Boundary integral method for the computation of the potential flow about ship configurations with lift and free surface effects

z

r

(Ji p-) C

ARCHIE

_{Technische Hogeschool}

Deift

NATIONAAL LUCHT- EN RUIMTEVAARTLABORATORIUM

NATIONAL AEROSPACE LABORATORY NLR

THE NETHERLANDS

NLR TR 85142 u

BOUNDARY INTEGRAL METHOD FOR THE COMPUTATION OF

THE POTENTIAL FLOW ABOUT SHIP CONFIGURATIONS

WITH LIFT AND FREE SURFACE EFFECTS

BY

C.M. VAN BEEK,W.J. PIERS AND

BOUNDARY INTEGRAL METHOD FOR THE COMPUTATION OF THE POTENTIAL FLOW ABOUT SHIP CONFIGURATIONS WITH

LIFT AND FREE SURFACE EFFECTS by

C.M. van Beek, W.J. Piers and J.W. Slooff

SUMMARY

A boundary integral or "panel" method is described for computing the three-dimensional, steady state, potential flow about ship configurations with or without lifting surfaces in the presence of a free surface in an initially un-disturbed sea. The low order method is characterized by a special modelling of

the free surface boundary condition, by an iterative procedure for solving the

resulting system of equations and by the possibility to perform calculations about configurations with lifting elements (rudders, hydrofoils, keels, stabi-lizing foils). A description of the method is given and results are presented of a limited convergence study which gives directives for the required exten-sion of the free surface grid and for panel distributions on the configuration as well as the free surface. Several examples of application are shown,

in-cluding both simple hull forms and complex (sailing) boat configurations.

Division: Fluid Dynamics Completed : 851115

Prepared: CMvB/.._ Ordernumber: 249.503/626.701

CONTENTS

Pa ge

LIST OF SYMBOLS 3

1 INTRODUCTION 5

2 MATHEMATICAL STATEMENT OF THE PROBLEM 7

3 SUMMARY OF RESULTS FOR A 2D-MODEL PROBLEM 8

4 METHOD FOR THE THREE-DIMENSIONAL PROBLEM 12

4.1 Double model formulation 12

4.2 Method of discretization 14

4.3 Method of solution 18

4.4 Forces and moments 21

5 RESULTS OF A CONVERGENCE STUDY 27

6 EXAMPLES OF APPLICATION 36

6.1 Some second DTNSRDC workshop examples 37

6.1.1 Wigley's parabolic hull 38

6.1.2 Series 60, CB = 0.60 39

6.1.3 Strut-like hull form 39

6.1.4 Vertical cylinders 41

6.2 Sailing boat configuration with different keels 42

7 CONCLUSIONS AND RECOMMENDATIONS 45

8 ACKNOWLEDGEMENT 47

9 REFERENCES 47

1 Table

APPENDIX A: Some comments on the free surface grid generation 51

42 Figures

LIST OF SYMBOLS

CL hydrodynamic heave force coefficient, related to ½ p U2 S

CD resistance coefficient (= D/½ p U2 S)

CM hydrodynamic moment coefficient, related to ½ p U2 S Lref

C wave-making resistance coefficient, related to ½ p U2 S

C side force coefficient (directed in z -direction), related to

Z5

_½pUS

S D resistance D. induced resistance Dw wave-making resistance U Fr Froude number L ref g gravitational constant

h stepsize along streamline

L reference length

ref

downstream extension of the free surface grid

normal vector, directed into the fluid, see Fig. i

p pressure

S wetted surface area of ship at rest

U total velocity in x-direction

U velocity at upstream infinity (see Fig. 1)

u perturbation velocity in x-direction, or in section 4.4:

x -direction

V total velocity in y-, y-direction

total velocity vector

y perturbation velocity in y-, y-direction

X

y Cartesian system of co-ordinates (see Fig. 1)

z X

y: Cartesian system of free stream co-ordinates (see Fig. 1)

z s

angle of yaw (see Fig. 1)

r vortex strength

upstream difference operator wave height

À wave-length

a source strength velocity potential

perturbation velocity potential

perturbation velocity potential (dimensionless)

dm double-model velocity potential

p density

Subscripts

b refers to body

bs refers to body sources

by refers to body vortices

d refers to doublet

dm refers to double-model

fs refers to free surface

fsd refers to free surface doublets

fss refers to free surface sources

i refers to i-th collocation point

L refers to streamline direction

n refers to normal direction

pr refers to pressure integration

s refers to source

T refers to Treff tz-plane

V refers to vortex sheet

1 INTRODUCTION

This report describes a boundary integral or "panel" method for

computing the three-dimensional, steady state, potential flow about

mono- and twin-hull configurations in the presence of a free surface in

an initially undisturbed sea. A FORTRAN program (called HYTROPAN) has

been developed based on this method. The method is a modification of the NLR low order PANEL method (Ref. 1). The computer program of the latter has been operational since 1969 and has been used for predicting the

(potential) f lòw about numerous complex aeronautical configurations.

The current "marine" version, which has been operational since

early 1984, is capable of treating a broad class of configurations

inclu-ding conventional displacement ships with or without lifting appendages,

catamaran and small waterline area twin-hull (SWATH) ships and hydrofoil

configurations. Hydrodynamically the range of useful applicability ex-tends from arbitrary thick bodies at low Froude number to slender bodies at Froude number of 0 (1). Fully submerged configurations, such as

sub-marines close to the water surface, are not yet admissible, but this

will require only minor modifications to the program.

The position of the ship is kept fixed during the calculation.

Sinkage and trim which will occur in reality are not accounted for in the present version. The conditions of tangential flow along the body

and smooth flow at the trailing edges of the lifting surfaces are

im-posed in a finite number of points on the body surface and along the

trailing edges respectively. The free surface boundary condition is imposed in a linearized form on the undisturbed (plane) free surface.

Upstream waves are prevented by the use of a specially developed,

up-stream difference operator in the linearized free surface boundary

condition. The method is capable of treating ships with or without angle

of yaw.

The flow calculation is split in two parts: firstly a calculation

with Froude number equal to zero (double-model problem) is executed,

followed by the generation of the free surface grid and secondly the

required number of non-zero Fronde number calculations. In both

calcu-lations the incompressible potential flow is described by the Laplace

equation and is obtained by a superposition of a) the undisturbed flow,

b) the flow generated by an unknown source distribution on the conf

igu-ration surface, c) (in the presence of lifting surfaces) the flow

vortex sheets, and d) (for the non-zero Froude number calculations) the flow generated by an unknown combined source/doublet distribution on the

(mean) free surface. The flat panels carry singularity distributions of constant strength.

The condition of tangential flow along the body together with the

condition of smooth flow at the trailing edges of the lifting surfaces (Kutta condition) and (for the non-zero Froude number calculations) the free surface boundary condition lead to a large system of linear equa-tions for the unknown source, vortex and doublet distribuequa-tions. This

system is solved by an iterative scheme, especially adapted to this kind of problem. Once the singularity strengths are known hydrodynamic

quantities such as pressure distribution, heave force, resistance and pitching moment can be calculated by integration of the pressure distri-bution over the configuration. Resistance is also calculated by means of

a far-field momentum approach.

