Calculation of the three dimensional free surface flow about a yawed ship in shallow water

Calculation of the Three-Dimensional

Free-Surface Flow

About a Yawed Ship in

Shallow Water

TEC4SCiE VTT

- Laboratorium voor

Zaojian Zou,1 Institut für Schiffbau der UniversitätHamburg - Scheepshydrornechajca

Archief

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_{TeL-Q1-7868T3-FcO15..7313}MkeIweg 2, 228CD D!ft

It is well known that the manoeuvring behaviours of ships in shallow water are quite different

than in deep water. In order to predict the manoeuvrability, the hydrodynamic forces and moments acting on the ship must be determined. Model experiments are the most reliable way to do so. However, due to the inherent drawbacks of the experimental method, there is a continual effort to develop theoretical and numerical approaches to predict the shallow water

effects on the ship manoeuvrability.

Based on the potential flow assumption, Zou and Söding (199) proposed a three-dimensional

Rankine panel method for calculating the free-surface flow about a ship at small drift angles. The circulation is introduced into the flow field by using a semi-infinite dipole/vortex sheet on the centerplane of the ship and imposing the Kutta condition along the trailing edge of the hull. In the present paper, the method is applied to calculation of the hydrodynamic lateral force and yaw moment acting on a yawed ship in shaFlow water. La the following, the basic theoretical and numerical aspects of the method are explained; the numerical results are presented and discussed for a Wigley hull and a Mariner type ship.

2. Formulation and Solution

We consider a ship in steady oblique motion at the undisturbed free surface in shallow water with constant depth h. The ship's speed is and the drift angle is ß, Fig.l.

V

h

////////

Fig.l: Coordinate system

The fluid is assumed to be inviscid and incompressible, and is disturbed only by the ship. The absolute velocity (relative to the earth) of the fluid can be represented by the gradient of a disturbance velocity potential. In the coordinate system o-xyz moving with the ship's speed (Fig.l), the velocity potential is (r, y, z), and the absolute velocity of the fluid is

=

2.1 Exact Boundary-Value Problem

The velocity potential must satisfy Laplace's equation

v2 = o

and the following boundary conditions:

in the fluid domain (l)

'Permaneni address: Wuhan University of Waier Transportation Engineering, 430063 Wuhan, P. R. China

1) Kinematic boundary condition on the free surface:

on z = ((z,y), (2)

where = (u, V, O), with u and y the forward and sway velocity of the ship, respectively; ç is the negative free-surface elevation.

2) Dynamic boundary condition on the free surface:

- i

-( V .V+

onz = ((x,y).

2

/

where g is the gravitational acceleration.

Eliminating Ç from (2) and (3), we obtain a single boundary condition on the free

(V).V

=g&

on z = Ç(x,y).

Kinematic boundary condition on the wetted ship surface SB:

on SB,

where fi = (ri1,n2,n3) is the unit normal vector pointing into the hull. Kinematic boundary condition on the bottom:

on z= h.

Disturbance-decay condition at infinity:

- (0,0,0)

as (z, y) - co.

Moreover, the velocity potential should satisfy a radiation condition at the far field (nowaves upstream) and a Kutta condition along the trailing edge of the hull.

The above formulation describes a boundary-value problem for the velocity potential. From the solution of this problem we can calculate the flow field and finally the forces and _moments acting on the hull:

=(F1,F2,F3) = JjPfidS.

M=(M1,M2,M3) =

ffP(xfi)dS7

where = (z, y, z) is the position vector, and p the fluid pressure given by

f-.

i

p=pV3.Vt

.VÇ.Vç5+gz 2

where p is the density of the fluid.

2.2 Decomposition and Linearization

The boundary conditions on the free surface are nonlinear and must be imposed on an

unknown boundary surface. To solve the resulting boundary-value problem by a panel method, a linearization of the boundary conditions and an iterative procedure are required.

Moreover, since the flow around a yawed ship is asymmetrical about the centerplane of the ship, to determine the flow directly by a panel method, it is required to discretize the boundary

46 Schiffstechnik Bd. 42 - 1995 / Ship Techno10 Research VoL 42 - 1995

surface on both sides of the ship. Thus the computational burden would increase substantially

compared to the wave-resistance problem.

Therefore, we divide the flow into a symmetric part due to the longitudinal motion and an antisymmetric part due to the lateral motion of the ship:

=

_±

, (11)

where and are the symmetric and antisymmetric velocity potential, respectively.

We assume that the drift angle is small; thus p « on the free surface. Consequently, the

original boundary-value problem can be linearized about the solution of the symmetric flow. It results from (3) and (4):

.(V+V) V.VgZ

= Z

g -

on z = Z(x, y), (12) {2[Ä

-

. V

+ B} . V.

