured resistance of

towed push-tows. I appears that the simplifications necessary to derive Eq.

(22) do not yield a Cp-value that is independent of channel dimensions. How-ever, for ship speeds below 95 percent of the speed limit, the differences in

CP -values of very different push-tows are small if the blockage factor is not varied too much (see Sub-section 3.2).

The resistance augment fraction of push-tows is generally moderate and rather constant. Only if the ship's speed closely approaches the speed limit does this fraction start to grow, thus preventing the convoy from attaining the

speed limit (see Sub-section 3.3).

The power-to-speed relation of push-tows can be described approximately by:

Ps

-(1 - w)(1 + a)

(V + u) [CF ip(V + u)2S + pgBTZ + C IpV2BT],

2 (28)

no

and order-of-magnitude values of w,a and Cp are given in Sub-sections 3.2, 3.3 and 4.2, if u and Z are calculated according to Thiele (Eqs. 1-3) and CF

ac-cording to the ITTC - 1957 formula (Eq. 22). no has to be determined from

tests with free-running propulsion systems.

The method of determining the engine power required to attain a certain convoy speed does not take into account the behaviour of convoys sailing close to

the canal banks. On flowing water, some tests indicate that no additional resistance due to the slope in the water level should be applied.

ACKNOWLEDGEMENT

The author wishes to thank Rijkswaterstaat (Dutch Public Works Department) for its permission to publish some results of a systematic research carried out at the Delft Hydraulics Laboratory, mainly concerning the ship-induced water motion

in and the attack on banks and bottoms of fairways.

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SYMPOSIUM ON ASPECTS OF NAVIGABILITY

SHIPS IN CROSS-CURRENTS

BY

J.W. KOEMAN

Research engineer, Delft Hydraulics Laboratory, Delft, the Netherlands

J. STRATING

Research engineer, Rijkswaterstaat, Traffic Engineering Division, Shipping

Branch, Dordrecht, the Netherlands F.G.J. WITT

Experimental psychologist, Netherlands Ship Model Basin, Wageningen, the

Netherlands.

SYNOPSIS

Studies of how ships are affected by cross-currents are often required to enable

the nautical properties of a waterway design to be examined. Rijkswaterstaatl) inaugurated a general cross-current study in 1972, the aim of which was to im-prove the quality of the research by improving the techniques used.

After making an inventory of the cross-current studies and the research

tech-niques in question,it became clear that therewas a special need to know more about

the factors affecting the behaviour of ships subject to cross-currents. In the research programme set up for this purpose physical phenomena were examined by means of model tests, special attention being paid to some important

simplifica-tions which are generally adopted when constructing physical and mathematical

models of the interaction between ships and cross-currents. In addition an

at-tempt was made to find a proper and useful mathematical description of the

inter-action mechanisms. Some mathematical models were tested to calculate the forces

acting on the ship and the movements produced by measured or calculated forces.

Finally, a study was made of some of the aspects of human control which

influ-ence the ultimate track of the ship.

The study is not yet complete; there will be a discussion of future research.

1. INTRODUCTION

When new waterways are being designed or changes made to existing ones, care

must be taken to ensure that the design is hydraulically and nautically accep-table. Information on the behaviour of ships in the actual situation planned

provides the only sound basis for assessing the nautical quality of the design.

This behaviour will often be largely dependent on the currents at the site in

question, cross-currents being the determining factor.

This means that at the design stage it is necessary to have detailed

informa-tion on the anticipatedcurrent flow pattern and how this will impede shipping.

It is safe to say that at present we do not have _{an adequate understanding of}

the factors influencing the behaviour of ships in cross-currents. _{Moreover, over} a period of time, various research techniques have been used, often with no more than a vague idea of their usefulness. This has meant that the research has been

fragmentary. Such inefficiency, which serves only to increase costs, prompted Riikswaterstaat,amajor commissioner of cross-current studies,to set up a

pro-ject team in collaboration with the Netherlands Ship Model Basin and the Delft

Hydraulics Laboratory. The team's brief was as follows:

"To discover the major factors affecting ships passing _{through cross-currents}

and to develop one or, if necessary, several optimal research techniques for

cross-current problems".

The area of application and the limitations of each technique developed in this fashion had to be given. What is optimal is determined by weighing the

effec-tiveness of the results against the study costs.

1) ..

