A rankine source method to capture forward-speed and 3-D effects in seakeeping

TECHNISCHE _UNIVERSITEJT Labo

SCheepshydrom.chajiJc

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Mekelweg 2, 2628 CD Defft

A

Rankine

Source

Method

to

Capture

Forward-Speh

015 78687

- Faic 015- 78133

and

3-D

Effects

in

Seakeeping

Volker Bertram, Michael Hughes

The linear ship _motion problem is still the seakeeping problem of principal practical interest as

it is the keystone for added resistance predictions. The most common tools to solve the ship

motion problem _are based _on strip theory. Strip methods solve _a number of 2-D flow problems

(for each "strip") and integrate the 2-D results _{to a} quasi 3-D result for the whole ship to

determine the motions. This approach is cheap, fast, and for most cases also quite accurate.

However, strip methods do not perform so well for high-speed ships, full hull-forms (tankers),

ships with strong flare, and generally for low _{encounter frequencies which typically occur} in

following _seas. Approaches to improve prediction of added resistance should _capture

-

3-D effects of the flow

3-D effects _are important for low encounter frequencies and full hull forms. 3-D diffraction

at the bow region of tankers contributes considerably to added resistance. Inherently 2-D

strip methods cannot capture this phenomenon.

-

Forward-speed effects

Strip methods include forward speed by the change in encounter frequency. But forward

speed enters the ship motion problem in additional _ways: the local steady flow field, the

steady _wave pattern of the ship, and the change of the hull form and wetted surface due

to squat (dynamic sinkage and trim).

We aim _to develop _a Rankine _source method to capture both 3-D and all forward-speed effects

for linearized seakeeping problems. However:

-

The method at present is limited to

r

₌

Uu/g>

0.25, where U is the ship speed, _w the

frequency of encounter, and g

=

9.81m/s2.

-

The consistent expression for added _{resistance has not} been derived yet.

-

No transom stern model has been incorporated in the seakeeping method yet.

-

Testing and debugging of the seakeeping code is _{not yet} finished.

The claim to be _a superior method has yet to be proven! Much work lies still ahead before

the current method will _grow into _a practical tool _to improve _our capability to predict added

resistance.

Consider _a ship moving with _mean speed U in _a harmonic _wave of small amplitude. For

T>

0.25,

diffracted and radiated _waves cannot propagate upstream. For

r

< 0.25, the method requires

an extension which could either use absorbing boundary conditions ("beach") or match the

near-field Rankine solution to a far--field Green function solution. The problem is linearized

around the fully nonlinear solution of the steady flow (derivatives of velocity potential (0),

wave elevation ç(°), wetted surface S including trim and sinkage and wave profile on hull). In a

first step, the steady flow is solved iteratively using higher-order panels (parabolic in geometry,

linear in _source strength), Hughes and Bertram (1995). The approach is _a straight-forward

extension of _{a wave} resistance code based _on first-order panels, Jensen et al. (1989). So here the

relevant _{equations are} given without further comment:

The particle acceleration in the stationary flow is:

For convenience _we introduce abbreviations:

In the whole fluid domain:

At the steady free surface:

On the hull: 2

a°)

₌

(V(°)v)v(°)

=

B

₌

a3g OZ = o

=

o

-

=

=

O

i is the unit normal _on the hull. Radiation and open-boundary condition _are enforced by

shifting sources vs. collocation points as published for 3-D applications by Bertram and Jensen

(1987). The method is _more grid sensitive than the old first-order panel code and requires _more

CPU time, but-can compute second derivatives _on the hull. Results for _a Wigley hull with

successive grid refinements for double-body flow showed quick _convergence of the contour lines

for all second derivatives. Computations for _a Series-60 including the free surface show plausible

results for all second derivatives, Fig.1., but _no other solutions (or experiments) _are available _to

prove validity of our computated second derivatives.

In _a second step, the diffraction/radiation problem is solved using almost the _same techniques

as for the steady flow computation except that now flat panels with linear source strength

distribution _are used. The instationary potential (') is divided into the potential of the incident

wave

çt',

the diffraction potential d, and 6 radiation potentials:

(1)

₊

_w

i=1

u are the motion ampliudes in the 6 degrees of freedom. It is convenient to divide ' and qY

into symmetrical and antisymmetrical _{parts to} take advantage of geometrical symmetry, e.g.:

y, z)

'(x, _y, z)

₊

(z,

y.

z)

+

y, z)

-

W(X,

y,

z) (2)

2 2

Bertram (1990b) derives the formulas which _are only summarized here. On _the free surface at

z

=

ç(0), the boundary condition is:

(-w

+

Biw)1

+

((2iw

₊

B)Vç5°

₊

₊

d)V1

₊

v(°)(v(°)v)v(1)

₌

0 (3)

We

=

W

-

kUcos /1j is the frequency of encounter, w

=

the frequency of the incident wave,

and k its _wave number. The symbol denotes _a complex amplitude.

Setting the steady flow to uniform flow simplifies (3) _to the usual Kelvin condition at z

₌

O.

However, this crude linearization loses _many physical properties. It cannot be recommended for

moderate to high speeds U _or blunt bodies.