+

[(T

.

V)V]

-= B

(v

. v ± gZ -

.

v)

-

.

[(w. V)V] + g

on z = Z(x,y), (13) where

-W=VV, Â=V(Vc.V

,

B=

J

gWV

and Z is the symmetric free-surface elevation due to the longitudinal motion of the ship. Correspondingly, the boundary condition on the ship surface is

Vp.

= vn2 Ofl _B, (14)

where j is the wetted ship surface during the longitudinal motion.

Once the symmetric flow is determined, the linearized boundary-value problem for the an-tisymmetric velocity potential can be solved. In the present paper, the symmetric flow is determined by an iterative procedure as for the nonlinear ship wave-resistance computation, Zou (199A). , Z and the dynamic sinkage and trim due to the longitudinal motion are

calcu-lated iteratively until the nonlinear boundary condition on the free surface is satisfied. In the following, emphasis will be put on the determination of the antisymmetric flow.

2.3 Solution for the Antisymmetric Flow

We solve the boundary-value problem by using a Rankine panel method. The ship surface and the portion of the free surface surrounding the ship are discretized intotriangular or quadrilateral

panels. The flow is represented by a Rankine source distribution on the ship surface and on a

horizontal plane above the free surface, and by a Rankine type normal dipole distribution on the

lateral plane of the ship and on a sheet downstream of the trailing edge. Under the assumption that the drift angle is small, the semi-infinite dipole sheet is put on the symmetry plane of the ship. On the other hand, since the flow is antisymmetrical about the symmetry plane of the ship, only one half, say, the starboard side, of the boundary surface and the flow field needs to

be considered. This side of the ship surface and free surface is paneled and covered with Rankine

sources of unknown strengths, whereas on the other side an antisymmetric source distribution

is used.

The singularity strengths are determined so that the corresponding boundary conditions on the ship surface and free surface and the Kutta condition along the trailing edge are satisfied

simultaneously. At the same time, the radiation condition is satisfied by the numerical technique

of "staggered grid" suggested by Jensen et aI. (1989).

The shallow water effects are taken into account by the method of images, i.e.. the bottom is treated as a symmetry plane. The velocity potential is represented by the singularity dis-tributions mentioned above and their images below the bottom plane. So there is no need to distribute singularities on the bottom.

The Kutta condition is applied indirectly by requiring equal pressure on both sides of the

trailing edge. This pressure-equality Kutta condition is initially nonlinear and should be satisfied

iteratively, see e.g. lIess (1974). however, by dividing the velocity potential into a symmetric part and an antisymmetric part, this condition can be linearized, Zoz (1994). It results from

(10) and (11):

= (15)

where the superscript (+ means the considered (starboard) side.

In the numerical procedure, (15) is imposed at the collocation points of the hull panels adjacent to the trailing edge. Correspondingly, the semi-infinite dipole sheet on the centerplane of the ship is divided horizontally into strips according to the hull panels adjacent to the trailing

edge. On each strip, the dipole strength is assumed to vary linearly from zero at the leading edge

to a non-zero value at the trailing edge and remain costant downstream of the trailing edge. This dipole distribution is equivalent to a vorticity distribution whose vertical component is

independent of the c-coordinate between the leading and trailing edge and vanishes downstream

of the trailing edge. In the numerical procedure, the vortex distribution instead of the dipole distribution is used. This vortex distribution is discretized equidistantly in the x-direction and

replaced by a system of discrete horseshoe vortices.

The antisymmetric velocity potential is determined by satisfying (13), (14) and (15). Then the total free-surface elevation can be computed from (12), whereas the lateral force Y (= F2) and yaw moment N (= M3) are given as

Y ₌ p(a) 2 dS, N

=

fJ

p(C) _{(s n2}

_{- y}

_n1)dS,

(16) where p(a) is the antisymmetric part of the pressure which is given by

p(a)=p_(v

_-

_{. V + u).}

₍₁₇₎

Numerically, the pressure integration in (16) is approximated by the finite sum of the pressure

acting on the immersed parts of panels.

3. Results

3.1 Wigley hull

For the Wigley hull, calculations were performed for the following cases:

= 0.267,0.316 and ß 50, with

= 1.5, 2.0, 2.5,,

where F = u//T is the Froude number, L is the ship length and T the ship draft at rest.

The numerical results are expressed in non-dimensional form as

Y N

Cy

pVS

and CN=

pVSL'

where S is the wetted ship surface area at rest, and V = /u2 ± y2.