Rijkswaterstaat is part of the Dutch Ministry of Transport and Public _Works

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1-2. METHOD

The project team began by compiling _{a list of all cross-current studies already} being conducted in the Netherlands. This _{was used to draw up a second list} show-ing all the existshow-ing study methods and _{those anticipated in the future. These} fall into three broad categories:

studies in a hydraulic model in which either _{the maximum measured current} speeds or the behaviour of free-running model ships _{determine whether the} design is nautically acceptable;

studies on a ship-manoeuvring simulator where _{the nautical acceptability}

of the design is determined by the ship's behaviour _{under human control,}

simulated on a computer; and

mathematical studies, where the emphasis lies _{on calculating the disturbed} flow pattern around the ship and the resulting _{external forces. Nautical}

acceptability is assessed using the maximum _{current speeds calculated or}

the behaviour of the ship subject _{to cross-current forces (using a} mathe-matical model for calculating the ship's movements).

After the first phase a research programme was set up, which was initially meant

to increase our knowledge and understanding. The _{research programme comprises} the following elements:

studies into the interaction of ship and _{current flow pattern (physical} scale model studies and mathematical/theoretical _studies);

studies using a ship-manoeuvring simulator _{to determine the human} influ-ence on the behaviour of a ship in cross-currents;

determination of the most functional mathematical _{model for calculating} ship movements in a cross-current.

In all these studies a 200.000 dwt VLCC _{or its mathematical representation was} used. The main characteristics of this vessel are sumiaarized in table 1.

The programme has now reached the stage at which _{these various studies are being} rounded off. The results will be discussed in the _{following section. Please see}

section 4 for the activities planned for the next stage of the programme.

3. INTERACTION BETWEEN SHIP AND CURRENT FLOW PATTERN

3.1 Studies into the interaction between ship _{and current flow pattern on a} mathe-matical basis:

The determination of the behaviour of _{a body in a flow is generally a rather} com-plicated problem as the flow cannot be described simply by a potential function

(Paradox of d'Alembert). Although _{a complete mathematical description can easily} be given, say a viscous incompressible flow and a rigid body, the solution of the

corresponding set of equations is hampered by the small ratio of the smallest and the largest flow scales of the flow fields under consideration. _{In the three}

di-mensional space this leads to very large sets of equations, too large to be solved by todays'computers. Application of a turbulence model to enlarge the smallest scales to be resolved by the computation grid, still requires _{too much computer}

memory and computation time. Consequently the present problem is one of chosing

suitable simplifying assumptions which still allow _{a meaningful result to be}

achieved.

The interaction of the cross current and the ship _{can be split up into the affect}

of the cross current on the ship, resulting in _{a change of the ship's course and}

the affect of the ship on the flow. Attention has primarily been focused on the last part of the interaction, mainly to examine the possibilities of describing

the flow field by a simple mathematical model.

The problem is dealt with in detail in

H.

The flow field can be divided in

three parts with respect to the affect of the ship:

a. The far field.

The affect of the ship in the far field can be neglected. The vertical component of the velocity will be small in respect of the horizontal

11 Delft Hydraulics Laboratory

Ships in cross current (in Dutch) W 252 - I, July 1976

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components and the flow pattern is determined mainly by the bottom roughness and the bed configuration. Much has been published on the mathematical descrip-tion and soludescrip-tion to this problem, since it is a well-known problem in hydrau-lics. In case of a non-layered flow, characterised by a small depth to width ra-tio, the velocity can be described fairly well using the depth averaged veloci-ties. The equations expressing the depth-averaged velocities are given together

with a number of applications in [21. This set of equations can be fairly straight

forwardly solved using a numerical method and applied to fairly large areas

with-out using too much computation time. The local three dimensional velocity profile, if required, can be estimated from simplified equations.

The very near field. In this field surrounding the ship the flow is three di-mensional. It encompasses the wake of the ship and a much smaller region upstream of the ship. Here the description of the flow depends wholly on the behaviour of the boundary layer along the ship's hull. The most favourable situation would be if separation of the boundary layer occurs only at the stern. Then the shear

stresses at the hull mainly contribute to the athwart forces, assuming a steady

state flow. In case of unsteady flow there will also be added mass effects.

