National Research
I* Council Canada
institute for Marine DynamicsCanad
Conseil national de recherches Canada Institut de dynamique marineSYMPOSIUM ON
SELECTED TOPICS OF
MARINE HYDRODYNAMICS
St. Johnts, Newfoundland
August 7, 1991
THE COMPUTATION OF WAVE PATTERNS AND WAVE
RESISTANCE INDUCED BY SUBMERGED BODIES
ADVANCING WITH STEADY FORWARD SPEED
R.E..Baddourf, J.S.Pawlowskit and S.W.Song
f NRC. Institute for Marine Dyna.mic3, St. John's. NF. MUN. Faculty of Enneering & Applied Science, St. Job.ns. NF.
:
-c
I
National Research CouncilCanada Conseil national de recherchesCanada
SYMPOSIUM ON
SELECTED TOPICS
OF
MARINE HYDRO
DYNAMIC S
St. John's, Newfoundland
August 7, 1991
-/5r-..
Institute fO( Maine Institut de dyrtarnique
/
THE COMPUTATION OF
WAVE PATTERNS AND WAVE
RESISTANCE INDUCED BY SUBMERGED BODIES
ADVANCING WITH
STEADY FORWARD SPEED
R.E..Baddourf, J.S.Pawlowskif and SW.Song
f NRC - rnstitute for Marine Dynamics, St. John's, NF.
MUN- Faculty of Engineering & Applied Science, St. John's. NF.
Abstract
The purpose of the present paper is to report on the progress in developing a Rankine source method
i.e. indirect boundary element method, for the
nu-merical sirnl3lat ion of 3- D surface wares. generated
by submerged bodies advancing with steady
hori-zontal speed and the evaluation of their correspond-ing wave resistance. The formulation of the
bound-ary value problem is based on the existence of a
velocity potential describing the flow 01 an inviscid,
incompressible fluid with a free surfe. The dis-cretization of the body surface and a part of the
fluid free surface, is based on first order triangular isopar arnetric boundary elements of Ra.nkine sources.
Triangular. three-node elements, which can repre-sent general three dimensional surfaces easily and accurately, are used in that respect. Linear distri-bution of the source density is assumed on each
ci-rnnt. Results of the computations fc'r an ellipsoid
and comparison with data published by different
au-thors ar provided.
1
Introduction
The computation methods of wave patterns and wave
re-sistance. induced by bodies advancing with steady forward
speeed at or in the vicinity of the free surface of an ideal
fluid, have developed significantly over the last two decades.
To simulate the steady ship motion problem, two numeri-cal approaches axe found in the techninumeri-cal literature. They are the Neumann-Kelvin source method and the Rankine source method. Studying, applying and furtherdeveloping
those two approaches are still major topicn of numerical ship
hydrodynamics today. TheNeurnann-Kdvin method solves
Neumann-Kelvin problem using a Green function which sat-isfies the Neumann-Kelvin linearized free surface condition.
Sources are distributed only on the surface of the moving
body. The Rank.ine source method, presented by Gadd
(1976) for solving the free su.race wave problem, was used bT
Dawson in 1977. Dawson linearized the nonlinear shipware problem in ternis of a double-body flow and solved the prob-lem by using a Rankine source distribution on the body and adjacent free surface. Qss'4"4lateral elements and consta.n: source distribution on each element were used. Those twc method.s are used to solve te specific problems
they were
developed for, i.e. the Neumann-Kelvin method is used to
solve the Neumann-Keith problem and the Rankinesourcn method is used to solve Dawson's linearized problem.
In the present study a Rankine source method with
tri-angular element discretition
and linearsource distribu-tion. i.e. a higher order l.nel method, is developed
to solve
both the Neumann-Kelvin and Dawson's p;ohlenis. In this
method, a part of the fr surface of the fluid and the stir-face of the body are discret.zed into triangular elements.