The method described in this report may also be regarded as an

extension of the method of Dawson (Ref. 3). The extensions consist of a)

the allowance for lifting surfaces, b) a combined source/doublet

distri-bution on the free surface, rather than a source distridistri-bution only,

together with a special difference formula ensuring numerical stability of the discretization at moderate and high Froude numbers as well as an improved accuracy relative to Dawson's method and c) an iterative

solution procedure for the resulting system of equations which permits a larger number of unknowns to be used together with a decrease in

computer time required to solve the system of equations.

After a survey of the governing equations (chapter 2) chapter 3 summarizes the results of a study in two dimensions on free surface dis-cretization schemes which underlies the present three-dimensional

method. In chapter 4 the three-dimensional method is described with

attention to the double-model formulation, the method of discretization,

the iterative solution procedure and the calculation of forces and

moments with emphasis on resistance calculation via a momentum approach.

Chapter 5 discusses results of a limited convergence study. This con-cerns the required extension of the free surface grid and the effect of

panel distribution on configuration and free surface. In chapter 6

several examples of application are presented. Firstly results are shown of computations on some testcase configurations, as specified for the second DTNSRDC workshop on ship wave resistance computations. Secondly

results are presented of calculations on two sailing boat configurations. Concluding remarks and recommendations are collected in chapter 7.

2 MATHEMATICAL STATEMENT OF THE PROBLEM

The HYDROPAN program is capable of computing the three-dimensional,

steady state, potential flow about complex ship configurations. The

where:

- x, y and z are co-ordinates in a right-handed Cartesian co-ordinate

system with the origin in the undisturbed free surface somewhere on the configuration centerline (Fig. 1), the x-axis is aligned with the configuration centerline, directed from bow to stern; the y-axis is aligned with the normal on the undisturbed free surface, positively directed upward; consequently the z-axis is located in the undisturbed

free surface

- _{is the angle of yaw: the angle between the x-axis and the velocity}

vector at upstream infinity; this angle is positive when the component

of U in the z-direction is positive (as in Fig. 1)

- ri is the value of y at the free surface (wave-height)

- n refers to the direction normal to the body surface (positively

directed into the fluid)

- g is the gravitational constant.

The free surface condition (4) is, after elimination of ri by means

of (5), linearized and applied at y = O (mean free surface) instead of

at the actual free surface.

The problem described above can be solved by means of distributions of singularities (sources, doublets, vorticity) on the body and the free

surface.

flow-field is described by the velocity potential which must satisfy

the following conditions:

= O inside the fluid (1)

= O on the body

n (2)

= (U cos , O, U sin ) at infinity (upstream and

infinitely far beneath the free surface)

gri+½(2-I42+2-U2) = O on the free surface

(3)

(4)

r - + = O on the free surface

xx y

z z

Before going into the details of the 3D-problem we will first summarize the main results of a study in two dimensions by the second author, [4], on discretization schemes for the free surface.

3 SUMMARY OF RESULTS FOR A 2D-MODEL PROBLEM

In the 2D-model problem of [4] the boundary condition on the free

surface is linearized with regard to U and imposed on the undisturbed

free surface (y0).

The objective of the 2D-study was to identify discretization

schemes for the linearized free surface boundary condition that are

stable at all Froude numbers and lead to a matrix structure that allows

an iterative solution technique for the resulting system of algebraic

equations. The latter is highly desirable from the point of view of

com-putational efficiency.

The 2D-model problem is illustrated by figure 2. It represents the

flow around a submerged body in the vicinity of a free surface. Part of

this undisturbed free surface is divided into panels of equal size. The

governing equation for the perturbation velocity potential p is

(6)

On the free surface the boundary condition, in linearized form, reads:

v+Fr2-O atyO

(7)

where y = and u = are the perturbation velocities normal and

tangent to the free surface. At infinity there holds:

V.p+O

forx+oo

(8a)

y + -

(8b)

Note that the problem is non-dimensionalized through the free stream

velocity U and a suitable length scale Lrf giving rise to the Froude

(9)

U

Fr =

gL

For reasons of simplicity the body was represented by a dipole-type of disturbance (axis in negative x-direction), approximating the flow

around a submerged circular cylinder. Boundary condition (7) can then be

wrítten as:

2 u

v+Fr

=g(x)

where y and u represent the perturbation velocities due to the free water surface only and g(x) contains the velocity components due to the

dipole-like disturbance.

A crucial aspect of the problem formulated above is that condition

(8a) expresses that upstream waves must vanish, while downstream waves

are allowed. Omitting (8a) renders the problem not unique: both upstream

and downstream travelling waves are possible. It is the crux of any com-putational method for the problem described above to build-in a

mechanism that properly controls the elimination of upstream travelling

waves and guarantees a unique solution. Dawson (Ref. 3) has solved this

problem in an elegant manner by replacing the terni by a backward

difference. He has demonstrated, both in 2D and 3D, that, when incorpo-rated in a panel method, this discretization leads to a solution with the correct physical behaviour. The problem and solution are similar to

one in compressible (transonic) potential flow where solutions with

(non-physical) expansion shock waves must be excluded [5].

To complete the formulation of the discrete problem we note that, according to Green's theorem, eq. (6) with boundary conditions (7) and

(8) can be solved by a distribution of singularities (sources, doublets)

on the free water surface y0.

Firstly a source distribution on the free surface is considered. Because both Dawson's and the basic NLR PANEL method (Ref. 1), from which the NLR HYDROPAN method is an adaptation, utilize constant source density panels, these will be considered only, although higher order

In the scheme proposed by Dawson the constant density per panel is

combined with a special 4-point backward difference operator. This

difference operator, which will be denoted by DWS is defined for

uniform meshwidth h by:

DWS

= ½(3 +

4) (10)

where and are standard 3-point and 4-point backward differences:

= ½(3ui_4u_i+u_2) (11)

= 1/6 (llu.-l8u.1+9u.2--2u.3) (12)

Fourier analysis shows that the discretization with source distributions

only becomes unstable for higher Froude numbers. This explains the

wiggles which Dawson reports to be present in some of his solutions.

A discretization scheme with a doublet distribution has also been

investigated. Once again the study was confined to constant density

panels. The tangential velocity can not be calculated directly with the

constant doublet density representation. However, utilizing the

knowledge that half the doublet strength i-i equals the potential one can write:

1 it

u

2 dX

where the derivative in (13) (just as the derivative is approximated

by backward differences. Here Fourier analysis indicates stability for

all Froude numbers. It can be shown that this conclusion is not

basically affected by the type of backward difference formula used. The

scheme described above is not necessarily the only stable scheme.

However, other schemes have not been investigated in depth by the

author.

The drawback of using a doublet distribution is that the structure

of the resulting matrix does not allow a convergent iterative solution

procedure. A combined source/doublet distribution scheme is found to

possess both required properties: stability at all Froude numbers and

good prospects for a rapidly convergent, iterative solution procedure.

with NLR 3,s 1,s 2

+(1_2,$)3

NLR 3,d 1,d 2

+(1_2,d)3

h2 h2 1,s =

-2Fr4 2,s =

-

6Fr4 h2 h2 Fr4

2,d3F4

pl = .2205

u = velocity induced by source panels

Ud = velocity induced by doublet panels

source strength a

doublet strength = -Fr2a

The best choice for the relation between source strength a and doublet

strength p is found to be:

ji =

-Fra

(14)

The combined scheme again uses backward difference formulae for

(both sources and doublets) as well as for

As shown by Dawson and in reference 4 higher order difference formulae are required in order to obtain acceptable accuracy of the

results for practical numbers of panels.