Boundary layer theory can be applied to calculate this stress distribution and the outer field can be described by potential theory. The behaviour of the boundary layer depends on the angle of attack of the flow. For larger angles the above picture is not true since the separation line will be situated at the bow. Here the pressure distribution along the hull will mainly contribute to

the athwart force. This problem is much more complicated to solve due to the recirculation zones as the field has now to be dealt with as a whole without using a potential function or parabolic equations. In [3] a review has been given of methods applied in literature to solve this problem. Very little has been done on 3-D flow and all attention is focused on the 2-D problem. This

suggests that a quasi 2-D description of the present flow field should be used,

for example by dividing the ship intc vertical strips. However there are two main objections:in the case of an oblique incoming flow it is possible that

there will be a strong interaction between the individual wakes of the strips.

And secondly the methods available to solve the 2-D flow past the strip are still inadequate: until now the Navier Stokes equations have been numerically solved only for simple body contour and low Re-number, while the application of a simple description (Kirchhoff) requires experimental data like the position of the separation point and the wake pressure. It was found that the resistance is strongly dependent on these data. Another 2-D description is obtained if the ship is seen as a vertical cylinder extending to the bottom. However then the fluid flow cannot pass underneath the ship but only along the bow and stern. An estimate of this effect can be obtained using a method derived by Newman [4], showing that in case of small keel clearances a large rate still flows under-neath the ship. The same objections apply here as to the foregoing methods. These results suggest that in the very near field a semi-empirical description

seems to be required.

The near field. A near field can be distinguished between the far field and the very near field in which the flow is still two dimensional but where the in-fluence of the ship can not be neglected. The introduction of this field is of

121 Kuipers J. and Vreugdenhil C.B.

Calculations of two-dimensional horizontal flow

Delft Hydraulics Laboratory, Report S 163 - I, October (1973)

Flokstra C.

Computational methods for the fluid drag of blunt bodies

Delft Hydraulics Laboratory, Report S 175-II, June (1977)

-141 Newman J.N.

Lateral motions of a slender body between parallel walls

J. of Fluid Mech, vol. 39, part I (1969)

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3-particular interest when a (semi-)empiricaldescription is applied in the very near field. Such an empirical description should in fact express the relation between the forces on the ship and the boundary conditions applying _{at the boundary}

be-tween the near field and the very near field and resulting from the

matchingcon-ditions. It is advantageous to know these conditions _{as close to the ship} aspos-sible. The exact equations to be used in the near field are not quite

clear.Pos-sibly a vorticity free description can be applied. However when the side walls of the fairway run through the near field this may not be sufficient. The non

trivial boundary conditions are supplied by matching conditions with the far

field. Extending the near fieldtotheship'shull and imposing _{a porosity}

condi-tion at the ship seems a suitable approach.

The accuracy of the predicted results is a matter of continuous concern. However there is the auspicious fact that the course of the ship is the most important

feature which is affected by the cross-current. Owing to the inertia of the ship

it can be expected that a certain freedom can be allowed concerning the accuracy

with which disturbing forces are calculated.

3.2 Studies of the interaction between ship and current flow pattern by means of model experiments:

The second part of the research programme deals with two different series of

mo-del experiments, both intended to study the physical phenomena which occur due

to the interaction between ship and cross-current and schematizations applied in

physical and mathematical modelling.

First a model investigation was carried out in order to achieve a better insight into the steady-state phenomena accompanying the passage of a ship through a

uniform cross current field in a constrained waterway. Attention was focused on

the following points:

the influence of the ship on the flow field, resulting in the establishment

of a near and a very near field (according to the division made in section

3.1); and

the influence of the flow on the ship, resulting in lateral forces and

mo-ments.

The validity of the following well-known assumptions _{were to be tested:}

estimate of the forces on a ship sailing through a homogeneous, parallel

cross-current by measuring the forces on the same ship sailing in still water with a drift angle that equals the angle of attack in the

cross-cur-rent situation;

superimposing the forces due to the current and the forces due to

bank-effects; and

estimate of the forces on a ship by measuring the forces on a flat plate

with the same lateral dimensions.

A very schematized situation was chosen for the investigation. A VLCC (see table 1) was towed through a canal with a bottom width of 412.5 m and vertical banks. A homogeneous, parallel cross-current velocity was generated in this canal.

