A Literature Survey into the 3-D Ship Hydrodynamic Problem in the Frequency Domain

(1)

A Literature Survey into the 3D

Ship Hydrodynamic Problem in

the Frequency Domain

Riaan van 't Veer, h.

i

I Report 1007 November 1994

fwJ Deift

FacuI' of Mechanical Engineering and Marine Technology

Ship Hydromechanics Laboratoiy

(2)

Chapter 1 Introduction

i

1.1 Ship motion theory i

Chapter 2 The Boundary Value Problem

3

2.1 The physical model 3

2.2 The mathematical model 4

2.3 Different solution aspects 5

2.3.1 Frequency domain solution 6

2.3.2 Multihull problem 7

Chapter 3 Solving the Boundary Value Problem

9

3.1 introduction 9

3.2 Linearising. the free surface . 9

3.2.1 The steady wave problem 10

3.2.2 The unsteady wave problem 12

3.3 Linearising the hull boundary condition 13

3.4 Integral formulation 14

3.5 The Kutta condition 17

3.6 Equation of motion 18

3.7 Solving the different potentials 20

3.7.1 The döuble-body flow 20

3.7.2 The steady wave flow 21

3.7.3 The unsteady wave flow 23

Appendix A A note on the hull boundary condition

25

(3)

Chapter i

Introduction

Li

Ship motion theory

Before the publicalion of the paper by St. Denis and Pierson (1953) about the application of the principle of superpostition to the ship-motion problem, ship

motions and wave induced loads were barely considered in the design procedure of

ships. Nowadays, the hypothesis that the response of a ship to irregular waves can

be considered as the summation of the responses to regular waves of all frequencies is generally accepted in the ship-motion field.

The first motion theory suitable for numerical computations which had ade-quate accuracy for the designer was the so-called strip theory, presented by

Korvin-Kroukovsky and Jacobs (1957). In the following decade this theory was modified and extended (see e.g. Gerritsma and Beukelman (1967) or Salvesen, Tuck and

Faltinsen (1970)) and nowadays the strip theory is widely used in ship design

practice.

However, since the strip theory is a two-dimensional (2D) ship motion the-ory it fails to give accurate results when the physical flow problem gains a more

three-dimensional (3D) character. Because of these limitations (e.g geometry, high forward speed) 3D methods have been developed since the early 70's. With the in-creased capabaJity of the computer facilities these 3D methods are now approaching

the design practice.

The last decade interest has grown in applying the multihull concept. Since

a multihull vessel has more than one hull, hydrodynamic interaction of waves

between the two hulls will occure and with forward speed the problem will become fully 3D. As shown by Van 't Veer (1993), strip theory will fail to give good results,

although with some artificial calculation scheme the ship motions of a catamaran can be predicted with reasonable accuracy.

(4)

Introduction 2

around ships. Most methods distinguish the ship resistance and the ship motion problems as two different potential flows, each can be calculated separately. The flow problem itself can be studied in the time domain or in the freqnency domain. When the frequency domain is used the incoming waves can only be regular sinu-soidal waves, thus limiting the application. However, when using a time domain method great numerical effort is necessary resulting in long calculation time. The research in the ship hydrodynamic problem is therefore restricted to the frequency domain methods.

In Chapter 2 the boundary value problem will be outlined and in Chapter 3 different solution aspects are presented in some depth. The reason to present the mathematical formulae, and not only to mention the references from where the formulae have been taken, was to get a clear understanding of the different

(5)

Chapter

₂

The Boundary Value Problem

2.1 The physical model

The problem considered here is that of a surface piercing vessel traveling with a

constant forward speed on a wavy surface. The fluid is assumed to be

incompress-ible, inviscid and irrotational. Therefore the flow around the vessel is described with a velocity ,pctential and the Laplace equation governs the fluid domain.

To obtain a well-posed problem, boundary conditions have to be formulated:

water does not. penetrate the ship hull (Neumann cindition)., water does not penetrate the water .surface (kinematic condition), there is atmospheric pressure at the water surface (dynamic condition), waves appear only in a sector behind the ship (radiation condition),

far away from the ship there is undisturbed parallel flow equal to the ship

speed U (decay condition),

water does not penetrate the sea bottom,

at the end of the strut or hull the flów separates (Kutta condition), and the ship is in equilibrium

in theory the calculation domain must be infinite. However, in practice this is impossible so the flow region is bounded and finite. Waves created by the shi.p

must pass through these boundaries without reflection. This is called the open

boundary condition.

The pressures on the hull can be calculated via the Bernoulli equation.

When the pressure distribution on the hull is integrated over the wetted surface the. forces acting on the vessel are obtained. These forces can be separated in a

(6)

The Boundary Value Problem 4

time independent part (steady part) and a time dependent part (unsteady part).

The steady part of the longitudinal force is the wave resistance and the steady part of the vertical force leads to ynamic sinkage and trim of the vessel.

2.2 The mathematical model

The equations are formulated in Cartesian coordinates with the z-axis pointing vertically upwards, see Fig. 2.1. The origin of the cartesian coordinate system is situated in the Undisturbed free water surface. The vessel has a constant speed, U, in the positive x-axis direction. Waves traveling in the water surface are attacking

the vessel with an angle /3. For head waves /3 equals 180, and for following waves

/3 equals zero. The normal vector is always pointing inside the fluid.

Figure 2,1; The mathematical model

The following conditions define the mathematical potential problem: The Laplace equation governs the fluid domain, which is presented by

= = O in the fluid (2.1)

where 'I' is the tôtal flòw potential.

Th kinematic free surface condition on the surface z = C(x, y, t),

(

± VII'.

