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OCR for page 417
Three Dimensional, Unsteady Computations of
Nonlinear Waves Caused by Underwater Disturbances
Y. Cao, W. Schultz, R. Beck (The University of Michigan, USA)
Abstract
Thre~dimensional unsteady nonlinear waves gen
erated by underwater disturbances (such as a mooring
sourc~sink pair or a moving body) are modeled us
~ng a mixed EulerianLagrang~an tune marching pros
cedure combined with a desingularized boundary in
tegral method. The waves computed by the present
method are compared with those for linear theory to
find the nonlinear effects. The wave resistance, lift,
moment and the pressure distributions on the body
are also calculated. We find that the wave patterns of
a spheroid and a relevant simple sourc~sink pair dm
turbance are very similar as long as the disturbances
are not too close to the free surface.
1 Nomenclature
CD
CL
CM
Cp
D
Dm
Fr
F
M
drag coefficient
lift coefficient
moment coefficient
pressure coefficient
diameter of spheroid
local mesh size
Froude number
hydrodynamic force acting on body
hydrodynamic moment acting on body
submerged depth of disturbance
h _
i;' ,k unit vectors
L
Ld
p

r
S
Sib
Sf
Sb
Sf
length ot mayor axis of spheroid
desingularization distance
1~' factor of desingularization

M moment acting on body
Nb node number on spheroid
Nit number of elements on spheroid in ~ direction
Nb number of elements on spheroid in x direction
Of node number on free surface
Nf
N,,f
n
node number on free surface in x direction
node number on free surface in y direction
outward normal of body surface into fluid
pressure
position vector from center of body
area of spheroid surface
body boundary
freesurface boundary
singular surface inside body
singular surface above free surface
417
t
V
V(t)
27
Xb
x
a(t)
ab
Q
time
velocity of body surface
velocity of disturbance
field point
position vector of free surface= (Of, Of ~ Of )
point on body
singular point of fundamental solution
desingularization exponent
startup inverse time constant
wave length of 2D linear wave
wave elevation
strength of sourcesink disturbance
steady value of a(t)
strength of source distribution above free surface
strength of source distribution inside body
velocity potential
fluid domain
2 Introduction
Since LonguetHiggins and Cokelet~ first developed the
mixed EulerianLagrangian method for twodimensional sur
face waves on water, variations of this method have been
used for a variety of nonlinear free surface problems in two
dimensions (Baker,2 Vinje and Brevig,3 etc). The meth
ods require at each time step: 1) solve a boundary value
problem in an Eulerian frame and 2) update the free sur
face points (which construct the free surface) by integrating
the nonlinear kinematic and dynamic free surface boundary
conditions with respect to time. More recently, the method
has been used for threedimensional nonlinear wave prob
lems. Dommermuth and Yue4 used this method to solve
several axisymmetric problems. Extensions to fully three
dimensional nonlinear unsteady waves are given in Dommer
muth and Yue5 using a spectral expansion procedure that
is limited to periodic problems without bodies. Jensen, Mi
and Soding6 solved the steady nonlinear ship wave problem
by using a simple source distribution above the free surface.
Cao, Schultz and Beck7 used the time marching procedure
for a preliminary study of the threedimensional nonlinear
wave pattern caused by a simple disturbance (a sourcesink
pair) moving under the free surface.
To make the time marching procedure practical, it is im
portant to have an effective solution method for the bound
OCR for page 417
ary value problem since it requires most of the computa
tion time. A boundary integral method is powerful because
it reduces the computational domain by one dimension. In
conventional boundary integral formulations, singularities of
the fundamental solution are placed on the domain bound
ary. When singularities of the fundamental solution are
placed away from the boundary and outside the domain of
the problem, a desingularized boundary integral equation is
obtained.