The length of the current field was 990 m. The mean current velocity was 0,85m/s, while the mean direction was perpendicular to the canal's axis. The water depth

was varied and had values of 33.0, 28.4 and 23.6 m, yielding depth/draughtratio's of 1.75; 1.50 and 1.25 respectively. The ship's speed was also varied and had values of 2.72 and 5.44 m/s. In order to investigate the effect of the banks the

ship was towed both on the canal's axis and with an offset of 125 m. The

length-scale of the model was 1 : 82.5.

The lateral motions of the ship were restricted and the forces needed for this

restriction were measured (see table 2). The velocity and the direction of the flow in the horizontal plane and the water level variations were measured at

dif-ferent points in the vicinity of the ship (see figure 1).

In all situations the disturbed flow could be characterized as follows:

the flow velocities under the ship were high in a direction perpendicular to the ship's axis. Even with the smallest keel-clearance more than 50% of the total amount of water passed under the ship's keel. This once again confirms the conclusion drawn in section 3.1, that the application of a 2-D

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description of the flow in the very near field by schematizing the ship's

hull as a vertical cylinder extending to the bottom, is not sufficient;

in the wake area the flow velocities near the channel bottom were high,

near the surface the velocities were small or even negative;

after the ship had passed, the whole wake system was transported downstream

and remained visible for several ship's lengths behind the stern.

The above description implies that the three-dimensional very near field (see

section 3) covered a considerable area.

The lateral forces and moments were greatly affected by the waterdepth, the

sail-ing velocity and the offset. Both the forces and moments increased as the

water-depth decreased and the sailing velocity and offset increased.

As far as the validity of the investigated assumptions was concerned, the

fol-lowing conclusions could be drawn:

the lateral forces on a ship sailing through a homogeneous current tally with those measured on the same ship sailing with a drift angle in still water, provided that the dimensions of the waterway in which the ship sails, are the same in both situations. There could be 100% difference between the forces measured in the cross-current in this constrained waterway and those measured under a drift angle in still water with the same depth but

infi-nitely wide;

a superimposing of the forces due to cross-currents (as measured on a ship sailing through the cross-current on the channel's axis) and the forces due to bank effects (as measured on a ship sailing with an offset of 125 m in still water) did not agree with the forces measured on a ship sailing with an offset of 125 m through the cross-current. The differences could amount

to 80%;

the forces measured on a flat plate differed up to 80% from those measured

on the ship in the same situation. See [5] for more detailed information.

A second model investigation was carried out in order to obtain information on

the applicability of a mathematical model used to calculate horizontal forces and yawing moment on a vessel sailing in a variable current field.

Hydrodynamic forces acting on the hull of a ship manoeuvring in still _{water can} be expressed using a Taylor-expansion in terms of longitudinal and _lateral

velo-cities (u,v), the rate of turn

(0,

_{the rudder angle (6) and their time} deriva-tives, all defined in a frame of reference fixed to the ship 1,61. Given the ship

bound derivatives used in this representation, together with _{the magnitude of u,}

v, r, _{6 and their time derivatives, one can calculate the forces} _{and yawing}

mo-ment acting on the ship.

The mathematical model which is used, is based _{on the assumption that the above} representation holds true in a variable current field provided _{that the} velo-cities u, v and the rate of turn _{r are replaced by corresponding quantities}

describing the motions of the ship relative to water

[71.

It can be shown that, as a first approximation

U = u - u V = v - v Dv CS r

=r

Dx

[5] Delft Hydraulics Laboratory

Omstroming van en krachten op _{een schip in homogene dwarsstroom (in Dutch)}

Report R 775, M 1315 (in preparation)

1 1 Abkowitz, M.A.

Lectures on ship hydrodynamics, steering and manoeuvrability Hydro- and Aero Dynamics Laboratory, _{report Hy-5}

[71 _{Pinkster, J.A.}

The measurements of flow forces on a captive model of a 425.000 dwt tanker

in a non-homogeneous _current

NSMB report-70-173-GST

(26)

5-In which

uc, vc: components of the undisturbed current vector at the position of the

cen-tre of gravity of the vessel in the fixed frame of reference.

CS

: gradient of the undisturbed current component at right angles to the

Dx

longitudinal axis of the vessel taken in the fore-aft direction.

In order to check the results of calculations using this relative motion concept,

model tests were carried out in variable cross-current fields with a captive

mo-del sailing at a constant speed along a straight line.

During the tests the longitudinal and lateral forces and the yawing moment on the model were measured and compared with the results of calculations.