Analysis has resulted in the following scheme:

22

p1h p1 h 2 NLR /h+Fr2 _ud/h = O i > n

v+(1++

)Fr u 4 3,s s 3, s Fr2 Fr V = 0, 1 i n - s

22

p1h p1 h

Fr2 NLR u 1h have been added to eliminate

The terms (--+

4)

Fr Fr 3,s s

the lowest order truncation errors of the source distribution for both

damping and dispersion. They are 0(h) and 0(h2) respectively, so

consis-tency is retained (the truncation error of the basic NLR PANEL method as well as of the HYDROPAN method is 0(h)). Eqs. (16) and (17) represent

consistent (2nd order) approximations.

A remark must be made with respect to the appearance of the require-ment v=0 for 1in in (15). Because of the use of backward differences the latter can not be applied directly to the first few panels. Moreover

it can be shown that for fundamental reasons an initial condition must

be imposed. The simplest way to handle this is to set the undefined

quantities, i.e. a and .j, equal to zero upstream of the forward boundary

of the panel distribution. However, this appears to be insufficient:

waves are not damped sufficiently in forward direction. Therefore an

initial condition has been applied in a more explicit manner, namely by

requiring v=0 for the first 5 panels.

The properties of the Dawson source scheme and the present combined source-doublet scheme are illustrated by figures 3 and 4. In these

figures numerical results for the 2D "submerged dipole" model problem

are compared with exact analytical results, [4]. The results obtained

with the Dawson scheme clearly reflect a large dispersive truncation

error. The latter is shown to be 0(h) in reference 4; the damping being

0(h4). For the NLR scheme the dispersion error is 0(h3), while the

damping error is also 0(h4).

The scheme selected for the present 3D method is similar to the NLR

scheme described above.

4 METHOD FOR THE THREE-DIMENSIONAL PROBLEM

4.1 Double-model formulation

For the three-dimensional problem use is made of the so-called

double-model representation (Fig. 5). In the double-model representation

the total potential is written as = + ', where is the

dm dm

and its mirror image with respect to the mean free surface at Fr = 0. '

is the potential due to the combined source/doublet distribution on yO and the additional source/vortex distribution on the body and its image required to satisfy the body boundary condition at non-zero Froude

numbers. For zero Froude number ' is equal to zero and the plane yO

acts as a plane of symmetry.

In order to linearize the free surface boundary condition, first the wave-height n is eliminated from (4) by means of (5), resulting in

the following equation:

½[

(2..42.42)

+

($2+2..42) ]+g

= O

xx y zx

z z

y zz

y

If (20) is imposed on y=O instead of on the free surface and if ' is

assumed to be small with respect to eq. (20) can, following

reference 3, be linearized as follows:

LL

+ g = 2

2

y dm dmLL

In (21) the subscript L denotes differentiation along a streamline of

the double-model flow at Fr = .0 in the plane y=O.

Introducing non-dimensional quantities through the free stream

velocity

Ç

and a suitable length scale Lref we write:

U

Fr-i,

vgL

ref

=U (cos-hp),

=Up,

=U (sine-hp)

x x y

y z

z

dmUdm

, 4

=U

(24)

dm

dm

dm

dm

X X

y

y z z

where tp is the perturbation velocity potential. The resulting set of

equations can then be summarized as:

= -n cos -n sin 8 on the body surface (boundary

n x z

condition) and, if applicable, at the trailing edges (Kutta condition)

Vc.P = (0,0,0) at infinity (upstream and beneath the free (27)

surface)

2(2 2

Fr2(4d2P ) + .P = Fr -cos 8

L L y _{dtnL dmLL} 4'dm4'dmL

S]fl

_8dmdmL

(28)

at the free surface

= O on the first 5 panels near the forward boundary (29)

of the free surface grid.

The fact that the free surface boundary condition is imposed on the plane y=O instead of on the actual free surface implies automatically

that the boundary condition on the body surface is imposed on the part

y O instead of on the part that is submerged by the actual free

surface pattern.

4.2 Method of discretization

Eq. (25) subject to boundary conditions (26), (27), (28) and (29)

is solved by expressing the perturbation potential (P in terms of a

dis-tribution of singularities on the body, on the mirror image of the body and on the undisturbed free surface.

The body and, automatically, its image are approximated by

quadri-lateral surface panels (constant source density) and, if applicable, by

a vortex system, exactly in the same way as in the NLR PANEL method [i] (Fig. 6). The free surface is partly covered by quadrilateral, combined source/doublet panels (constant source and doublet density) on y=O.

Although not strictly necessary for Fr .0, retaining the double-model

representation has the advantage that the extent of the singularity

dis-tribution on the free surface can be kept bounded, as will be shown in

section 4.4. The required extension of this singularity distribution is

dealt with in chapter 5.

Before the flow problem can be solved the panel distribution on the

free surface (free surface grid) must be generated. This implies

per-forming the Fr = .0 double-model calculation by means of the NLR PANEL method and tracing streamlines in the (symmetry) plane y=O. In the HYDROPAN program this double-model calculation and also the generation of the free surface grid are completely integrated in the code, so that no separate programs have to be run. The intersections of the stream-lines with a set of parabola constitute the cornerpoints of the free surface panels, see for instance figures 7a and b. (Because Figs. 7a and b represent ai example with symmetry with respect to the xy-plane the program was run with the symmetry option, so only half a configuration is requested and only half the free surface grid is generated.) The panels and, consequently, the source/doublet distributions adjacent to the body are extended to the body centreline for numerical reasons. The

forward and af t edges of such a free surface panel are kinked in such a

way that the configuration waterline contourpoints are located at these panel edges. This has been found to reduce numerical errors. The

strength of the constant source/doublet distribution on the part of the panel inside the configuration is taken equal to the strength on the

part outside the configuration.

As in Dawson's method the streamlines are intersected by parabola

(because the latter approximately coincide with the direction of

wave-crests and -troughs and consequently the resulting panel distribution will also be approximately adapted to this direction. This can be

ex-pected to be favourable for the numerical results).

At this point it is only remarked that, for numerical reasons, the parabola intersect the configuration in the waterline contourpoints. The extension of the grid, as well as the panel distribution in transverse direction and the distribution upstream and downstream of the boat will

be discussed in chapter 5.

The double-model velocity components 4d and are calculated

in the collocation points (centroidal points) of the panels. The

collo-cation points of the kinked panels adjacent to the configuration are defined as the centroidal points of the part of the panel outside the

The system of equations to be solved is constituted by the eq. (26), (28) and (29), expressing the boundary conditions to be satisfied

on the body and on the free surface. As in the 2D-model problem the strengths of free surface sources and doublets are coupled by the factor -Fr2. The velocities induced by a constant density panel are expressed in terms of the equivalent singularity of a ring vortex along the panel edges. The exact integration formula for the induced velocity is used

irrespective of the distance between inducing singularity and

collo-cation point. The velocity induced by a free surface source panel is calculated in the same way as in the NLR PANEL method: using exact inte-gration for the shorter distances, a quadrupole source for intermediate

distances and a monopole source for larger distances.