V)(z

-

C(x,y,t)) = 0 on z = C(x,y,t) (2.2) also written as,

± II'C, - W + Ç = 0

(2.3)

The dynamic free surface condition,

(7)

The Boundary Value Problem 5

The Neumann condition on the hull wetted surface,

s The.radiat'ion condition at infinity,

hm (V'I') = (U,O,O) on

z = S

(2.6)

(x2+y2 -I-z2)-400

s The sea bottom condition

on z=h

(2.7)

Since both the dynamic and kinematic condition has to be satisfied on the free surface these two conditions are cmbined, resulting in:

g'IJ +Wu+2VW

on z= (x,y,t,)

(28) This condition itself is non-linear and it has to be applied on the yet unknown free

surface

2.3 Different solution aspects

The ship motion problem can either be solved in the frequency domain or in the time domain. if the problem is studied as a pure linear problem, that is when all the boundary conditions are being linearised, the frequency and the time domain solution are. related by a Fourier transforma Depending on the actual problem

un-der consiun-deration a choice must be made between the time domain or the freqùency domain solution method. If for example the vessel is precluded to large amplitudes

of motion leading to a changing geometry of the underwater part of the vessel the time domain solution approach is the most useful. However, time domain

solu-tions will always involve a time stepping process and will theréfore ask much more

numerical effort than a frequency domain method, thus when the problem can be studied in the frequency domain this method is the more favourable.

To solve the boundary valúe problem (time domain or frequency domain) most methods present the problem in the integral formulation and apply a Green's function. This Green function can be a simple Rankine source or a more complex

Green's function based on the Havelock-Kelvin source.

The flow around the vessel described by the boundary value problem can be divided in two different problems, the ship resistance problem and the ship motion

(8)

The Boundary.Value Problem 6

problem. The total flow ptitential is therefore decomposed into three different

potentials,. as:

'P(x)

-

(x)

₊

_«x)

± (x,t) _(2.9) where (x) is the double-body potential, qf(x) is the steady wave potential, and

(x,t) is the unsteady wave potential. The .ship. resistance or steady problem is described by the steady wave potential, and the ship motion or unsteady problem is described by the unsteady potential. The double-body potential describes the base flow around the vessel sailing at forward speed in flat water, not creating

waves.

2.3.1

Frequency domain solution

.

In the frequency domain approach it is assumed that the vessel is subject to regular

waves and that the response of the vessel is periodic with respect to the frequency of encounter The siiperposition principle of St Denis and Pierson is valid, so all the responses to the iñcoming waves are linear. The ship motions can be studied in the frequency domain using the steady state limit of the ship response.

Since the complex Green function satisfies all the boundary conditions except

for the hull boundary condition, the flow around the vessel is solved by using a source distribution over the wetted surface of the vessel only. To calculate the

hydrodynamic forces on a ship advancing, at a forward speed in waves the non-zero

speed Green function has to be used. Application of the complex Green function

in the frequency domain is discussed by Chang (1977), i.nglis and Price (1981) and

others It is however still difficult and time consuming to calculate the complex

Green function.

The Rankine source .was introduced in the ship resistance calculations (fre-quency domain) by Gadd (l'976) and Dawson (1977).' The Rankine source is much

easier to evaluate than the complex Green function but since the Rankine source itself does not satisfy the boundary conditions, the. sources have to be distributed on the hull wetted surface and on the free surface as well. It is obvious that the calculation domain can'not be infinite and thus an artificial boundary is created at the borders of the discretised free surface. At these boundaries the waves must pass without being reflected onto t'he calculation domain again (open boundary condition); This extra condition, together with the' vadiatión condition, must be implemented numerically which makes the r ethod more difficult. Recently the Rankine source met'hod has been 'used to solve the unsteady ship motion problem in the frequency domain, see Nakos ('1990). Nakos' method is restricted to high speed flows (r = Uw/g 0.25') since no waves are allowed to propagate forward o'f the body.

(9)

The Boundaiy Value Problem 7

The nonlinearity ofthe derived free surface condition, equation (2.8), makes

the boundary value problem difficult to solve. Most methods therefore linearise the govern equations and solve a linear set of equations, it is possible to designan

iterative process in which the nonlinear free surfàce boundary condition is satisfied

at the end of the iteration process. At each iteration step a set of linear equations is solved for the velocity potential after which the free surface is updated. Such

schemes are outlined by Ni (1987), Jensen (1988), and Raven (1992) fór the steady ship wave problem only.

As already mentioned, a numerical radiation condition has to be

imple-mented when the Rankine source method is used. Several finite-difference

operators have been proposed, introducing numerical damping into the solution. -The best known is the four-point upstream finite-difference (FD) operator, used

by Dawson (1977) in the steady ship wave problem. Using this four-point FD

scheme introduces unwanted dissipation and the wave lengths are underpredicted by about 5%. Others. researches introduced different FD operators based ón 3 or 5 points upstream giving better results (Ni (1987), Kim and Lucas (199t)).. The open boundary condition can be implemented by use of a two-point FD operator which strongly dampen the waves downstream. Another method has been used by Jensen (1988) who shifted the source points to a plane above the collocation: points and by doing this enforced both the radiation and the open boundary con-dition. Very promising results were obtained. A similar method has successfully been used by Raven (1992). In the steady .and unsteady problem Nakos '(1990)

has used a bi-quadratic representation of the source distribution and he found: very

little dispersion in his solution.

2.3.2

Multihull problem

Over the past decade interest has grown in application of advanced, vessels (cata-maran, SWATH) for special purposes. (e.g. 'oceanographic research, oil-drilling platforms, férries). Only a few calculation tools nowadays are able to calculate the motions or 'loads on these vessel's' since they are often stretching thé existing

theories or 'computer programs outside- t'he limits applicable to these theories.

Of-ten the physical problem is more complicated than for an ordinary monohull and phenomena which .are not incorporated in t'he theory oçcur 'for these vessels. An

example is the' catamaran hull fOrm where Waves from one hull can reach the other hull and' thus flow interactions are appearing. This has already been mentioned by Hadler, Lee, Birmingham and Jones (1974) who studied the catamaran seakeeping behaviour.

(10)

mo-The Boundary Value .Problem 8

tions of a multihull vessel but in most cases the motions where far over:predicted in the resonance peak. Better motion predictions could be made using an artificial calculation scheme with the strip theory (Van 't Veer (1993)), but the agreement with model test results is still not completely satisfactory. Kashiwagi (1993) used the concept of Newman's unified slender-ship theory to predict the motions of a catamaran in waves and the agreement with model test of a Lewis-form ship were

reasonable.