and Rankine sources are distributed on the mesh
formed
by the triangular elements. On each element the
source
density is assumed to be linearly distributed. A compari-son between the present method of solving the
Neumann-Kelvin and Dawson's prolemss is presented. As
an
appli-caton a submerged ellipsd in rectilinear, uniform
motion
in otherwise still fluid is considered. For this
case a com-parison is presented with the theoretical solution by Farrel (1973) and the numerical results by Doctors andBeck(1987) which were computed by esimg the Neumann-Kelvinsource method,
2
Theoretical Formulation of the
Problem
It is convenient to describe the problem by fixing the body with the coordinate system and letting thefluid flow in the
present analysis the body is always fixed with the
coordi-nate system and the fluid flows in the pcsitive x direction. By assuming the fluid flow around the body to be poten-tial, the ship wave problem can be maihesiatically described by a boundary value problem with La.l..ce's equation to be satisfied everywhere in the fluid domain and the boundary
conditions to be satisfied on both the free surface and the body surface. The boundary value probm is then written
using fixed on the object coordinate system z. y, z as:
I V2 = 0
in the fluid domart
Ion'the
body s.rfw2
8z8z
on :=
(1)I
I as
where z. y. define the coordinate system with z upwards
from the still water surface and z in the direction of fluid velocity U; ' is the free surface elevatn; ff refers
to the
unit normal to the body surface, pointiig into the fluid; gis the acceleration due to gravity; is the velocity potential
within the fluid domain; and U represeits thevelocity of the fluid far away from the body, and U=-Ui, where U6 is the body velocity. We have aLso to
emse that the body is
not generating waves which propagate fcrwa.rd of the body.Equations (1) describe the exact ptentia.l ship
waveproblem and it is nonlinear. Both theNeumann-Kelvin and
the Rankine source methods are developed to solve the ap-proximate problems which are Iinearizarions of this exact problem but with different approximaticus.
By assuming the potential ô to be that ofa uniform flow
1 plus a perturbation d" i.e.
. =
Uz -8, equations (I)
could be linearized into the followin linear boundary value problem which is the so-called Neuma.ni-Kelvin problem: See e.g. Baar and Price(1988))
in the fluid domain
on the b.ody surface
(2)
U2+g=0
onIthi-0
a.s Jz+y2+:, z<0
where n7 represents thex component of and the generated waves must propagate downstream.Alternatively, by assuming the poventia.l
to be due
to the flow of fluid surrounding
the cc'responding
double-body plus a perturbation, i.e.
+ -
', equations (1) could be linearized into another linear problem which isDawson's linearized problem: ( See
Dawson(1977), Pawlowski( 1990))
2
= 0
in the fIuzLi domain
i.'=
0an the body surface
+ B + =
C an
as Iz+y2-', zSO
whereA
-
(8+
2B 23C 2
28+
-
at
8P
-(is tangent to a streamline of the corresponding double-body flow on =
0; and I is the double-body potential
which is obtained by solving the double-body problem. We
have also to ensure that the generated surface waves are
propagating downstream.
The double-body problem is defined by:
=0
in the f(u:.d domain
ñ. I
= 0 onthe body surface
an on
St
as
iz--y2---z2--.00, z0
Dawson's linearized problem is originally writtenin terms
of total velocity potential 0 . however it can also be writ-ten in terms of a perturbation o' as in the Neumann-Kelvin
formulation, (see Ogiwara 1983)).
It is important to note that the applicationof
Neu,mann-Kelvin formu.lation requires not only the small wave condi-tion, because of the linearization in z direction, but also the
so-called thin ship condition. On the other hand, in
Daw-son's problem the linearization is taken only in direction and the exact condition is kept in the z and y directions.
The influence on the fluid velccit'r in direction of the
ve-locity changes in z and p directions are considered in the
free surface condition. Therefore no thin ship condition is required in its application.
3
Numerical Method for Solving
Dawson's Problem
The basic numerical procedure presented by Dawson for
solving his problem are here adopted. However, in the present study triangular elements and line3.r source distributions are used, which are different from the quadrilateral elements and constant source distributions used by Dawson(1977).
The algorithm begins with solving the double-body prob-lem. The purpose of using the double-body shape is to
gen-erate a steady flow around the body with zero velocity in
other words. the flow generated by the double-body moving
steadily ut a uid of jnfinite extent, is equivalent to the flow
generated by a single body moving in a semi-infinite fluid
when the free surface disturbance is neglected or = 0 on = 0.
To satisfy equations (4), the double-body velocity
po-tential 4 at any point in the fluid domain (including the
boundanes) is expressed in terms of sources distributed
lin-early over each element of the mesh on the body surface. This potential is written as:
"'
1 13,=rz,+J (
+ )o(q)dS,.
s, r(p,q)
r(p,)
where p is any point in the fluid field including the body
surface: j refers to the element number; ne is the total
num-ber of ekinents on the body surface (the real body only), and r(p.q) is the distance front p to the integration point (zq,Y,,;) on element
j
and r(p,) is the distance from pointp to the irua.g point (eq,Yq, q) of point (zq, Yq' Zq) with
re-spect to the plane = 0.