For the right-hand side of eq. (28) the usual 3-point central dif-ferences along the streamlines are used (3-point backward for the last panel of every strip).

For the left-hand side the discretization scheme has been chosen in analogy with the 2D-model problem, tacitly assuming that the 2D-results

concerning stability and accuracy still hold. Results for some

confi-gurations at high Froude numbers (section 6.1.3 and 6.1.4) indicate that the stability is indeed maintained; the accuracy has not been thoroughly

investigated, but, on the face of it, deterioration with regard to the 2D-situation is very likely. The left hand side is thus written as:

(P +Fr2

n1bvL)1

3,bv d y 3,bs

22

p1l p1 h _2tNLR

cfSSL)11

(1+ -- +

)ir O Fr Fr 3,fss Fr o2tNLR

(mfsdL)F1

3,fsd where: NLR 3,bs - 1,s 2 + 3 NLR 3,bv = 1,d 2 + 3

NLR

81,s2+(1_

)3

3,fss - 2,s

NLR

3,fsd 1,d

2 + '82d

3

2' 3 = usual two and three point upstream differences

h12 h12 1,s -2Fr4 2,s = 6Fr4 h12 h12 1,d = - Fr 2,d = 3Fr4 p1 = .2205

h = distance between two consecutive collocation points in

stream-wise direction

h1 = distance between the collocation point where the boundary

condi-tion is applied and the adjacent collocacondi-tion point upstream

bs = velocity in streamwise direction induced by the source

distribu-L

tion on the body and mirrorred body in free surface collocation

points

bv = velocity in streamwise direction induced by the vortex

distribu-L

tion of the body and the mirrorred body in free surface colloca-tion points (if present)

fss = velocity in streamwise direction induced by the source

distribu-L

tion on the free surface in free surface collocation points = velocity in streamwise direction induced by the doublet distri-L

bution on the free surface in free surface collocation points,

calculated by means of:

fsdL =

4

=

4

3T1/h (37)

where .' is the doublet strength.

Sfs Ffs Vfs

Sk Fk Vk

where:

- Sc, Sfs and Sk represent the influences of the body and mirrorred body

sources on, respectively, the body surface collocation points, the

free surface collocation points and points along the trailing edge of lifting parts of a configuration where the Kutta condition is imposed

("Kutta points")

- Fc, Ff s and Fk represent the influences of the combined free surface

sources/doublets on, respectively, the body surface collocation points, the free surface collocation points and the Kutta points

- Vc, Vf s and Vk represent the influences of the vortices on,

respec-tively, the body surface collocation points, the free surface

collo-cation points and the Kutta points

- o, a and r represent the unknown strengths of, respectively, the

body sources, the free surface sources and the body vortices

- Rc, Rf s and Rk represent the right-hand sides of, respectively, (26)

(body surface collocation points), (29) and (28), and (26) (Kutta points).

o

a fs r

Normal velocities induced by the doublet distribution in free surface

collocation points and all velocity components induced in body

collo-cation points by the free surface doublet distribution are calculated by

means of the ring vortex representation.

4,3 Method of solution

Satisfying eqs. (26), (28) and (29) in the collocation points on body and free surface results in a system of M linear equations in M unknowns. This is solved in an iterative way similar to the iterative procedure of the NLR PANEL method (Ref. 1). An iterative procedure is desirable for computational efficiency in view of the fact that large numbers of unknowns can be involved in complex configurations. The

system of equations may be written as:

Sc Fc Vc Rc

Rf s

Rk

As in reference 1 Sc contains diagonal blocks, each representing the influence of sources of a strip of body panels and its image on the collocation points of the same body strip. These blocks are dominant in

Sc.

Within Ff s there are no diagonal blocks, because the elements of Ff s do

not only contain contributions of free surface sources, but also of free surface doublets. The latter affect the dominancy of the diagonal blocks

for the non-zero Froude numbers.

In the iteration process Sc is replaced by a submatrix A containing only the diagonàl blocks A:

A1

O

-O O A --O

2-A=

I I I I O in

Bearing in mind this structure of Sc and writing (38) as follows:

Sc B a Rc

(40)

C I D

afSr Rfsk

where:

- B contains the elements of Fc and Vc - C contains .the elements of Sf s and Sk

- D contains the elements of Ff s, Fk, Vf s and Vk

- a r contains the elements of a and r

fs fs

- Rfsk contains the elements of Rf s and Rk,

lt becomes clear that the system of equations can be solved by exactly

the same procedure as that of reference 1. This can be sunnuarized as

follows: starting from a given estimate af$r(in_1) and a(m a

correc-tion Mf r and Aa(in) _{is obtained from:}

(in) = (-CA B +

D)1 (RfSk(m) -CA

Rc(m))

ta r f8 (in) A (Rc(m_l)_BAO r(m)) Aa = fs (39) (41)

where: (m-1) Rc

=Rc

m=1 (m-1) = -(Sc-A)Aa in>1

RfSk(m)

_{= Rfsk}

_m1

=0

m>1

The solution of the system of equations is then given by:

a

a r = a r(m)

fs in fs

It is clear that the effectiveness of this iterative solution

procedure

depends on the

ratio of

the sizes

of

the block Sc and the other blocks.

However, in the cases where an iterative procedure is preferred to a direct solution procedure, i.e. the cases with large totals of unknowns, the number of configuration panels can be up to 3 times as high as the number of free surface panels and the presence of the iterative solution

procedure will then be profitable.

Within the HYDROPAN program it is possible to perform an unlimited number of Froude calculations for one configuration and one angle of yaw

in one computer run. This offers the possibility for a reduction in the

number of required iterations by prescribing the solution vector of the last Froude number calculation as the starting vector for the next one

(assuming that the Froude numbers are specified in a monotonous

increa-sing or decreaincrea-sing order) instead of starting each Froude number

calcu-lation with the null-vector. Only the first Froude number calcucalcu-lation

will start with the null-vector.

The number of iterations required to obtain a sufficiently accurate

solution depends on the configuration. Typical numbers are:

- for a simple configuration without lifting surfaces (such as the

DTNSRDC configurations presented in the examples): ± 7 for the first

Froude number, ± 5 for each next one.

- for a complex configuration with lifting surfaces (such as the sailing

boats presented in the examples): ± 15 for the first Froude number, ± 9 for each next one.

4,4 Forces and moments

Once the singularity strengths are known induced velocities can be calculated and through that the pressure distribution on the body sur-face as well as the wave heights of the free sursur-face. Pressure integra-tion over the part of the body beneath the mean free surface leads to hydrodynamic characteristics such as pressure resistance, heave force

and pitching moment.

In case of the presence of lifting surfaces the induced resistance is calculated separately by means of a far field approach (Trefftz-plane analysis; see e.g. Ref. 6). The induced resistance is defined as the

resistance associated with the trailing vortices. It can be written as:

Di = p [(6v)2+(6w )2]dS (44)

where 6v and 6w are the perturbation velocities in the Trefftz-plane

ST(x=oo) induced by the infinitely long trailing vortices and their double-model image. It can also be written as:

D.