Kring and Sclavounos (1991) used a 3DRankine panel method to investigate

the motions of a multi-hull ship sailing in waves They argued that due to the

interaction between the two hull a cross flow exists, which must be modeled by

the shedding of a thin wake and the enforcment of the Kutta condition at the

stem of each stern. With this additional condition the wave elevation downstream

improved, but upstream the wave pattern and pressures on the hull were not

(11)

Chapter 3 Solving the Boundary. Value

Problem

3.1 Introduction

In solving the total flow problem the total velocity potential is decomposed into

three parts the problem is divided into smaller parts by decomposing the total

velocity potential into,

'Ii(x, t) = I(x) + «x) +

(x, t) (3.1)

where c1(x) is the double-body potential, «x) is the steady wave potential, and ço(x, t) is the unsteady wave potential. The steady problem, also called the ship resistance problem, is described by the potentials 1(x) + q(x). The unsteady ship wave problem, also called the ship motion problem, is described by the unsteady potential (x,.t). In the steady problem the double-body flow is of leading order and since small motion amplitudes are considered, the unsteady potential (x, t)

is also small compared to the double-body potential (x).

3.2 Linearising the free surface

The boundary value problem outlined in the previous Chapter is difficult to so1v

since the free surface condition (equation (2.8)) is highly nonlinear. Therefore .the

free .surface condition must be linearised so that the boundary value problem can be solved. In the literature several linearisation processes are presented leading to

different numerical schemes. In this section the linearisation methods are presented

(12)

Solving the Boundary Value Problem 10

3.2.1

The steady wave problem

The steady wave potential, «x), describes the wave flow round a vessel sailing with forward speed in flat water whereby the vessel itself is restrained: from motions.

An important quality criterium for wave resistance calculations is the choice

of the free surface boundary condition. Several methods are presented in the

literature.

Neumann-Kelvin linearisation

In this classical approach the non-linear free surface condition is linearised using the Kelvin condition at the undisturbed water surface (z = 0). The incoming

flow is Ux, parallel to the ship longitudinal axis, and far behind the ship the

disturbances due to the vessel can be neglected. The govern equation for the free

surface reads:

= Ux + «x,.t)

/ o

(3.2)

H--U--j +g--=0

_ot _OX/ _oz

This is a simple approximation for the base flow, and many researchers have pre-sented free surface conditions leading to better flow predictions.

When the body boundary condition is collapsed onto the centreplane of the vessel the thin-ship approximation is used. In that case the beam-to-length ratio is

assumed to be small. However, it is more accurate to use the actual wetted surface

as the body boundary. Together with the linearised free surface condition about a uniform incoming flow the. basis is gi.ven for many ship resistance calculation

schemes.

In the ship motion problem Chang (1977) was the first to introduce the

Neumann-Kelvin formulation. In three dimensions the method however becomes complicated due to the difficult kernel of the integral equation to be solved. The 2D problem is easier to solve when the slender body approximation is used. The slender body theory, assuming the. draft and beam to be small compared to the length, leads to the strip-theory definition.. These theories are widely used in the

ship motion problem, giving very reasonable results when applied within thelimits

of the theory.

. Double-body linearisation

In the double bo.dy ap.proach the. underwater part of the vessel is mirrored in the undisturbed free surface plane (z = O) building a closed double body. The flow around the double body is assumed to be of leading order compared to the

(13)

disturbance flow. The method to calculate the double-body flow has been presented

by Hess and Smith (1962).

One approach is to use the double-body flow as the base flow on which

the steady wave flow is superpositioned, thus:

) (3.3)

Dawson (1977) was the first to introduce this approach in the calculation of the

ship resistance. His linearised free surface condition reads as,

(qi)i

- gq =

24?I (3.4)

where the subscript I denotes the derivative to be taken along the streamline of

the double body flow.

As pointed out by several authors (e.g.. Jensen (1988)) this linearisation

process is not very consistent. Jensen presents a consistent Dawson-like

linearisa-tion which includes an extra term,

(+2iqi'U2) not seen in the Dawson

equation which is due to the Taylor expansion,

(-

+ Ii4i) + FlI4lçbl - gq -

+2i4 - U2) = o

(3.5)

Nakos (1990) also solved the steady wave flow with the double-body flow, as the base flow and the equation he presents equals the one presented by Jensen (only

Jensen used a different coordinate axis definition). Nakos' free surface linearised condition reads as (z-axis upwards), .

V.V(V.Vq)+

on z=O(3.6)

with the steady wave elevation:

(= (V4 .

-

U2)0

-

(V (3.7)

s A.rbitrary flow linearisation

Another approach is to use an iterative scheme in solving the nonlinear

boundary condition. At each iteration step a linearised boundary condition (using the dynamic boundary condition) is used to calculate the velocity potential and

the wave elevation Z is obtained from the previous iterations step using the kine-matic boundary condition. Thus, both the dynamic and the kinekine-matic boundary

(14)

conditions are fulfilled in each iteration step. Jensen (1988) used this method,. resulting in the free surface condition,

+

vvw]

+

VWV(V)2

-g(P + 'I1) + (C --

Z)[.VV(V)2 g}

O on z = Z (3.8)

C - z -

[(V)2 + 2V

ViI' - U2] - gZ (3 9)

-

gVV

When in this last equation (3.8) the double body approach is adapted (F = O at Z = O), equation (3.6) is obtained. Note that Jensen used a z-axis pointing

downwards.

3.2.2

The unsteady wave problem

The unsteady wave problem describes the ship motion problem. The ship motiòns are calculated in the undisturbed free water surface, thus there is no link between the unsteady and steady wave potential. The unsteady problem is more difficult to calculate than the steady problem since the body boundary condition becomes

more complicated.

Yasukawa (1990) extended the Dawson linearisation to the unsteady prob-lem. The total velocity potential was divided in the three components as presented by equation (3.1) resulting in the unsteady Dawson-like free surface condition,

where

ço + +

+ 2jçj + gço = O

on z = 0 (3.10)

The steady wave potential presented by Yasukawa (1990) is equal to the one

pre-sented by Dawson (1977).