The boundary condition on the average position of the hull is:

ñV1

_+-ñ

₊

[(ñ _x

₊

_x ff)

_-

iwe(+

_X

ii))

₌

O (4)

with

₌

(ñV)V(°),

_{= u1, u2,} u3}, _{a =}

1,

a2,3}T.

Diffraction and radiation problems for unit amplitude motions _can be solved independently using

Rankine _sources (panels). _{Each source} fulfills automatically Laplace's equation and the decay

condition

that

the disturbance of the flow vanishes _at infinity. For the diffraction problem, all

u are set to zero. For a radiation problem, the relevant motion amplitude is _set to 1. All

other motion amplitudes, ç5', and ç5d _{are set to zero. Then the boundary conditions at the}

free surface (3) and the hull (4) _are fulfilled in _{a collocation} scheme. Radiation and _open-

boundary condition _are enforced by shifting the Rankine _sources vs. collocation points, Bertram

(1990a). The collocation scheme forms 8 systems of linear equations in

the

unknown complex

source strengths. The symmetrical (resp. antisymmetrical) _{modes share the same matrix of}

coefficients. Only the right-hand sides differ. Therefore, the 4 systems of equations for the

symmetrical (resp. antisymmetrical) _{cases are solved simultaneously using Gauss elimination.}

Now only the motions _u remain to be determined.

The first-order forces P and moments

Pi

acting _on the body result from the body's weight and

from integrating the _{pressure over} the _average wetted surface S:

(1)

f(pW+pd)dS+(fpindS)u+fp«&x

_P

(1) f(PW ₊ d)( _{x ñ) dS}

₊

(f'(

x ñ) dS) u (5)

s

=

.{0, 0, mg}T (m is the ship's mass), _{is the center of} gravity. The other symbols denote:

=

¡

(0) _dS

¡P(0)(

x ) dS

=

p

((Vo))2

-

U2

-

p

p(2

+

v°v)

pt!)

_p(iJ

-

v°vø')

p(1

₊

Vçt°Vç)

The relation between forces, moments and motion acceleration is:

3

(12)

e,

etc. _are the moments of inertia and the centrifugal moments. Combining (5) _and (12) forms

a system of linear equations for _u (i

₌

1,

..., 6) quickly solved by Gauss elimination.

Acknowledgement

This research _was sponsored by the German Association for Research DFG.

References

BERTRAM, V. (1990a), Fulfilling open-boundary and radiation condition in free-surface problems using

Rankine _sources, Ship Techn. Res. 37/2

BERTRAM, V. (1990b), Ship motions by _a Rankine _source method, Ship Techn. Res. 37/4

BERTRAM, V.; JENSEN, G. (1987), A _new approach _to non-linear _waves generated by _a body moving

steadily at a free surface, IUTAM-Symposium on Non-linear Water Waves, Tokyo

HUGHES, M.; BERTRAM, V. (1995) A higher-order panel method for steady 3-d free-surface flows,

IfS-Report, Univ. Hamburg

JENSEN, G.; BERTRAM, V.; SODING, H. (1989), Ship wave-resistance computations, 5. mt. Conf.

Num. Ship Hydrodyn., Hiroshima

(1)

=

m(t+

Stt _x m(± _x

₊

e

-e

-e

-e

e

e!/Z

-e

-

-e

e

I I

0 1 2 3 ₄ 5 6 ₇ 8 9 10 11 12 13 14 15 16 17 18 19

Fig.la: Contour lines of ç5. for F,,,

=

0.26; contour levels

at

-1.0, -0.8, -0.5, -0.2, -0.14, -0.08,

-0.02, 0.04, 0.10, 0.16, 0.22, 0.28, 0.34, 0.4, 0.6, 1.0, 1.5, 2.0, 2.5

o

1

Fig.lb: Contour lines of for F,,,

₌

0.26; contour levels

at

-1.5, -1.0, -0.5, -0.3, -0.25, -0.225,

-0.20, -0.175,

-0J5,

-0.125, -0.10, -0.075, -0.05, 0.0, 0.05, 0.1, 0.15, 0.2, 0.3

Fig.lc: Contour lines of / for F,,,

₌

0.26; contour levels at -1.0, -0.8, -0.5, -0.2, -0.14, -0.08,

-0.02, 0.04, 0.10, 0.16, 0.22, 0.28, 0.34, 0.4, 0.6, 1.0, 1.5, 2.0, 2.5

t I t I t I -

10 [1 12 13 14 15 16 17 18 19.

Fig.ld: Contour lines of

c for

F

=

0.26; contour

levels

at

-3.0, -2.0, -1.0, -0.8, -0.6, -0.4, -0.2,

0.1, 0, OJ, 0.2, 0.4, 0.6, 0.8, 1.0, 2.4, 3.6, 4.8, 5.0

Fig.le: Contour lines of for F,,,

₌

0.26; _contour levels

at

-3.0, -2.0, -1.5, -1.0, -0.8, -0.6, -0.4,

-0.2, -0.1, -0.05, 0, 0.05, 0.10, 0.15, 0.2, 0.4, 0.6, 0.8, 1.0

I t I I I t t