= -

p I ds

1 2 _j an

S flS

VT

where M is the jump in potential across the trailing vortex sheet SV and-p-the velocity component "normal" to the vortex sheet in the

(45)

Trefftz-plane induced by the trailing vorticity and its double-model image. Here one should be aware of the simplification in the method: the vortex sheet (in fact: a set of discrete vortices) is assumed to be parallel to the undisturbed flow direction far downstream where the Trefftz-plane is located. In other words: displacement and roll-up of the vortex sheet is neglected. It can be argued (see e.g. Lock, ref. 12) that the errors involved should be small, except for situations

involving high lift and low aspect ratio.

At zero Froude number the resistance calculated by this procedure

(Di ) should be equal to the resistance obtained by pressure

inte-gran°over the body (D

), if the truncation errors are zero.

Experience from aeronauticar application is that it is generally not. It has been found that the induced resistance determined through the far field Trefftz-plane procedure is generally more accurate than the

resistance obtained through pressure integration, because the latter often suffers from a large truncation error for manageable numbers of panels.

Although the truncation error in the non-zero Froude number

pressure resistance (D ) calculation will be different in general

prF4A

from the truncation error fn the zero Froude number case (D ), it

prF _{_}

is likely that the value of D will be improved by correcing it as

prFO

follows:

D

=D

-D

).

prFO

prFO

_1Fr=O

prFO

The wave-making resistance obtained by pressure integration is therefore

computed as:

*

D

=D

-D

w pr

prFrO

1FrO

*

In spite of the correction of D into D , the values of

*

prFO

prFO

D and thereby D can be rather inaccurate. Therefore the

pr w

wavking resistance

k'l

also calculated in an other way by means of a

momentum approach. The momentum conservation equation in streamwise (x-) direction:

ff

p (U-U ) (.)dS +

f!

(p pgh) n dS = O (48)

s s s

is applied to the control volume depicted in Fig. 8a. Note that eq. (48)

strictly speaking represents the momentum conservation equation minus

the mass conservation equation multiplied with U. In this way one corrects implicitly for the spurious, additional momentum which results from numerical inaccuracies in the surface integration. As a consequence

of the latter

if

pv, dS will not be zero in general. Without this

control volume

correction the spurious, additional momentum would result in a spurious, additional resistance of the configuratíon.

Now make the following assumptions:

- the front plane, aft plane, both side planes and the lower plane are

located infinitely far from the configuration. The velocity in these

- across the vortex sheet SV both normal velocity and pressure are

continuous.

- the vortex sheet is parallel to the undisturbed flow direction far

downstream where the Trefftz-plane is located.

Then eq. (48) reduces to the following expression for the total

resis-tance D:

D = - J-J p n dS X

S s

= - p f! (U-U) VdS - p f! (U-U) UdS (49)

S S

fs T

- ½ p f!

(u2_.) dS

ST

If one recollects that the body plus the vortex system and their images

are symmetrical with regard to the plane Sf one can express the

resis-tance in terms of the velocities induced by the various components as follows:

D=_Pff(ub +u

+u )v

dS-pffu (U +u )dS

s by fs fs fs fs

S

fs ST

-½p!!(-2U u

-u2 -y2 -2v

y -y2

fs fs by by fs fs

-w2 -2w

w

-w2)dS,

by by fs fs

where the index bs refers to the configuration sources and their images, by to the configuration vortices and their images, fs to the free

surface sources plus doublets. Note that, because of the double-model

linearization, the vortex system contributes to only near the vessel

(Sf) because the vortices trail off to infinity in streamwise

direction.

The expression (50) for the resistance can be simplified by reali-zing that the fictitious flow generated by the source/doublet

follows:

distribution in the mean free surface only (plus free stream) also

satisfies conservation of momentum. Hence, there must hold:

-

Paff

Uf Vf

dS -uf Vf dS - p

f!

uf (U + uf) dS fs fs. ST i T

where S is the shaded area inside the hull, not included in S

fs. fs

i

(Fig. 8b) where the extended distributions of the free surface panels adjacent to the configuration are located. Combining (50) and (51) yields:

D=-p

f!

(u,+u.0 ) y

dS+p If

u y dS y fs fs fs Sf fs. i -

½ p f! (.-

2 U u - u2 - y2 - w2 ) dS = 0, fs fs fs fs S

-½pff

(y2 -w2 -2v

V

-2w

w ) dS by by by fs by fs ST

The first two terms in the last integral represent again the induced

resistance.

If q, and * are scalar functions of two variables and if they can be

differentiated at least twice, one can write according to the chain-rule:

+ 4. + +

If

V . q) V I) dS =

If

(Vq, . Vip + q)v21p) dS. (53)

S S

After applying Gauss' theorem in two dimensions on the left-hand side

eq. (53) yields:

q, ds = Ii.

(q).ip + q,v2ip)

dS. (54)

C S

Let q, be th potential due to the free surface source/doublet

distribu-tion

fs'

the potential due to the Vortex system plus its image and

34 ds = ¡f (

.p +

V24i ) dS ds -s fs y fs y outer contour fs n _STflSV ST (55) 1pv

-

will be zero on the intersection of S with both side planes and the

3n T

lower plane because of the infinite distance from the vortex system to

these intersections. Due to the mirrorred vortex system

i will also be

zero at the intersection of S with S and therefore the first contour

T fs

integral vanishes. Because of the continuity of

f and

i-

over S the

second contour integral will also vanish. Finally, ip will satisfy in

D D.

w i

(57)

where the first two integrals are (by definition) the wave-making

resis-tance. The first integral should be evaluated over the infinitely

ex-tended mean free surface, the second integral is also dependent on

con-tributions from the source/doublet distribution on the infinitely

ex-tended free surface. In practice the source/doublet distribution and the

integration area on the free surface are limited to a finite area and

therefore an error will be introduced in the calculation of the

wave-making resistance. The latter can be written as:

the Trefftz-plane the two-dimensional Laplace equation. Eq. (55) reduces

to:

fs dS = ¡f (vfS Vb + wf

Wbv) dS = 0. (56)

ST _T

Eq. (52) can now be written as:

D=-p

¡f (u,+u) y

dS+p

¡fu

y

dS-½p

- ds,

y fs fs fs

-

pff (ub _{+ 11b}

(y

+ y ) dS y fs fs C U fs u + p If (u + u ) (y + y ) dS, fs fs fs fs s c u c u f s. i

where Sf refers to the part of the mean free surface which is in practice

covered by source/doublet panels (see Fig. 8b); Sf refers to the part

outside Sf which is uncovered by panels; Uf Vf refer to the

velo-city compon:nts induced by the source/doublet panels located on Sf;

Uf Vf refer to the velocity components induced by the (imaginary,

in practice not present) source/doublet panels located at Sf.