In the frequency domain Nakos (1990) solved the ship motion problem with

a more consistent linearisation process. For the unsteady ship motion problem,

Nakos (1990) derived,

Ç'ti + 2V

. V1 + V

. V(V

. V) +

V(V

. V) +

zz

+V

.v)

0

with for the wave elevation:

C

= -i

+V

.

on z = 0 (3.11)

(15)

When the vector identity for a scalar c1 and a vector field H is introduced (see also Jensen (1988)),

(VV)H

=

1H1 (3.13) and substituted in the by Nakos derived free surface condition, it is seen that the difference between the Yasukawa and Nakos derived free surface condition is the term _zz(çOt + VIV), which as in the steady problem comes from the Taylor

expansion around the wave elevation.

3.3 Linearising t;he hull boundary condition

As was presented in section 2.2, one of the boundary conditions to be satisfied is the Neumann condition on the wetted surface of the hull. By introducing an

'infinitesimal oscillatory displacement vector measured from the mean position of the vessel (a), this boundary condition can be written as:

ô'F a.

=.n

ôn

at

on (B) (3.14)

This hull boundary condition has been linearised by Timman and Newman (1962)

using a Taylor. expansion around the nean position of the vessel. They derived

the following,

-=

_{+ V x (o x V)i .}

on (B) (3.15)

where is an infinitesimal oscillatory displacement vector measured from the mean

position (.) of the vessel and il is the normal vector on the hull pointing inside

the fluid.

Applying the double body approach to equation (3.14) results in:

=

_{- V( + q)}

_on _(B) _(3.16)

And for the linearised condition the fôllowing equation is derived (Timman and

Newman (1962), see also Appendix A),

(16)

Introducing the rn-terms from Ogihlvie and Tuck (1969) and some other notations:

(ei,2,&) =

(4,5:,6) =

(ni,n2,n3) = (n4,n5,n6) L x (mi,, m2, m3)

(j V)V4,

(m4,m5,m6)=(i.V)(xV)

it can than be shown that the hull boundary condition for the unsteady wave flow can be written as:

=j(ni

±

ejmi) on (B). (3.19)

where ¿ is written as _{= E +}

l x

L.

From the formulation of the rn-terms it is seen that the hull boundary

condi-tion for the unsteady problem is forced by the double body potential. Calculacondi-tion of the rn-terms involves second derivatives and is therefore subjèct to numerical errors. Nakos (1990) shows that with the application of Stokes' theorem the com-putational effort can be reduced, and only first derivatives are needed. Another common approach is to neglect the perturbation of the steady velocity potential by assuming a uniform incoming flow Ux. The only non-zero rn-terms involved are than, m5 Un3 and m6 = Un2.

3.4 Integral formulation

Using Green's second identity the boundary value problem, described in the pre-vious Chapter, can be expressed in an integral formulation. As already mentioned it is possible to distinguish two different Green's functions (a Green function is

describing the influence at a collocation point x due to a unit source strength

positioned at the source point x8) for the boundary value problem, the Rankine

source given by,

i

G(x,x3) =

(3.20)

I - x31

and the more complex Green function, Wehausen and Laitone (1960i),, for infinite waterdepth (p.476) given by,

(3.18)

i i

G(x,x3) = - -

- + W(x,x3)

(17)

with: R1 = [(xe - x3)2 _{+ (lie - ya)2 + (z -} z3)2]12

R2 = x)2 + (Yc - ya)2 +

(z +

z8)2]h/2

IL(x, x) ₌ [J fo + #fL, + f fL2]f(O,k)dkd0

f(9, k) = g(O,k)/[gk - (w + Uk cos0)21

g(9, k) .k[k(zc+z8)+i!«xc_x3) cos9] cos(k(y _{y) sin O)} 7 = cos(i/4ß) for wU/g 1/4, else = O

L1 and L2, integration paths depending on O and /3

The more complex Green function automatically satisfies the radiation

con-dition at infinity and the free surface boundary concon-dition, but numerically this Green function is very time consuming to evaluate (see for this approach for ex-ample Inglis and Price (1981)). The Rankine source does not satisfies these two boundary conditions but is on the other hand very easy to calculate. Therefore,

when the Rankine source method is used, not only the hull wetted surface () but also the free surface (FS) must be distributed with singularities. The latter

method is followed.

Using Green's second identity and the Rankine function results in the for-mulation ( is used for a potential, not necessary the double body potential).,

-ll

(8)G(xC,x3.)

(xs)'3))

_dS (3.22)

FS

where x is a collocation point on the surface S and x3 is the position of the

singularity. Equation (3.22) is also known as the direct formulation. When a Neumann condition is prescribed on the boundary surface S, a/an is known,

equation (322) results in a Fredhoim integral equation of the second kind, which

can be solved.

A more general formulation of the boundary value problem is obtained when

the integral is expressed using source and doublet terms. A superposition f the

inner and outer expression for the flow problem yields the equation,

=

-_Lf

dS -

Lp(xc) (3.23)

4ir J afl(x3) 2

where the source strength is defined by o =

-

, and where the doublet

strenght is given by

jt

=

- 4'.

The potential of the inner problem (here

'outside' the boundaries) is ' and for the flow itself (here the water flow) is .

in equation (3.22) en (3.23) it is already assumed that the outer boundaries are situated at infinity which removes there contribution from the integral. When the inner potential ' is chosen as zero, thus = 4, equation (3.23;) results in the

(18)

The normal derivative of equation (3.23) is necessary to solve a problem for

which the Neumann condition on the boundaries is prescibed. The intergral reads

as:

¿i(x)

=

--ll

((X8)0G(xcxs)

+ ¡t(x3)

(aG(xx8)))

dS+

8fl(xc) ôfl(xc)

afl()

afl(3) 2 (3.24)

To obtain a unique solution for the problem the values of o and ¡t, or a relation between both, have to be prescribed.

A method often quoted in the ship hydrodynamic problem is to use¡t = O,

or I? = on the boundaries. Equation (3.24) results than in a Fredholm integral of the second kind,

ôW(x)

=

¿

a(Xs)cX3)d5(xs)

+ o(x)

(3.25)

FS,B

from where a source distibution is obtained. The potential itself is in this case

described by the equation:

'Ifl(x) = cr(x3)G(x,x3)dS(x3) (3.26)

FS,

When in this approach the more complex Green function is used, it has been shown

by Brard (1972) that a line integral must be included, resulting in:

B

where C is the contour integral over the intersetion of the hull wetted surface with the free surface ((FS) n ()).