In this equation o'(q) is the linear function based on
"m(q). o':(q'. o'3(q), where o'j(q), o'(q), o'3(q) are the source
density ma.itudes at the locally numbered three
verticeswhich define panel q. The double-body velocity potentialis written as:
& = Uzi
1°fs,r(i,q) + (..))TJdSs
(6)where 3 represents the velocity potential at node number
i: o represents the source density at node point j; T, is a
tent function centered at node j, i.e. 7', = 1 at node j and
= 0 at the nodes surrounding j ;Si represents the contin-uous compact support of T, i.e. the surface of the elements having common vertex at node
j;
r(i, q) represents thedis-tance from nc.de point ito the integration point (zq,yq,Zq)
and (r.,.y.
) E 5,; r(i,7) is the distance from node ito
the image point with respect to = 0. of (zq,yq,:q). The
dista.nces ri.q) and r(i,)
are given as:(5)
r-(i,q) = /(z1
- z)2 + (y - y,)2 + (; - zq)2
(7)r(t.) = /(zi_q)2+(yi_g,)2+(;_iq)2.
(8)By ktting equation (6) satisfy at each node (i = 1, ...,nn) the impermeability condition in equation (4), a linear
sys-tem of nn equations can be obtained for the an unknowns o',, i = 1,2.3. ...,nn. The linear system couldbe written as:
[Alto']
U[nJ.
(9)A is the influence coefficient matrix whose entry a1 rep-resents the influence at point i produced by a unit source
density at node j. The unit
source density at point j islinearly distributed on all the elements surrounding
pcnt j.
i.e the source density at node j is equal to I and is ser' at
the surrounding nodes.
a,, represents the z compcnt of
the unit normal of the body surface at point i. The bMar system is solved by using the Gaussian elimination meod.
A standard linear system solver is adopted.
The integration in equation (6) is performed by
ne of
two methods depending on the collocation point i be.zon
eement j or not. When the collocation point i is nc
'. the
integration element
j
a numerical integration procedt_e is u..sed. When the collocation point i is located on theinte-gration element j (i.e. when i is any vertex of the
ent)
the integrand becomes singular, however it is integraEe. A theoretical formula is derived to calculate the integrals.
In Dawson's linearized problem, the free surface
cz-eidi-tion is written in terms of the streamlines located on the
plane z = 0. Therefore these streamlines must be
ger-ated and a surface mesh based on these streamlines. rnust be produced. In the present study the streamlines the double-body flow, on z
= 0, are generated by solv.x the
ordinary differential equation which defines
a strearr on
0, that is
'10) where z, y are the coordinates of a point on the streamline; i arid i are velocity components in the double-body & in
z a.nd y directions respectively, which can be expressed in
t.ms of the source density
o', i
1, 2, 3...nn, distzibtedou the body surface mesh and obtained from soFcin.c the
double-body problem.
The first node on each streamline and
the increents
.z. in the z direction are predefined by the overal! e ofthe free surface mesh and the size of the elements, wii are Froude Number dependent.
Having solved the double-body problem. gener1z.e the streamline-bounded free surface mesh, and compnsed the coefficients A, B. and C in terms of the double-bodr
pc-ten-tial the total potenpc-ten-tial could now be obtained by scring Dawson's linearized boundary value problem (3). 11 the
same manner, as for the double-body problem,
the
ptur-b.ation potential ' ,where= 4+' is expressed in :er.s of
a distribution of sources on the discretized body surface andpart of the free surface. By satisfying at each nodal point
the body surface (impermeability) condition or the fr sur-face condition of problem (3), and by expressing () by a suitable four-point finite difference equiva.lent, we obtain a
linear system to be solved for the unknown source d.sity distribution o',;i = 1,...,nnt ; in the form:
AB RB
=
AS
RSThe upper part of the influence coefficients matrix _4BJ represents the influence at the surface of the body produced by the sources distributed on the body surface and the free surface. The entries to this part of the matrix axe computed
by the equation which is obtained from the solid body
im-permeability condition. The corresponding part of the right
ha.nd side [RDj contains Un, terms. The tower part ci the
influence coefficients matrix [ASJ represents the influence at the node points of the free surface produced by the sources
on the body surface and on the free surface. The entries to this part of the matrix are computed by using the free s-urface boundary condition equation whose known terms, expressed in terms of the double-body solution potential, make up the right hand side part [RS]. The integraLs used in computing fab1j and [aJ,j are *imihr to the ones i.sed in solving the double-body problem, i.e.
when i is a,
the theoretical formulation is used; when i not on 5, the
numerical integration is preferred. The routine for solvin.g the linear system generated in conjunction with the double-body problem is also used here to solve the system of linear equations (11).