Fig.2 shows the calculated lateral-force and yaw-moment coefficients as functions of water depth. For F,,. = 0.316 and hIT = 1.5 (corresponding to a Froude depth number F,,.h = 1.032), the hull touched the bottom in the calculation, hence there were no results for this case. From

Fig.2 we can see that the hydrodynarnic force and moment increase as the water depth decreases,

except for the yaw moment at F,,. 0.316.

As example, Fig.3 shows the calculated ship wave contours for the Wigley hail at F 0.267 and h/T = 1.5 (F, = 0.872) during 5° oblique motion. Correspondingly, Fig.4 illustrates the wave profiles along the hull for this case. We see the expected enlarging of the Kelvin angle in shallow water. The free surface is deformed significantly in shallow water. Therefore, treating the free surface as a rigid plane seems questionable.

CyxlO2

CxiQ2

) ç t t, i i/ r i ¡i i t, t' ti / i

'ti/f

/1/1/ it//f 1//f u/i j/i< if! t _'/ "S (t t___' ' \

r

' n F,,. = 0.267

--a-- F =0.316

0.0 o s-_. 2.0 2.5 0.0 0.0 _1.0

h/T

Fig.2: Shallow water effect on lateral-force and yaw-moment coefficients

for Wigley hail at 5° drift angle

'.. 's \'Ç .5., S'

tt

'\\\S. I"5". \,S' 1.0 0.5 ,.

_."

_"t t

lit

_lit,

_{--.... 'S. I}

It ,'

-'-.. "t, I

lit

___ L' 't I

t(/ t

'S,,

\

\ I

-

_'

_\

I I

liii! /

L t' t

/iii!' ,---'-.., '\

I r i/i/I! / _-. L' _'t \ I I \IIII i

ii\IIIi! r'. .

f t 'S'., 's \ t' t' / I 't _{'t t'} i f \'SS\\ I i i )iIiit'itl i f s\\\\ t't i _tJi\\\\t,\ /

r-.\\\\\t' H

,_._t) t\',tt\\ _{'"".\\t'\\ /i t} .t't .\.\\ _{t'.' "\'\'\\t'i i} i l'tI' :' '""'t'ts\\'ti i I

)fliti'tt, ''2-.Y"..' S':s5.S'iÑtìt,tttçi /,(ç?Iii;It

__i\\\$/

-' -' -.

:-'

/Ii/\'. ' --' i') ),JlJ';f i ) ,,o,'I//i t '

'i

'B ////f(i

littiit\'t't t i

,'///,'t iii

I /I1¼H Hl i / / f/i / / II ii // / j j

\\\iiIIiii\

..'/////li

I lIti I .,,' / / / t / I I hut' 't _--

/ /

/ t f I

I/I'

L'

_., /

/ / j i tI\ s

/

j',/ j i /

i¼/''-__-' /

_{/ i I} J I/It S -, ,/ _/ i .-.--- _{/ / I} H /\'s... _.Í _/ i

fi,i

t. '-.---

I

_/ Il t' "t

-

/ I I t' I ----/ I I t 't n _{F,,. = 0.267}

--a-- F =0.316

2.0

h/T

h

r

= 1.5

- 0.0

2.5

I

X

Schi.ffstechnik Bd. 42 - 1995 / Ship Technology Research Vol. 42 - 1995 49

0.5

y

Fig.3: Wave contours for Wigley hull at F,,. _{= 0.267, ß = 5° and} Distance between the contour lines is 1.2 x iü- L.

2.0

-1.0

-0.01

/L

Pressure side Suction side

0.0

/

0.5

_ilL

Fig.4: Wave profiles along Wigley hull at F = 0.267, ß = 5° and = 1.5

3.2 Mariner ship

The Mariner ship is usually used as a standard ship to investigate the manoeuvrability, hence

computations were performed for this ship. For purpose of comparison, the numerical results

are expressed in the following non-dimensional form:

V'-

_and

-u_ j

_pL-u

_pLu

where, due to the linearization,

3Y

Y Y

0N

N N

Y=-_=:_ and N==z----.

_{av y 3v V uB}

Fig.5 shows the lateral-force and yaw-moment coefficients calculated by the described method

for the Mariner ship at F, = 0.155 and S = 5° in deep and shallow water. These results are

compared with experimental results of forced oscillating test and oblique tow test, Fjino (1968), and with numerical results of the slender-body theory including the first-order free surface effects,

Zou (1990). We can see that although there are quantitative differences between the numerical and experimental results, qualitatively the shl1ow water effects are predicted by the proposed method. Moreover, as illustrated by the comparison with experimental results, regarding the

calculation of the lateral force the three-dimensional panel method shows no advantage over the

body theory; on the other hand, the three-dimensional panel method and the slender-body theory overestimates and underestimates the yaw moment respectively.