To obtain a solution, the integrals over the surface S are represented by

a summation of integrals over N panels, which represent the surface. The first-order panel methods use flat panels to represent the surface and a constant source strength distribution over a paneL Such a method is described by Hess and Smith (1962) and was introduced in the ship resistance calculation by Gadd (1976) and Dawson (1t977). Nakos (i99O extended the use to the ship motion problem.

if the source distribution over each panel is taken as constant, then equation

(3.25) can be written as,

V(x) = o(x) +

;x3)dI..s3(x3)

(3.28) i=1

(19)

Equation (3.28) represents a set of linear algebraic equations. Once the source distribution is known, the potential function itself can be calculted in each point of the fluid using,

'

_4ir

iSj

3.5 The Kutta condition

In Chapter 2 it has been mentioned that a Kutta condition has to be implemented if the 3D problem is studied for a multihull vessel. This is due to the cross flow

between the two hulls, which is an effect of the interactión between the hulls.

Bertram (1993) studied the steady problem for a SWATH vessel and Kring and

Sclavounos (1991) studied the steady and unsteady problem for a catamaran vessel, both using a Kutta condition to, model the effect of the cross flow. Since the cross

flow effect is not very strong a linearised Kutta condition was used.

A wake was modeled as a thin sheet of panels arising from the stern of each.

hull:. For a wake sheet it is not possible to sustahi pressure thus on each opposite

panel in the wake surface the pressure must be equal, having only an opposite normal. if the wake sheets are named S, and S, then the pressure cndition is:

p(x,t)Is = p(x,t)-

(330)

Since the pressure on each side of the wake must be equal the cross flow must be zero, resulting for the steady case in the condition,

= 0 (3.31)

or equally it is possible to state that the jump potential ,being the difference in potential at each opposite point on the wake sheet, is constant along streamlines

downstream (Kring and Sclavouños (1991)), resulting in,

=0

on (3.32)

where Sw is the jump surface as _Sw

S, - S,,. For the unsteady potential

the condition is different since there can be a shed vorticity (the vorticity at a point

at a constant distance behind the trailing edge is not constant in time), leading to the condition for the unsteady potentiai,

x x3)dS(x3)

(3.29)

(20)

Sölving the Botindary Value Problem 18

3.6 Equation of motion

The pressure distribution on the hull wetted surface can be calculated using the

Bernoulli equation,

p

p(oj +

-

U2 _{+ gz)} _on _(B) _(3.34)

Using the Taylor expansion of the pressure on mean position of the vessel leads to (Nakos (1990)):

U2 +gz)

-

p(pt + VVp)

- p((aV)( VV + gz)

on ()

(3.35)

In this equation the higher order term VjV4 is included which is not very

con-sisted with the linearised free surface condition. However, according to Nakos and Sciavounos (1994) inclusion of these term often makes a sigñif cant difference.

Integration of the pressure over the hull wetted surface . results in the

forcing of the vessel,

F,=_JjpndS

(3.36)

()

where the subscript j represents the forcing direction. The steady part of the

pressure (V( ± )V( +

) - U2 + gz) is taken into account to calculate the

wave resistance and the trim and sinkage of the vessel with respect to the origin of coordinate axis. The pressure due to the unsteady potential must balance the ship motion problem.

The unsteady potential (x, t) can be divided into two different components, leading to the formulation of the radiation problem and the diffraction problem. In

the radiation problem the vessel is assumed to oscillate in undisturbed Water, so

the unsteady radiation potentials p are due to the motion of the vessel in each

mode with unit amplitude = i,..., 6. In the diffraction problem the vessel is restrained from motions and the incoming wave potential diffracts (or scatters at the vessel resulting in the diffraction force. In reality the vessel is not restrained

from motions, thus the diffraction problem is introducing oscillatory rnotioñs of the

vessel. Thus in the calculations, the incoming wave potential o and the resulting diffraction potential p7 have to balance each other. The total unsteady velocity potential is now given by:

=

{[(o + 7) +

(3.37)

(21)

where, the incoming wave is given by: o(x, t)

Wo (3.38)

with k =

We = wo - kU cos(ß), and as the wave amplitude. When

the unsteady potential is substituted into the free surface boundary condition the radiation and diffraction problem results in separate boundary conditions, so it is

pössible to solve the radiation and diffraction problem apart from each other. It is common in the ship hydrodynamic problem to write the equation .of

motion in the following form:

+ Ak)ik + Bkzk + Cjk?lk] =

j=1,...,6

(3.39)

where Mk = component of.the generalized mass matrix for the ship, Ak = added-mass coefficients,

Bk = fluid damping coefficients,

Gjk = hydrostatic restoring coefficients,

F2 = complex amplitudes of the exciting force or moment, We = encounter frequency,

'1k = displacement in the origin of the coordinate system.

In the frequency domain approach the oscillations of the vessel are periodic with the

encounter frequency (jk = r1k&), thus the time exponential term can be omitted from the motion equation, Straightforward combination of equation (335) and equation (3.39) results in the expression for the added mass, fluid damping and restöring force (Nakos (1990)) for j, k = 1,.

Ak = _R jf (iWeçOj+ VVça)nkdS

(3.40)

Bk =

Jj(iwecj

+ VV)nkdS

(3.41)

6'jk = p

Jf(c. V)(VVF

+ gz)ndS

(3.42)

The pressures arising from the diffraction potential (diffraction force) and

the incoming wave potential (Froude Kri'loff force) are the exciting force, being:

F = _pQJJ[iw(po +

) + VV(o + p7)]njdS

, j = 1, ...6 (3.43)

(22)

Solving the Boundary Value Problém 20

3.7 Solving the different potentials

The total' flow around the vesse! is described with three different potentialsas was

presented in the introduction of this chapter. First of all the. double body flow is solved. When the double flow is known the steady and the unstead'wave potential can be calculated. Since the interaction between the steady and unsteady flow was

neglected the free surface condition was balanced in a steady part, e.g. equation

(36), and an unsteady part, e.g. equation(3.Ii). it is therefore. not necessary to

solve the steady wave part to obtain the solution of the unsteady part.