The water depth amounted to 1.2 times the draught of the model (see table 1). The width of the model basin corresponded to 3300 m in reality so that bank effects

were negligible.

The length of the model basin corresponded to 4950 m in reality. The distance

travelled by the model during which measurements were made amounted to 3600 m.

The current direction was at right angles to the direction the vessel was travel-ing in. The two current fields in which tests were carried out are shown in fig. 2 along with examples of the results of calculated and measured current forces. The complete programme consisted of 16 tests, 8 tests being carried out for each

current pattern.

For each current pattern four speeds were tested viz.: 5 kn., 7 kn., 9 kn. and

12 kn.

For each speed, tests were carried out at zero mean effective drift angle and at 30 mean effective drift angle. The mean effective drift angle is the angle

be-tween the longitudinal axis of the vessel and mean relative flow vector.

From the results of model tests and calculations the following preliminary

con-clusions can be drawn:

the main characteristics of the measured current forces and moment in most cases can be reasonably well predicted by the calculations in so far as they concern the magnitude of forces and moments. The mean absolute error in the calculated values of the lateral force is in the order of 20% of the maximum measured value. For the yawing moment this difference amounts to

approximated 40% of the maximum yawing moment;

the calculations are to some extent shifted in phase in relation to the measured values. The measured forces tend to develop later than is

calcu-lated. This effect is smallest for the lateral force and largest for the yawing moment. This is probably due to the fact that in reality the flow around the vessel is not stationary while the calculations assume the flow to be stationary at all times. Also the difference between the frequency at which the ship's derivatives have been measured and the main frequency

of the cross-current variations can be a determining factor.

3.3 Comparison of some mathematical models of manoeuvring characteristics under

cross-current influence:

The goal of this study is to examine the use of various mathematical models. In this respect three mathematical models of the chosen VLCC (see table 1) at a

depth/draft ratio of 1.2 have been considered. These are: a linear model according to Abkowitz [6]

a non-linear model according to Abkowitz [61

a linear model resulting from an approximation of hydrodynamic forces by means of impulse-response-functions 181

[81 Cummins W.

The impulse response and ship motions DTMB 1661, 1965

(27)

A full description of these models is given by Huysmans [9].

External disturbances were restricted to a non homogeneous cross-current field

schematized as follows: -X=0

vm

-

7

-x.2

vem [ c 1 - cos( - x

V(x) -

. 20] (m/s) for 0 x ₁ 2 1 Vc(x) = 0 for 0 x 1

Thus the variables 1 and Vcm completely characterize the _{cross-current}

mathema-tically. Variation of the current pattern was brought in by giving the variables

1 resp. Vcm the following values:

The interaction between ship and cross-current has been _{neglected. The current}

forces were incorporated in the mathematical models _{through the relative motion} principle (see section 3.2).

The models were compared using the covered paths of the ship _{resulting from the}

various cross-current situations as given above and a fixed rudder algorithm.

To study the affect of this last aspect it was assumed that the manoeuvrability of the ship can be covered satisfactorily by the use of the following rudder

setting:

sin bit (co = .05 radis.)

The analysis of the covered paths is as follows:

with 6 = 150 we associate _{a maximum travelling distance in x-direction}

(= Xmax) and a maximum travelling distance in y-direction ( Y

-= -max)

with 6 = 0, we associate a maximum off-set

with 6 = 150sin wt, analogous to 6 _{= 00 (= Fmax)}

These variables, Y_{-max, Xmax, Dmax, Fmax, will be taken as a function of the}

cross-current variables 1 and Vcm.

The following conclusions can be drawn from _{the calculations (for a complete}

re-view of the results see Huysmans [91):

the linear equations of lateral motions _{according to Cu} _{ins [81, can be} expressed as follows: . -M(V ur) =

_{,v .v}

+ Y r.r + K2(t - T).v(T).dT + fK3(t - T).r(T).dT+ Y.,1*/ + [9] Huysmans R.H.M.

Schepen in dwarsstroom, mathematisch _{modelonderzoek (in Dutch)}

NSMB report 08522-3-GT (in preparation)

(= Dmax) 1. 6 = 15° 2. 6 = 0° 3. 6 ₌ 15° 1 = 2500 m, 3250 m, 4000 m Vcm = .5 m/s, 1.0 m/s, 1.5 m/s