Rearranging yields: D = - p fI (u + u ) y dS + p f! u y dS w bs by fs fs fs S

+S

c S c c fs fs fs C u

j

-

p If (u.1, +u ) y dS+p ff (u y + u + u y ) dS. s by fs fs fs fs fs fs fs S +S u Sf C u u C u u fs fs C u i

Here the first two integrals represent the wave-making resistance as

calculated by HYDROPAN and the last two integrals the error due to the

boundedness of the free surface grid. The integration over Sf in the

first integral needs to be performed only on that part Sf ofuSfSU

(shaded area, see Fig. 8b) where the integral supplies a noticeable

con-tribution to the wave-making resistance. Through numerical experiments

(see chapter 5) it can be shown that the error is approximately

inver-sely proportional to Ls, where L5 is the distance from the stern of the

D

-pff (ub +u )v

dS+pffu

y

dS=

w s by fs fs fs Sf Sf i

+v

)dS

y fs fs Sf C U C

configuration to the downstream edge of the free surface grid (see Fig.

8b). The decrease of the error with increasing grid extension can also

be made plausible by the following reasoning. The velocities u..b and

tend to zero (far) away from the configuration. Also, the contribution

of Vf Ofl Sf will only be significant in the vicinity of the

u c

boundary between S and S and the velocities u and

y

induced

fs fs fs fs

c u u u

at Sf will decrease with increasing grid extension, while uf and

i

Sc

y

induced at S will tend to constant values. It should be

fs fs.

c i

emphasized that the fact that one can suffice with a bounded free

sur-face distribution directly results from the double-model approach.

Within the HYDROPAN program the "momentum" wave-making resistance is calculated via the first two integrals in (59). The integrals over

the additional areas Sf and Sf are evaluated by firstly dividing the

areas in a sufficient number of panels. Care is taken that the area Sf

is large enough to take all noticeable contributions to D into account.

Then the required velocity components in the collocation points of these

newly generated panels are calculated and the integrals are evaluated.

Experience up to now suggests that the wave-making resistance ob-tained by this procedure is more accurate than the one obob-tained by pressure integration over the configuration. Therefore all resistance numbers presented in the examples of application of chapter 6 are based

on this momentum approach procedure.

All forces and moments presented in the chapters 5 and 6 are, un-less otherwise stated, made dimensionun-less by the dynamic pressure, the wetted surface of the entire configuration at rest and (for the moments)

the reference length.

5 RESULTS OF A CONVERGENCE STUDY

In this chapter the extensions of the free surface grid, as well as the influence of panel distribution and panel density on hydrodynamic characteristics are examined. This convergence study is by no means of a

thorough mathematical nature, but may provide some insight in the

rela-tions between the various paneling parameters and the hydrodynamic characteristics. The results can serve for guidance in generating panel

Firstly the required extensions of the free surface grid are

investi-gated. A priori the observation can be made that noticeable disturbances with regard to the mean free surface y=O due to the motion of a ship are

only confined to a cone with half a vertex angle of 19°28' (see Fig. 9

and Ref. 7). This implies that it should not be necessary to locate free

surface panels (far) outside this cone: the resulting source/doublet strengths on such panels should be zero. Therefore the upstream

exten-sion can be limited to a short distance (half the waterline length of

the configuration) and the downstream and transverse extensions are

coupled in such a way that the cone is entirely located within the free

surface grid, so it can be concluded that the extensions of the free

surface grid are determined by only one parameter for which we choose

the downstream extension Ls

A second general observation can be made in relation to figure 10.

Imagine that the initial free surface grid is bounded by the solid line.

Although the three-dimensional wave pattern is quite complicated,

be-cause of the presence of two systems of (each other intersecting) waves

(see Ref. 7), some degree of periodicity in streamwise direction can be

observed. This alternating sequence of wave crests and troughs will be

modelled by an alternating sequence of free surface source/doublet

values. If the grid is now extended by half a wave-length (dashed line) this will result in the addition of transverse rows of singularities of

the same sign, for instance sources and related doublets. These

addi-tional singularities induce a certain change in velocities near the

con-figuration and thereby in pressures and hydrodynamic characteristics

(forces, moments). If the grid is again extended by half a wave-length

(dash/dot line) a similar change in hydrodynamic characteristics will occur, but this time of reversed sign and of somewhat smaller value,

because of the increased distance to the configuration. This process

suggests that there exists a relation between calculated hydrodynamic

characteristics, such as resistance, and grid extension of the following

kind:

n

CD = CD + (Ls/) sin (2

L/)

o

Here CD is the resistance value in case of an infinitely extended free

surfacegrid, and n constants, and X the wave-length. The term

(Ls/)n

models the damping of the surface waves, sin (2rr

Ls/)

the oscillations.

In order to assess the usefulness of eq. (60) a few test calcula-tions have been performed on a simple, parabolic Wigley hull conf igura-tion which is described by the following formula:

2

X

z = .8 (1- g: (l-y2)

-8x8 , -ly0.

The hull has been paneled in 33 cross-sectional strips of 8 panels each on half the configuration (see Fig. lia). Because the higher Froude numbers to be run for a configuration will be decisive for the extension of the grid, the Froude number in the test calculations have been set at a value representative for the high-speed regime: Fr = .45 (based on the

waterline-length). The yaw angle was zero.

The test consisted of three calculations with different grid exten-sions (L/Lf = .53, 1.03, 1.54) in order to calculate the unknowns

CD , and n. In each calculation the transverse extension has been

adpted to the streamwise extension. The value of A resulted directly

from the calculations (and is indeed independent of the grid

exten-sions): A/L = 1.20. Two additional calculations at grid extensions

ref

LS/Lf = .32 and 1.27 (Ls/A = .26 and 1.06) have been performed in order to check the applicability of eq. (60). The generated free surface grids are shown in Fig. lib. The panel distributions on overlapping

parts of the grids are identical.

It should be remarked that at the time these computations were

per-formed the calculation procedure for the resistance by means of the momentum approach (see previous chapter) was not yet available. There-fore all resistance numbers (and all other forces and moments) for this free surface grid extension study have been calculated by means of

pressure integration over the configuration, eq. (47).

The results are presented in fig. 12a, where the resistance is plotted against the grid extension. The agreement between the resistance as calculated by the flow computation and as fitted by eq. (60) is

reasonably good, even for the smallest grid extension X/Lref = .26. This suggests that eq. (60) can be suitably used for determining the required

grid extension. By plotting CD against (L/A)Tl, n = -1.28 one can easily

draw the envelope of the resistance and thereby determine the grid extension which is at least required to obtain the resistance with a

certain accuracy. For instance if one requires a one per cent accurate value, the minimum grid extension should be approximately half a

wave-length.

The hydrodynamic heave force and pitching moment as a function of the grid extension are presented in Fig. 12b and c. Only for conformity

they are also presented as a function of

(Ls/A)28 no efforts have

been made to find any analytical expression for CL and CM. These hydro-dynamic characteristics show nevertheless a similar behaviour as the

resistance. One can conclude from the figures that the heave force and pitching moment will also be calculated with a one per cent accuracy if the streamwise extension is at least half a wave-length. In the

remainder of this chapter the concept "required grid extensions" refers

to the grid extensions which lead to a one per cent grid extension error

in calculated forces and moments.

It should be emphasized that the value of CD is not necessarily

the "real" value of the "double-model" wave-making resistance, because

the panel distribution used on hull and free surface grid (of which only

the required location of the outer bounds have been established so far)

is not necessarily sufficiently dense. Therefore the influence of panel

distribution on hydrodynamic characteristics has also been examined

(again for the Wigley hull at Fr = .45).