.3.7.1

The double-body flow

The. method' to calculate thed'ouble flow is .dès.cribed in detail by Hess and Smith

(l962) The .underwater part of the vessel is mirrored in the undisturbed free

surface z = O and the total bod.y surface is discretised with NB panels, On each panel a collocation point (x) .is chosen. In 'the double body problem 'there is no free surface. On the underwater par.t of the vessel the Neumann condition has to

be satisfied.

A matrix of influence coefficients is calculated using

= n

Vj

where n1 is the normal vector on the ith panel and is. the vector velocity at a collocation point of the ith element due to a unit source density on the jth. element.

The matrix is thus filled usin.g thenormal derivatives .of the Green's function. For points' sufficiently far away from the collocation point under consideration .the

source' distribution over a panel can be approximated .by a source 'strength in a source point. in the panel to simplify the calculations. The normal velocity at the ith collocation point due to the whole body is thus,

Na

> A,

j=i

The main diagonal terms in 'are just equal to 1/2, making the. matrix A domi-nant in the main diagonal. i,he total velocity at ,a point must equal the negative of the normai component of the onset flow (here U) at that point, thu,

=n.0

i='l,...,NB

(3.45)

This is a set of linear equations replacing the integral formulation given in equation

(328). Solving equatión (3.45)' provides the source density on each panel. If

necessary the potential itself can be obtained by means of equation .(3.29). The NB

vn1 (3.44')

(23)

//

total velocity vector at a collocation pc'int on the body can be calculated using: NB

V=V,o,+U

i=1,...,NB

(3.46)

1=1

3.7.2

The steady wave flow

As a result of the forward .speed of the vessel the free surface gets disturbed and a steady wave pattern, occurs. The wave pattern is formed in the so called Kelvin sector behind the ship. The steady wave potential has been solved by many re-searcher using different methods. If the Rankine method is used the free surface and the underwater part of the vessel' are discretised using respectively NFS and

NB panels. On each panel a Neumann condition can be specified.

If the free surface condition from Nakos (1990) is used, equation (3.6),, the

govern linear free surface condition on z O is written in the form:

-g

= VV(VV) + V

. V(VV) - zzVV +

V(VV)

-

-.

U2) (3.47)

In a collocation point x. on the undisturbed free surface the double body velocity can be calculated using the expression:

N

= >JVrj + U

(3.48) j=1

On the hull wetted surface (B) the Neumann condition of no water pene-trating the ship's hull must be satisfied, leading to:

n Vq = O

on (B) (3.49)

since the double body flow already compensates the onset flow.

Nakos (1990) solved the. problem using the direct formulation, that is

equa-tion (3.22). As a result f the Neumann condition on the hull surface the góvern equation reads:

«xc) -

G(x, x3)dS +

«x)

ac(x,)

= 0 (3.50)

Using equation (3.47) the (x3) in equation (3.50) can be. discretised in «x3) values. Nakos (1990) used a hi-quadratic spline function herefore. The discretised integral formulation (3.50)' can now be written in the matrix form:

/ A[BJEB] I 'j2

(À.FB

_\

u 13 uj

\

I

/

/ A[B) I 't'i 1,[FS] 'rl,

(24)

where the matrix coefficients are build using the Green function and it's deriva-tives.

The use of the direct formulation to solve the problem as was proposed by Nakos (1990) is only possible by virtue of the linearisation process to the surface

z = 0.

If the linearisation was performed on another surface the double body

potential derivative in the z-direction () would not vanish from the problem.

This term is than introducing second derivatives of the potential ' and thus q can not be expressed explicitly.

Therefore Jensen (1988') used the more general integral expression, that

is equation (3.28), to solve the steady wave potential. As a result the source

distribution ¿i(x) is obtained and not the steady potential4j itself.

In discretised' form the problem was written as,

The coefficients Ck can be cal'culate,d usign the Green function, it's derivative and

the known double body potential. To obtain the coefficients for the free surface

Jensen (1988) introduced the following free surface condition,

(2a + BV)V'W + (VV)V'P

- g'P = 2aV +

B'((V)2

+

U2 _{-1- gZ) (3.53)}

where a and B are given by Jensen and are only a function of the double body potential. The arbitrary flow 'level is Z. The unknown total flow potential 'P was substituted via the equation,

NB-(-NpS

=

oVGA

for j' = 1, ...,NFS ('3.54)

where A is the panel area. The last equation is according to Jensen (1988) not

very accurate in the flow near the body 'surface. When equation (3.54) is used in equation (3h53) the coefficients ck are obtained' and the problem can bç solved. With the Bernoulli equation (3.9) the new free surface for the next iteration step can be derived.

Other panel methods often use some kind of finite difference scheme to calculate the derivatives of the velocities. The first to introduce this' method was Dawson (1977): who used a four point upstream finite difference scheme. Using such a scheme introduces numerical damping into the solution. The derivatives of the velocities are expressed in terms of a using the finite difference scheme and after solving the linear system the values for a are obtained on the. hull and the

free surface.

NB+NPS

(25)

3.7.3

The unsteady wave flow

The unsteady flow is due to the motions of the vessel and as a result of the incoming

wave. Since the motions are assumed to be harmonic the unsteady wave potential is written in the form:

= (po( )C6t (3.55)

where We is the wave encounter frequency.

The solution for the unsteady wave potential can be obtained by using the same method as was presented for the steady problem. If the linearised condition on z = O from Nakos (1990) is used, equation (3.11) is rewritten in the form:

g'p

- WÇO + 2jWeV.7VçO + Vc1 V(VV) +

V(VV) - zz(iWCp + VV')

The hull boundary condition in the radiation problem is given by:

0'Pk

----=zwnk+mk

for k= 1,...,6

(3.57)

Un

and for the diffraction problem by:

f358

ôn

k.