The first use of a desingularized method is the classi
cal work by Von Karmas for the flow about axisymmet
ric bodies using an axial source distribution. The strength
of the source distribution is determined by the kinematic
boundary condition on the body surface. Kupradze9 pro
poses locating the boundary nodes on an auxiliary boundary
outside the problem domain. Heisted studies some numer
ical properties of integral equations in which the singular
points are on an auxiliary boundary outside the solution
domain for plane elastostatic problems. Han and Olsonii
and Johnston and Fairweather~2 use an adaptive method in
which the singularities are located outside the domain and
allowed to move as part of the solution process. This adap
tive method requires considerably fewer singularities than
the number of boundary nodes, but it results in a system of
nonlinear algebraic equations for both the strength and the
location of the singularities. For unsteady nonlinear waves,
Schultz and Hong~3 use the desingularization technique in
two dimensions. McIver and Peregrinei4 ~5 show that a two
dimensional overturning wave can be well modeled by only
a few singularities outside the flow domain with desingu
larization. Webster uses a triangular mesh of a simple
source distribution inside the surface of arbitrary, three
dimensional smooth bodies and improves the accuracy of
solution. Cao, Schultz and Beck7~7 compute the unsteady
waves caused by a simple sourcesink pair using a source dis
tribution above the free surface as in the steady nonlinear
computations of Jensen, Mi and Soding6.
In the following sections, we describe the problem for
mulation in section 3 and a more detailed discussion on
the desingularized boundary integral method in section 4.
Finally we present the results of the computations for the
waves caused by a simple disturbance moving below a free
surface with forward speed and the results for a fully sub
merged spheroid moving below a free surface.
3 Problem Formulation
For an irrotational, incompressible flow in an ideal fluid,
the Laplace equation is the governing equation for the ve
locity potential ¢:
A¢=0 (in Q)
The boundary conditions are:
Dt Zf + 2 Vat Vat (on Sf)' (2)
~ = V'> (on Sf), (3)
~ = V ~ n (on Sb), (4)
Van O (es x ~ Do), (5)
where Q is the fluid domain. Equations (2) and (3) are the
dynamic and kinematic conditions on the free surface Sf in
Lagrangian form. Here, Xf = (Xf, Of, Of) is the position vec
tor of a fluid particle on the free surface, Do iS the substantial
derivative following the fluid particle, n is the unit normal
vector of the body surface pointing into the fluid. din is the
normal derivative on So, and V is the velocity of the body
surface, which we assume is given. The quantities are nondi
mensionalized by setting the gravitational acceleration, the
fluid density and an appropriate length scale equal to unity.
For the unsteady problem, initial conditions are required.
Here, we study the flows generated by disturbances starting
from rest. Therefore, at t = 0, we require that ~ _ O and
the free surface elevation ~O. The coordinate system is
shown in Fig. 1.
Free Stream _
~ _
H
D'C=FF I ~ I l
1~1i~'
1
~L
Fig. 1 Problem definition and coordinate system
4 Solution Procedure
The initial boundary value problem (15) and the asso
ciated initial conditions are solved by the mixed Eulerian
Lagrangian method. In this method, the following boundary
value problem with a Dirichlet condition on the free surface
and a Neumann condition on the body surface is solved in
the Eulerian frame at each time step:
/~= 0 (in Qj,
A= To
~ =V.n
{fin
Vet ~ O
(on Sf),
(8)
(9)
where ¢0 and Sf are known from the previous time step.
After solving the boundary value problem, the velocities of
the fluid particles constructing the free surface can be cal
culated and the free surface conditions (2) and (3) can be
integrated with respect to time following the fluid particles
to update their potentials and positions which serve as the
boundary conditions at the next time step. This procedure
is repeated as time goes.
418
OCR for page 417
There are many methods to solve (69~. The method we
use is the desingularized boundary integral method. Simi
lar to conventional boundary integral methods, it reformu
lates the boundary value problem into a boundary integral
equation. The difference is that the desingularized method
separates the integration and control surfaces, resulting in
nonsingular integrals. There are two versions of the method:
direct and indirect. In the direct method, the integral equa
tion is obtained from Green's second identity evaluated on a
surface (control surface) somewhere outside the problem do
main and the integration surface is the problem boundary.
In the indirect method, the solution is constructed by in
tegrating a distribution of some fundamental solutions over
a surface (integration surface) outside the problem domain.