Firstly the influence of anel distribution on the free surface grid has

been examined. Keeping the panel distribution on the hull constant, the

streamwise panel distribution on the grid between bow and stern is also

constant due to the intersection of the parabola with the configuration

waterline contour points (see Fig. 13). The remaining parameters which

can be varied are then: the transverse width of the free surface strips

and the streamwise width of the free surface panels upstream of the bow and downstream of the stern. It should be remarked that it is assumed that the required grid extensions are not affected by the panel distri-bution within these grid boundaries.

Comparison of the calculated wave-making resistance values

(pressure integration) with the experimental wave-making resistance

value corrected for sinkage from Shearer & Cross at Fr = .452 (Ref. 8)

showed that best results are obtained when the transverse width of the

free surface strip adjacent to the hull is small. The width of this

strip at the upstream boundary (B, see Fig. 13) of the grid is taken

number of streamwise panels on the hull, see Fig. 13. The streamwise panel lengths of the panels upstream of the bow and downstream of the stern should be of the same order as the largest streamwise panel

length. The influence on the resistance of the transverse width of a

free surface strip is rapidly decreasing with the distance of the strip to the configuration. The ratio of the widths of two consecutive strips is therefore taken as high as 1.5 (see Fig. 13). Lack of experimental data prevents comparison of other forces and moments. Comparison of Figs. lib and 13 shows the difference between the original and the modi-fied grid. The downstream (and thus transverse) extension of the latter is larger than necessary to obtain the one per cent accuracy in hydro-dynamic characteristics. Note that dimensions in the two computer plots can not directly be compared, because the plot program uses different

scaling factors for each plot.

Next the panel density on the configuration and thereby on the free

surface grid has been examined (Wigley hull, Fr .45). Again it has

been assumed that the required grid extensions and also the favourable characteristics of the grid within these grid boundaries (panel dis-tribution), as found in the previous paragraph, are not affected by changes in the panel density. The type of panel distribution on the con-figuration is still a cosinus distribution in streamwise direction and

an equi-angie distribution in cross-sectional direction (see Fig. lia). Three panel densities are examined: 8 (streamwise) * 2 (cross-sectional),

16 * 4 and 32 * 8. This resulted In the following grid panel densities:

20 (streamwlse) * 3 (transverse), 33 * 4 and 64 * 8 (see Fig. 14, note again that dimensions between the three configurations can not be

compared directly). The extension In upstream direction of the coarsest grid (20 * 3) is larger than strictly required (more than one boat-length instead of half a boat-boat-length), because otherwise to few panels upstream of the bow would be available in order to satisfy properly the boundary condition (29). The extension In transverse direction is also larger than required (two strips would suffice), because for numerical reasons (3-point transverse differences of doublet strengths in order to obtain velocity components on the grid panels) at least three strips are required. A sequence of for instance 16 * 4, 32 * 8 and 64 * 16 panels

The results of the calculation for the three grid densities are

shown in Fig. 15. The hydrodynamic characteristics resistance, heave force and pitching moment are plotted as a function of the reciproke of the square root of the number of panels on the hull which is a measure

for the panel width h. Because these calculations have been performed with a more recent version of the HYDROPAN program which contained the

procedure for calculating the wave-making resistance by means of a

momentum approach, both resistance numbers are available. It can be

noticed that the difference between both resistance numbers is almost independent of the panel density. The upper envelope in Fig. 15a

com-prises various experimental, residual resistance data (Ref. 8). However,

these experiments have been performed with the model free to trim and

sink. The experimental values have therefore been corrected with

calcu-lated corrections for sinkage and trim (Dawson and Gadd, Ref. 8)

resul-ting in the lower envelope which is also the envelope wherein the

HYDROPAN results should be located. The momentum-resistance for the

highest panel density satisfies this requirement. The tendency for the

value of momentum-resistance to be more correct than the value of the pressure-resistance has also been noticed for several other conf

igura-tions. Therefore the resistance numbers presented for all the examples

of application in the next chapter are momentum-resistance numbers.

As illustrated by Fig. 15 resistance values of reasonable numerical

accuracy are obtained for about 32 streamwise panels on the hull (for a

calculation at Fr = .45). However, the influence of Froude number was

not investigated. Variation of Froude number implies variation in free

surface wave-length and, when the grid is kept constant, variation of

the number of free surface grid panels per wave-length. Fig. 4 suggests

that at least 20 panels per wave-length are required in order to

accurately represent the (2D-) wave pattern (for Fr = .4, related to a

suitable length scale in this 2D-model problem). Assuming that this

2D-result is also valid in a 3D-situation at arbitrary Froude number,

and bearing in mind that the accuracy, with which the free surface wave

pattern is represented, affects the accuracy of the calculated

hydro-dynamic characteristics on the configuration, a criterion can be found

for the minimum required number of streamwise free surface grid panels.

Because the streamwise panel distribution on the free surface grid

(between bow and stern of the configuration) and hull are coupled, one

establishes automatically a criterion for the minimum required number of

has proven to be reasonably useful in 3D-situations: 2ir Fr2 = XILf the requirement of at least 20 panels per wave-length then results in the following criterion for the minimum number of

streamwise panels on the hull: N 20 ₂

min 27r Fr

For Fr = .45 this implies: N . 16, while the 3D-HYDROPAN

calcu-min

lations yielded: N. ± 32. This suggests that, at least for Fr = .45,

the panel density on the hull is more important than that on the free surface. However, additional investigation is required into the matter of required panel density as function of Froude number. For the time

being the following criterion seems usable: N . = max

_20

32}.

min 2ir Fr2'

The panel distribution on the hull in cross-sectional direction and the distribution on appendages, such as keels and rudders, should be adapted to the streamwise panel distribution on the hull in order to obtain a regular panel distribution on the entire configuration.

It should be remarked that the results concerning grid extensions and panel distribution on the free surface grid (for fixed panel density on

the hull) have been obtained at Fr = .45. It is not a priori clear that

these results will also be valid for other Froude numbers. If, in the absence of further information, they are assumed to be valid for other Froude numbers, the required downstream grid extension for a Froude

number of, e.g., .9 would be .S*ÀF9.

The (limited) convergence study described above was done for a conf

igu-ration without lifting surfaces. It can be expected, however, that different conclusions will be obtained for configurations with lifting surfaces. In particular the required downstream grid extension of the free surface grid may differ from the non-lifting case, because of the interference between free surface and trailing vortices. Four test cal-culations have been performed on a configuration with lifting surfaces (63 ft sailing-yacht configuration, no rudder deflection, 660 panels plus 12 vortex strips, see Fig. 16) in order to examine the influence of

the free surface grid extensions for such a case. As for the Wigley hull the panel distribution and density on configuration and free surface grid are not necessarily optimal, only the grid extensions are examined.

The flow conditions are: Fr = .45 (based on waterline-length), = 4

degrees, no heel angle. The four downstream grid extensions are:

LS/Lf = .28, .59, 1.13 and 1.74 (see Fig. 17). The wave-length A is

again: AIL = 1.20.