Following the procedure form Nakos (1990) the integral representation for

the unsteady wave problem is:

-

C(x, x3)dS

+

ç(xs)

(x,x)

_dS i (3.56)

ô(x3)

G(x,x3)dS

(3.59) B

where (x3) is obtained from equation (3.57) for the radiation problem and from equation (3.58) for the diffraction problem. Using the free surface boundary

con-dition and a discretising scheme the potential (x3) can be discretised in the

potentials themseif.

The matrix representation fOr the radiation condition reads now as (for

k=1,...,6),

[B)[BJ [BIEFS] [B]

(

\ I (co

) '\

(zWnk+mk\

.

I\AS]B

AS]FS])

9[FS])) =

o

₎

for z, = 1,...,NB+Nps

(26)

and for the diffraction problem the matrix formulation is,

A]SI

_'\((çc'7

IB] )

)

[FSIEFSI ₍ IFS] J )i / , [B] I P° )n 0 for

i,j=1,..,NB+NFs

(3.61)

where the matrix coefficients A are build using the Green function and it's

deriva-tives.

The unsteady flow has been solved by Yasukawa (1990) using the same

method as Dawson (1977) applied for the steady problem. A finite difference

scheme was used: and numerical damping was introduced in the unsteady free

surface condition by virtue of a so called Rayleigh viscosity j. His govern equation

reads (the time dependent term was used):

w2ço + 2iwiçoj + 4?çoii + - iwço = O on z = 0 (3.62)

The Reyleigh viscosity is employed to introduce the radiation condition of so called k2 waves while the radiation condition for the k1 waves is modelled by the upstream finite difference scheme.

(27)

Appendix A

A note

on the hull boundary

condition

A. i

Introduction

The linearised' hull boundary condition as derived by Timman and Newman (1962)

is presented here. When the vector notation from Nakos (1990) is introduced it is shown that in his equation (11.3.4) a typing error exist.

it is also possible to write t:he linearised hull boundary condition in a very well known expression using the rn-terms from Ogilvie and Tuck (1969). This is pre-sented at the end of this Appendix.

A.2 Linearised Hull boundary condition

The cartesian coordinate system = (x1, x2, x3) is moving through the fluid with

constant velocity (the mean forward velocity of the ship) in the s1 direction and, with the 53 vertically upward. A second coordinate system is defined as ₌

where ¿ is defined as an infinitesimal oscillatory displacement vector measured from

the mean position of the vessel () The coordinate system

' = (x(t), x(t),x(t))

is thus fixed with respect to the vessel. Thus,

= X -'- X'

The time derivative of this oscillatory vector is,

ô ., (ôx1' ôx2' ôX3'

The kinematic boundary condition has to be satisfied on the body itself. We

(28)

Appendix A 26

the equation F(x(t), x(t), x(i)) = O. In the inertial coordinate system (in which

the motions are periodic) this equation is G(xi, x2, x3) = O.

The exact boundary condition which holds on the actual surface of the body is,

O=F

-=VF+V VF+

_(

_Oxi

V1F)_f(--

_Ox2

V1F)_/(

_Ox3

The next step is to linearise this hull boundary condition to the mean wetted

surface (e), using the Taylor expansion,

-I-(a V)V

+0 ((a.

V)2V)

(S) (S) (S)

The problem is linearised since the amplitude of the oscillatory motion is assumed

to be small comparable to the wave length. The following condition on () is than obtained,

O = -

. V3F +

[v

+ (a .

V)V]

₊

vlF) _Y(- VF)

vi1)]

When the second order terms (O(a2)) are neglected, the following is obtained,

V.VXIF.VIF(a.V)V.VXIF+

v.

(-- vF) +(--

_Oxi

ViF +

(-

v2i')]

Ox2

j

\0x3

F)]

OF Ox' OF Ox' OF Ox'

V

+

_OxT

+ (OF Ox' 0F Ox' 0F Ox' \

OxÇ Ox1 + OxOx1 + Ox Oxi)

+

(OF Ox 3F Ox

OF Ox\

+

k (OF Ox OF 3x OF 3x +3 + 7 + ---- j

( + +

(29)

Appendix A 27

This last equation is comparible with equation (3) in Timman and Newman (t962).

Since VF is a vector normal to the body surface, it follows that

The vector identity',

is introduced and since the fluid is incompressible, V V1 = 0, and since the

normal velocityon the wetted surface is zero, n V1 = 0, the next equation can be derived, which is comparible with equation (4) in Timman and Newman (1962),

The total velocity potential ' can be divided into three components. Here the

approach of Nakos (1990) is introduced, and thus the total velocity potential is. written as 'I' instead of & Thus,

'I'(, )

_{= () + «) +}

i)

where I() is the double-body flow, q) is the steady wave flow, and ço.(,t) is

the unsteady wave flow.

On the mean position of the vessel, (s), the body boundary condition must be

satisfied and thus,

=.0

an

Thus the linearised hull boundary condition can be written as,

When this equation (A.3) is compared with equation (1L3.4) in Nakos (1990), it is seen that the minus sign in brackets is a plus sign in Nakos (1990). The latter is thus wrong.

1see e.g.: Gradshteyn and Ryzhik (1980.), equation 10.31.7

(30)

Appendix A 28

A.3 Introducing the rn-terms

An alternative form of equation (A.3) can be obtained when the vector ¿ is written

as,

and is substituted in equation (A.3).

The time derivative term in equation (A.3) becomes,

-. a

-

-. n = _(e + ì X x) . n a -.

=.n+(íxx)

n -.

ía

-. -.

=---n+xx+1x--(a

..\

=n+

_t,--

XX) fl

a(

.a

.

=n+-- (xxn)

The second term from equation (A.3) is evaluated into,

_[(.v)v_(v.v)]

.

-

[(+

x

y)

v -

(V . _)] .

=_.[.v)v_(v.v)ei+

il [((fl

x .

y)

v -

(V . X 11±12

The term Ii is,

ii = -. [. V)V - (V

V)]

= -.

V)V]

+ o

In the first term of 1 the vectors ¿ and il can be interchanged since the' rotation of the fluid, V4, is zero, (potential flow). The second term of I is zero because when

the term (V.

_V)eis written out it is found to be just V, and thus from the body

(31)

Appendix A 29

boundary condition that no flow can go throught the wetted surface (il. V1 = O)

the second term of I becomes zero.