4
Solution of the Neumann-Kelvin
Problem
The Neumann-Kelvin problem is solved by using the Rank-inc source method. SRank-ince the Neumann-Kelvin free surface condition is linearized by a uniform fluid flow plus a wave perturbation. no image surface or streamlines
on : = 0 are
needed to solve the problem. The free surface is discretined by straight lines parallel to the direction of the uniform sow. The size of the surface mesh and the size of the elements aredetermined in the same way as for the Dawson's problem so-lution.
The Neumann-Kelvin problem m terrns.of the pertrs.rba-tton 0' is given by equations (2), where U is the fluid velocity far away from the fixed body.
The perturbation ' can be expressed by Rankine sources
linearly distributed on the elements of the free surface and
the body surface. That is we write:
ô' =J (p,q)T()dSs
where S, is the area of element J, net is the total numb-er of elements which include the elements on the free surface and
on the body surface.
The total velocity potential is then in the form
(13)
As in the proposed solution of Dawson'sproblem,by ki-ting 'o satisfy at each node point, the body surface or the fluid free surface conditions given in (2), and this tiine ex-press (u) by the four-point finite difference formula.. we
can write nnt equations similar to the linear system 411) obtained for Dawson's problem. The upper part of the
in-finence coefficient matrix [ABJand corresponding right band
side vector IRBJ are identical to their counterparts obtained in solving Dawson's problem. However, due to a different free surface condition the lower
part of the matrix [AS i
not the same, while the corresponding riglit hand side RSJ ts identically equal to zero.1
412)
4
5
Free Surface Wave Evaluation
The free surface elevation can be evaluated by using the velocity components u4, v. r. on the plane = 0. The free
surface elevation at each node point of the free surface mesh
m i = nn + 1, nfl + 2...nn, where nn defines the number
of nodes on the solid body mesh, is given as:
=
.[u (iz,2+v42 +L:2)I;
i = nn + 1,nn+2...nni
(14)
where i refers to a grid point on the free surface mesh. The perturbation velocity components u,
v, r can be
calcu-lated directly in terms of the source densities obtained by
solving the linear systems (11).
8
Wave Resistance Calculation
The wave resistance can be calculated by integrating the pressure over the area of the solid body surface submerged
in the fluid. It is expressed as:
R=_>.f PqNzqdSq
(15)j=1
where Nzq represents the unit normal component of element
q in the 2 direction; P5 is the pressure on element J, it is
the linear function of the pressures Pji, P and p
at the
three vertices. These locally nunibered
pressure values p,
p and 33 can be converted into globally numbered pressure values p. in the same way as are the source densities a',. The pressure at each node is calculated by:= [U2_u+u,..
1;i=1.2.3...nn
(16)where it4 = U + u, v, = ti. and w, =
In the present work, the wave resistance coefficient is
computed as:
Cw=R;.pU2S,
(17)where S is the total surface area of the body and p is the
fluid density.
7
Numerical Results and
Compar-ison
The submerged prolate spheroid cases, considered
theoreti-cai.ly by Farrel( 1977) and nuxnexically, with Neumann-Kelvin
source method, by Doctors and Beck(198fl are here
anal-ysed using the present metbod..
The dimensions of the ellipsoid are a = .0m, I'= 1Mm,
where a represents the semi-m.a.jor axis and ô represents the
semi-minor axis of the ellipsoid. Two cases with submer-gence depths d = U.3226c and d = O.k are studied, where
The total number of elements used in the computation is
2244. with 228 elements on the surface of the ellipsoidand
2016 elements on the free surface It should be mentioned
that the above elements are only distributed on half of the
ellipsoid surface and on half of the free surface respectively, since the symmetrical condition about planey =0 is used.
The Neumann-Kelvin and Dawso&s linearized problems for the defined above submerged prolate spheroid,are solved
by the present method and the wa-e resistance coefficient C'. is computed for a range of Froude numbers Fd, defined
as = with L& representinga characteristic body length taken as equal to 2a. The comparison with the the-oretical solution of Farrel and the numerical results by a
Neumann-Kelvin method of Doctors and Beck of the same Neumann-Kelvin problem are presented in Figs. 1-4.
The comparison shows that the present computation for solving the Neumann-Kelvin problem matches the results of Doctors arid Beck and also matches the theoretical solution by Farrel in most of thecases except Lfl Fig 3. In Fig. 3 the
present computation matches the results of Doctors & Beck, however both the present computation and the computation of Doctors & Beck differ with Farrel's results corresponding to the case in Fig. 3. No satisfactoryexplanation were given to resolve the difference in the papee published by Doctors
& Beck( 1987).