An important aspect of the shallow water effects is the influence on the sinkage and trim (squat). It is well known that the squat increases significantly in shallow water. Since the squat changes the form and area of the ship's lateral plane under water, it affects directly the hydrodynamic lateral force and yaw moment acting on the ship. In order to investigate the

influence of the squat, calculations were performed for Mariner ship with and without squat due

to the forward speed.

Fig.6 shows the calculated lateral-force and yaw-moment coefficients in dependence on the water depth for the Mariner ship with and without squat at F = 0.194 and = 5°. It can

be seen that the calculations without squat underestimate the hydrodynamic force and moment distinctly, and that the discrepancies between the results with and without squat increase as the water depth decreases. Therefore, with regard to the investigation of ship manoeu'rabLity in shallow water, the influence of the squat should not be neglected.

SO Schiffstechnik Bd. 42 - 1995 / Ship Techno1o' Research VoL 42 - 1995

S'

0.01

Yt x

20

20

10 -o 1.0

Fig.5: Lateral-force and yaw-moment coefficients for Mariner ship

in deep and shallow water at F = 0.155

O

With squat e-- Without squat

50 30 20 0

N X

20 IS 10

-

S-0 1.0

Fig.6: Lateral-force and yaw-moment coefficients for Mariner ship

in deep and shallow water at Fi,. = 0.194 and fi = 50

4. Concluding Remarks

A three-dimensional panel method using Rankine singularities is applied to calculation of the

free-surface flow around ships at small drift angles in shallow water. The flow is decomposed into a symmetric part due to the longitudinal motion and an antisymmetric part due to the

lateral motion of the ship; the original boundary-value problem is linearized about the nonlinear solution for the symmetric flow. Hence the proposed method can be connected to any numerical procedures which are available for computing the nonlinear ship wave resistance in shallow water

to calculate the lateral force and yaw moment acting on the yawed ship.

The comparison of the numerical results with the limited experimental results which are

available has shown that the proposed method can predict qualitatively the shallow water effects.

The underestimation of the lateral force by the proposed method and the slender-body theory at < 2.5 indicates that the potential flow methods can only predict the linear part of the

hydrodynamic force due to the lifting effect; they cannot include the nonlinear component due

to the cross-flow drag which obviously becomes important in shallow water even if the drift angle is small. Therefore, in order to estimate accurately the lateral force acting on the ship in shallow water, potential flow methods using more rational flow separation model or viscous flow methods

25 20 IS 10 s 20 lo

Schiffstechnik Bd. 42 - 1995 / Ship Technolo' Research Vol. 42 - 1995 51

O

N x lO

4°

20

-30 20 10 -10 O

° _{3-D panel method Slender-body theory}

Oblique tow test - -- - Forced oscillating test

1.0 1.3 20

h/T

2.0

h/T

should be used. On the other hand, the proposed method overestimates the yaw moment not only in deep water but also in shallow water cases. This overestimation is probably due to the numerical procedure used to satisfy the Kutta condition. The pressure-equality condition at the trailing edge seems to lead to an incorrect prediction of the hydrodynamic force distribution along the length of the hull.

References

FUJINO,M. (1968), Experimental Studies ori Ship Manoeuvrability in Restricted Waters, Part 1, International Shipbuilding Progress, Vol.15. No.168

HESS,J.L. (1974), The Problem of the Three-DirnensionalLiftirzg Potential Flow and its Solution by Means of Surface Sin gular'ity Distribution, _{Computer Methods in Applied Mechanics and} Engineering 4

JENSEN,G., BERTRAM,V., SÖDING,H. (1989), Ship _{Wave-Resistance Computations, 5th} International Conference on Numerical Ship Hydrodynamics,_Hiroshima

ZO1J,Z.J. (1990), Hydrodynamische Kräfte _{am manôvrierenden Schiff auf flachem Wasser bei}

endlicher Froudezahl, Institut für Schiffbau der Universität _{Hamburg, Bericht Nr.503}

ZOIJ,Z.J. (1994), Berechnung der Potentialstrômung um einen schräg fahrenden Schiffsrumpf auf tiefem und flachem Wasser, Institut für Schiffbau _{der Universität Hamburg, Bericht Nr.541} ZOU,Z.J., SÖDING,H. (1994), A Panel Method for Lifting Potential Flows around

Three-Dimensional Surface-Pierczng Bodies, 20th Symposium on Naval Hydrodynamics, Santa Barbara