The term '2 is,

12 = il [((a x

.

y) v

+ [cv 'c7)( x

=

(lì x

).

[&. V)V]

± n

[_ x (V

V) + lì X (V

V)x]

=.([(n.V)V] xx)+n.[O+xV]

=

.

([(n. v)v]

x + . (V

xn)

= .

([(n

.

V)V]

x x + V x

[(n. V)xJ)

= -.

[(n

V)(x x V)]

In this derivation the. same arguments. are used as before with the term '1. The

vector identity (a. V)(b x c) = c x (a. V)b + b x (a. V)c can be found in e.g.

.Timman and Reyn (1965).

Substitution of the time dependent term, and the terms Ii., and '2 in equation

(A.3) leads to the following equation,

(A.4)

Introducing the so-calléd rn-terms from Ogilvie and Tuck (1969) and vector

nota-tions, (ni,n2,n3.) = Ti

(n4,n,n6) =

x ii (m1,m2,m31) (rn4, ms,.m6)

₌

. V),(

x V)

=

[(n. V)V4]

x x + V x n

equation (A.4) can be written in the well known form,

(32)

Bibliography

30

Bertram, V. (1993), A computational fluid dynamics method for swath ships, in 'Proc. 2nd mt. Conf. FAST', pp. 253-261.

Brard, R. (1972), 'The representation of a given ship form by singularity distri-bution when the boundary condition on the free surface is linearized', Journal of

Ship Research 16(1), 79-92.

Chang, M. S. (1977), Computations of three-dimensional ship-motions with

for-ward speed, in 'Proc. 2nd mt. Conf. on Numerical Ship Hydrodynamics', Berkeley, pp. 124-135.

Dawson, C. W. (]J977), A practical computer method for solving ship-wave

prob-lems, in 'Proc. 2nd mt. Collf. on Nunïerical Ship Hydrodynamics', Berkeley,

ppi 30-38.

Gadd, G. E (1976), 'A method of computing the flow and surface wave patterñ around full forms', Trans. Royal institution of Naval Architects 113, 207-219. Gerritsma, J. and Beukelman, W. (1967), 'Analysis of the modified strip theory

for the calculations of ship motions and wave bending moments', Journal of Ship

Research 14(156), 7-24.

Gradshteyn, I. S. and Ryzhik, I. M. (1980), Table of integrals, series, and products, Academic Press.

Radier, J.. .B., Lee, C. M., Birmingham, J. T. and Jones, H. D. (1974), 'Ocean

catamaran seakeeping design, based on the experiences of USNS Hayes', Tràns.

SNAME 82, 126-161.

Hess, J. L. and Smith, A. M. O. (1962), Calculation of non-lifting potential

flow about arbitrary three-dimensional bodies, Technical Report Report No. E.S.

40622, Douglas Aircraft Co., Inc.

Inglis, R. B. and Price, W. G. (1981), 'Calculation of the velocity potential of a translating, pulsating source', Trans. Royal institution of Naval Architects

(33)

Bibliography 31

Jensen, C. (1988), _{Berechnung der stationären Potentialströmu.ng um ein}

Schiff unter Berücksichtigung der nichtlinearen Randbedingung an der

Wasser-oberfläche, PhD thesis, University of Hamburg. Bericht 'Nr. 484.

Kashiwagi, M. (1993), Heave and pitch motions of a catamaran advancing in waves,

in 'Proc. 2nd lut. Conf. FAST', Yokohama, Japan, pp. 643-655.

Kim, Y. H. and Lucas, T. (1991), Nonlinear ship waves, in 'Proc. 19th Symposium on Naval Hydrodynamics', Washington, pp. 439-452.

Korvin-Kroukovsky, B. V. and Jacobs, W. R. (1957), 'Pitching and heaving

mo-tions of a ship in regular Waves', Trans. SNAME 65, 590-632.

Kring, D. and Sciavounos, P. (1991.), A new method for analyzing the seakeeping

of multi-hull ships, in 'Proc. ist lut. Conf. FAST', Vol. 1, Trondheim, Norway,

pp.' 429-444.

Nakos, D. E. (1990), Ship Wave Patterns and Motiöns by a Three Dimensional Rankine Panel Methos, PhD thesis, Massachusetts Institute of Technology. Nakos, D. E. and Sclavounos, P. D. (1994), 'Kelvin wakes and wave resistance of

cruiser- and transom-stern ships', Journal of Ship Research 38(1), 9-29.

Ni, Si-Y. (1987), Higher order panel methods for potential flows with linear' or non-linear free surface boundary conditions, PhD thesis, Chalmers University of

Technology, Göteborg, Sweden.

Ogilvie, T, F. and Tuck, E. 0. (1969), A rational strip theory of ship motions:

Part 1, Technical Report 013, Dept. of Nay. Arch. and Mar. Eng., Unliversity of

Michigan.

Raven, H. (1992), A practical nonlinear method for calculating ship wavemaking

and wave resistance, in 'Proc. 19th Symposium on Naval Hydrodynamics', pp.

60-72.

Sal'vesen, N., Tuck, E. O. and Faltinsen, 0. (1970), 'Ship motions and sea loads',

Trans. SNAME 78, 250-287.

St. Denis, M. and Pierson, W. J. (1953), 'On the motion of ships in confused seas', Trans.. SNAME 61, 1-53.

Timman, R. and Newman, J:, N. (1962), 'The coupled damping coefficients of a symmetric ship', Journal of Ship Research 5(4), 1-7.

Timman, R. and Reyn,

J. W. (1965),, Vectoranalyse, Delftsche Uitgevers Maatschappij, Delft, The Netherlands.

(34)

Bibliography 32

Van 't Veer, R. (1993), Catamaran seakeeping prediction, Technical Report 980-S,

Delift University of Technology.

Wehausen, J. V. and Laitone, E. V. (1960), Surface Waves, Vol. 9 (Fluid Dynamics) of Encyclopedia of Physics, SpringerVerlag.

Yasukawa, H. (1990), 'A rankinepanel method to calculate unsteady hydrodynamic

A Literature Survey into the 3-D Ship Hydrodynamic Problem in the Frequency Domain