In Fits 1-4 the computational results of the presentmethod in solving Dawson's problem are also compared with the
re-suits of Farrel, Doctors & Beck in solving the Neumann-Kelvin problem. For the high Froude number cases, Fig.
I and Fig.
2, the present computation to solve Dawson'sproblem is in good agreement with both results of Farrel
and Doctors & Beck for the Neumann-Kelvin problem. But
in the low Froude number cases. Fig. 3 and Fig. 4 differ-ences are found between the
present computation to solve Dawsons problem and the results by Farrel and Doctors & Beck in solving the Neumann-Kelvin problem.
Numerical experiments also show that the location of the wave front, generated by the presence of the body does not depend on the location of the free surface mesh as long as D1 L,,,. Here D, denotes the distance between the left most edge of the surface mesh and the front of the body,
and L
=
2rF4Lo, represents an estimate of the surfaceundulation length for a Froude number Fd and characteristic
body length L . Fig (5) shows that the
generated surface
waves are the same when
different leftmost edge locations are used
8
Conclusions
A Rankinesource method is developed in the present study to solve both Neumann-Kelvin
and Dawson's linear
prob-lems. Good
agreements are found betweenthe results of
the present computationswhen solving the
Neumann-Kelvin
problem and the results obtained theoretically by Farrel and numerically by Doctors & Beck in solving the same
Neuman.n-Kelvin problem. Differences are found between
the present method a-hen used
to solve Dawso&s lineaproblem and the results by Farrel and Doctors & Beck fo the Neumann-Kelvin problem. The differences in the nu
merical results could be induced whensolving ba.siculiy dif ferent formulations. These variations become more apparen at low Froude number.
References
Baar,J.J.M and PriceW.G. 1988 Developments in the cal-culation of the wave-making resistance of ship.
Pr-oc. R.
Soc. Land., A 416, pp.115-147.
Dawson, C.W. 1977 A practicalcomputer method (cc solv-ing ship-wave problems,
Proc. 2nd mt. Conf.
Nunz.. Ship Hydi-od. Berkeley, pp.30-38.
Doctors, L.J. and Beck. ELF. 1987 Convergence Prcoerties of the Neumann-Kelvin Problem for a Submerged Body, J.
Ship Research, Vol.31. No4. pp.227-234. Cadd,G.E. 1975 A method of computing the flow
and
sur-face wave pattern around hull forms, The Rojal
Inittutian
of Naval .4rchitects.
Vol.118, pp.207-219.Farell, C. 1973 On thewave resistance of a submerged
spheroid
J. Ship Research. VoLIT, .Vo 1, pp.1-li.
Pawlowskj, J.S. 1990 On Dawson's method, In preparation. Ogiwara,S. 1983 A method to predict free surface &w ..round ship by means of Rankine sources, J. of Kansal Soc Nay.
.
2.
.
.
.
2+ ,.,.J/
£''" 4'S.ck
o flEa..I-i
2(d/cO.3266 or H/L-O. 16)
Fig. 1. Ware resistance coefficient C, at high Froude n*nb (d/c=O.3266)
.. a
.:..
l41III
1:UIII
SN
5.75(d/c-O.5 or H/L-O.245)
Fiç 2. Ware resistance
coefficient C,, at high Froude nb
(d/c=O.5)
6
2
2
(d/c-O.3266 or H/LO. 16)
Fig. 3. Wave resistance coefficient C, at low Froude number
(d/c=O.3266) 0.37
(d/c'..O.5
or H/L-'O.2450)
1.2* Ulit
1.21 5.31 1.34 Pd T2:
Fig. 4. Wave resistance coefficient C1.,. at low Froude number (d/c=O.5)
2
x
+ £ 6t.I/
a..i.'
Ppua1-a i.a
x
+ £ 0'VP.,,
a..,.,, E1..i
F,.:.. F#m'..
Pd S. Ul.a
1.14. a
Fig. 5.a. Comparison of surface meshposition and wave startin.g point (3-D view
4Y4A
Fig. .5.b. Comparison of surface mesh position and wave starting point (side views
!["Wörterbuch der griechischer Papyrusurkunden mit Einschluss der griechischen Inschriften, Auschriften, Ostraka, Mumienschilder u. s. w. aus Ägypten", IV Bd., Emil Kiessling, Berlin 1944 : [recenzja]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)