The integral equation for the distribution is obtained by sat
isfying the boundary conditions on the problem boundary and
(control surface).
The effectiveness and accuracy of desingularized bound
ary integral methods have been examined by Schultz and
Hongi3 for twodimensional problems, Websteri6 for three
dimensional steady flows, and Cao, Schultz and Beck7~7 for
threedimensional unsteady waves caused by simple under
water disturbances. The following are advantages of the
desingularized boundary integral method:
More accurate solutions may be obtained by a desin
gularized boundary integral method for a given trun
cation.
The kernels are nonsingular, so special care is not re
quired to integrate the singular contribution. Simple
numerical quadrature greatly reduces the computa
tional effort by avoiding transcendental functions.
Fewer nodes may be required since simple quadrature
eases the restrictions of a flat panel.
There is more flexibility since higherorder Green func
tions or fundamental solutions can be more easily in
corporated.
The indirect desingularized boundary integral method
has two more advantages when compared to the direct one:
· Integrals can be replaced by a summation if the desin
gularization distance is sufficiently large. This makes
the computation even simpler.
· The indirect method may result in smaller errors due
to truncation of an infinite boundary.
Desingularization also causes some difficulties associated
with uniqueness and completeness. However, if the singular
point is located away from the boundary a distance pro
portional to the local mesh size, the singular point will get
closer to the the surface as the mesh becomes finer. In the
limit, the desingularized formulation becomes identical to
the singular formulation.7
Because of its advantages, we use the indirect desingu
larized method. We construct the solution using a source
distribution on a surface (Sf) above the free surface and a
source distribution on a surface (Sb ~ inside the body surface:
419
¢(X)=~S'af(Xa)~_~ ids
~ISb ~za~
(10)
By applying the boundary conditions, (7) and (8), we obtain
a boundary integral equation for the unknown strength of
the singularities, Afros) and ab~x2,j,
+ ,, ab(Xs) ~~ i ~dS = 550(Xf) (on Sf) (11)
S' a/(~'~S);3n (a D _ _ At) dS
+ , ,, at ~3 (I ~ _ _ if) dS = V · n (on Sb), (12)
where Us is the integration point on surfaces Sf and Sb, Of
is the control point on Sf, and xb is the control point on Sa.
After af~x8) and abbot) are determined, the fluid parti
cle velocities on the free surface can be calculated. Then the
time marching procedure integrates the free surface bound
ary conditions.
The pressure on the body surface is evaluated using the
Bernoulli equation:
_p = Bi + 2 ~V¢~2 + Z = d'+ + ~ 2 V ~  V) · V) + Z. (13)
where ~ = (~93' + V V)¢ is the substantial derivative of the
potential at fixed points on the body surface. The second
form is more useful when following points fixed on the body
moving with velocity V.
The forces and the moments on the body are calculated
by integrating the pressure over the body surface:
J. J/Sb (14)
and
M = ~ Isper x node (15)
where r is the position vector of the body surface point to a
reference point (usually the center of the body).
5 Numerical Implementation
In the results presented in this paper, the submerged
disturbance (either a simple sourcesink pair or a spheroid)
moves in thex direction smoothly starting from rest to
a final speed. The strength of the sourcesink pair is also
smoothly increased from zero to a final value.
The free surface conditions (2) and (3) are in the fixed
coordinate system. This has an advantage since no spa
tial derivatives are required on the free surface which helps
reduce numerical reflection from the truncated boundary.
OCR for page 417
For large time simulations, the computational window moves
with the disturbance. At certain time steps, some fluid par
ticles are ignored downstream and new particles with zero
values of potential and elevation are axlded upstream. The
initial free surface grid (at t _ 0) is equally spaced in the
x direction and the spacing is increased algebraically in the
y direction. The moving computational window makes it
difficult to have a nonuniform grid in the ~ direction with
finer spacing near the disturbance.