First of all it is remarked that Fig. 17 shows a major shortcoming of the free surface grid generation procedure. The transverse dimensions of the panels of the free surface strips adjacent to the hull are seen to

vary considerably between bow/stern and amidships, leading to

undesi-rably if not unacceptably strong variations in lateral panel width. This behaviour has been found to be typical for configurations with a large beam/length ratio and significant yaw angles. It is clear that this phenomenon will be unfavourable for the results of non-zero Froude number calculations. Appendix A contains some additional comments on

this subject including a possible solution to avoid this problem. This

solution is not yet incorporated in the HYDROPAN program.

As in the non-lifting case it has been checked if the total resis-tance (wave-making resisresis-tance plus induced resisresis-tance) could be

ex-pressed in a relation like (60). The three largest extensions have been

used to determine the parameters CD a and n. The shortest extension

can serve as a check on the validity of the expression for CD. The

result is presented in Fig. 18a. The agreement at LS/Lr = .28 between

the result from the HYDROPAN calculation and the result from the

ana-lytical expression is reasonably good. The value of the parameter n is

in this case -.91, resulting in a less damped curve. This implies that,

not surprisingly, a larger downstream grid extension is required for a

one per cent accurate resistance value: at least 1.25 wave-length is

required.

The shape of the curve for the wave-making resistance is almost

identical to the shape of the total resistance, because the induced

resistance is found to be almost insensitive to the grid extension (see

Fig. 18a). This insensitivity can be explained on the basis of Fig. 19.

Far downstream the flow exhibits still a strong degree of symmetry with

regard to the x-axis. Changes in índuced velocity components near the

configuration resulting from the extension of the free surface grid will

therefore be much larger in the x- and y-direction than in z-direction.

Because the vortex strengths mainly depend on the Kutta condition (zero

normal velocity at the vortex sheet), these strengths will remain almost

unchanged. This results in an almost unchanged induced resistance.

The variation of the other forces and moments as a function of the

grid extension is presented in Figs. 18b up to 18f inclusive. The

hydrodynamic heave force shows a variation similar to the resistance,

Fr2 , 32} - 20

by the fact that the changes in velocity components at the port- and starboard sides of the configuration do not differ significantly and consequently the changes in side forces at both sides will almost cancel

each other. Also CM _{and CM} show a damped sinusoidal path.

CM exhibits a diffrent, bu also damped path.

It can be concluded from these results that all calculated forces and moments will not be significantly affected by free surface grid truncation if the downstream grid extension is taken at least 1.25

wave-length.

The results obtained in this chapter can now be summarized as follows: - location of the outer boundaries of the free surface grid:

upstream boundary: half a boat-length upstream of bow

downstream boundary: C * À where is the wave-length downstream of

the stern at the actual Froude number.

Non-lifting configuration: C = .5.

Lifting configuration : C = 1.25.

transverse boundary: at least at such a distance that the

Kelvin-cone is entirely embedded by the grid. - panel distribution on the free surface grid:

width of the strip adjacent to the hull should be taken small (pro-portional to the width of the ship and inversely pro(pro-portional to the

number of streaniwise panels)

ratio of width of consecutive strips can be taken as high as 1.5 streamwise length of panels upstream of the bow and downstream of

the stern of the same order as the largest streaniwise panel length on the hull

- panel density on configuration:

minimum number of streamwise panels on the hull:

N. =max

min

the panel distribution on the hull in cross-sectional direction and the distribution on appendages, such as keels and rudders, should be adapted to the streamwise panel distribution on the hull in order to obtain a regular panel distribution on the entire configuration.

These recommendations have been followed at the examples of

6 EXANPLES OF APPLICATION

In this chapter results are presented for two kinds of conf igura-tions. One class consists of simple geometrical hull forms. The

calculations on these configurations offer the possibility to compare the HYDROPAN results with published experimental and other numerical results. The other class consists of two fairly complex sailing boat configurations with lifting surfaces (keels and rudders) for which

towing-tank results were available to the authors.

All calculations have been performed at NLR on a CDC-Cyber 180-855

series mainframe computer.

The range of Froude numbers for all configurations has been chosen such that comparison with other numerical and/or experimental results is

possible.

The efficiency of calculations using the HYDROPAN code is best if

all non-zero Froude number calculations are run in one computer run.

This results from the fact that the velocity induced by an arbitrary singularity of unit strength in an arbitrary collocation point is of

course independent of the Froude number and has thus to be calculated

only for the first non-zero Froude number and can be saved for all other non-zero Froude numbers. For all Froude numbers one has to set up the matrix for the system of equations which does depend on the Froude number, see eq. (28), solve the system of equations for the unknown

singularity strengths and calculate velocities, pressures and hydro-dynamic characteristics. This difference in the amount of computational work between the first and each next non-zero Froude number manifests

itself in the CPU times required for the first and each next Froude

number. These CPU times and amount of core memory used in the calcula-tions are presented for all examples of application in table 1.

Running all non-zero Froude numbers in one computer run implies the usage of one and the same configuration paneling and free surface

paneling for all Froude numbers, i.e. the configuration paneling (panel

density) and free surface paneling (grid extensions and panel distribu-tion) should meet, for all non-zero Froude numbers, the requirements

established in the previous chapter.

The locations of the outer boundaries of the free surface grid have been taken to be sufficient for the highest Froude number. The panel distribution on the free surface grid is coupled to the panel

distribu-tion on the hull. The panel density on the configuradistribu-tion causes some problems. Because one has to use one panel density for all Froude

numbers and because the lowest Froude number to be run was considerably below .32 for all configurations, the number of streamwise panels on the hull should be N

= 2 Fr2 . However, this would lead to a very

large total number of pan 8t configuration and free surface grid,

well beyond the central memory capacity of the NLR Cyber (without

modifying the I/O of the current code).

Instead of splitting the calculations up in a run for the lower Froude numbers and one for the higher it was decided to perform all cal-culations in one computer run with the same panel density, thereby

sacrificing some accuracy at the lower Froude numbers.

Because the downstream grid extension is input for the HYDROPAN program, it is necessary to know the wave-length beforehand. This has

been estimated by use of the 2D-formula: AIL = 2r Fr2, which has

ref

proven to be reasonably useful for 3D-wave-length determination.

6.1 Some second DTNSRDC workshop examples

A number of simple geometrical configurations has been defined as testcases for this workshop (Ref. 9). Calculations with the HYDROPAN program have been performed on the following configurations: Wigley's parabolic hull, series 60 block coefficient 0.60 hull, the simple

strut-like hull form (with the sharp end ahead as well as the round end ahead) and two vertical cylinders. All configurations are symmetric with regard to the xy-plane and because all yaw angles are zero only half the configuration and free surface grid have to be considered. Froude

numbers are related to the waterline-length of the configurations. For the Wigley hull and the series 60 configuration comparisons with towing-tank results can be made. For the strut-like hull form and

the cylinders only some other numerical results are available to the authors for comparison purposes. For the Wigley hull and series 60

con-figuration experimental results are available for both free and fixed trim and sinkage. This is specified in more detail in sections 6.1.1 and 6.1.2. As far as can be concluded from the papers in Ref. 9, all numeri-cal results used for comparison purposes in sections 6.1.3 and 6.1.4 have been obtained, like HYDROPAN, with the configurations in fixed posi-tion.