Collocation is used to satisfy the boundary conditions on
the surface grid. The solutions are constructed by replac
ing the integrals over surfaces Sf and SO in (10) by isolated
sources on the surfaces. The sources are placed approxi
mately perpendicular from the node points on the bound
aries at a distance Lo determined by
Lot = Iy(Dm)
(16)
where lo is a parameter that reflects how far the integral
equation is desingularized, Dm is the nondimensional local
mesh size (we choose Dm as the square root of the average
of the areas of the four elements around the node point)
and ~ is a parameter associated with the convergence of the
mesh refinement. An appropriate ct lies between O and 1
to ensure the convergence of the mesh refinement and the
uniqueness and completeness properties of the solution of
the integral equation. We have examined the influence of la
for two problems: 1) a simple potential problem in which a
dipole is below a ~ = 0 infinite flat plane and 2) a prelim
inary study of the nonlinear waves by a simple sourcesink
disturbance. It was found that good solutions could be ob
tained for 1 < la < 3 and the solutions are not sensitive to
the variation of lo in this region. More detailed discussion
on the selection of cat and lo can be found in Cao, Schultz
and Beck7. We found that lo = 1 is "optimal" in considera
tion of the condition of the resulting algebraic system. This
value is therefore used in the present computations for the
free surface desingularization.
For the example with the submerged body, the mesh size
on the free surface is usually larger than that on the body
by about 10 times and the differences among the influence
matrix coefficients are large, so that the resulting system
for of and ab is likely to be poorly conditioned. To avoid
this, we split the system into two, one for of and the other
for at which are alternately solved using LU decomposition
for each subsystem. Each set of equations is much better
behaved and more accurate solutions can be expected. An
other advantage of splitting is that the coefficient matrix for
ab does not change with time and needs only to be inverted
once for the entire time simulation. The matrix for of does
not change during the iteration between the body and the
free surface and only needs to be inverted once for the cur
rent instant of time. Of course, the matrix for of changes at
next instant of time. In contrast, the sourcesink pair distur
bance has fewer unknowns and can be solved very efficiently
with a GMRES minimization procedure.7
The pressure on the body is evaluated at the node points.
The substantial derivative of the potential ~ in (13) is calcu
lated using a fourpoint forward difference scheme. A fourth
order RungeKuttaFehlberg method is used in the nonlinea*
free surface integration. An initial time increment is set, but
is modified by the RungeKuttaFehlberg subroutine where
appropriate.
6 Results
6.1 Numerical aspects
In the two examples presented in this section, the dis
turbance velocity is given by V(t) = Fr(1ems) in thex
direction, where Fr is the Froude number. For the source
sink disturbance, the strength of the source and sink is given
by arty = aO(1en. The problems are assumed to pos
sess symmetry about the xz plane.
The free surface is discretized using Nf nodes and Nf
nodes in the x and y directions, respectively, to form Nf =
Nf x Nf free surface nodes. For the body problem, the
spheroid used by Doctors and Becker is chosen. The diameter
t~length ratio D/L is 0.2. The basic grid on the surface is
shown in Fig. 1. The grid lines are spaced uniformly in the
circumferential direction. The grid lines have a cosine spac
ing in the longitudinal direction. The body has a grid with
Nb elements in x direction and Nb elements in the circum
ferential direction, resulting in Nb = (Nb _ 1) x (Nb + 1) + 2
body nodes including the two end points. To improve the
computational farfield behavior, we add negative images of
the disturbance singularities.7
The pressure is integrated over the body in (14) and (15)
using Simpson's rule first in the circumferential direction and
then the longitudinal direction. The usual hydrodynamic
coefficients (CD, CL, and CM) are obtained by multiplying
F · l; F · k and M ·.1 by 2/(SFr2), where S is the area of the
spheroid surface. The pressure coefficient, Cp, is defined as
2p/Fr2.
We require the ratio of the element size in :e direction to
the wave length to be less than 1/10 to resolve the waves.
The nondimensional wave length ~ is estimated by ~ =
2,rFr2 using twodimensional linear theory.
6.2 Waves generated by a sourcesink pair mov
ing below a free surface
In this example, the length scale is chosen to make the
depth of the submerged disturbance unity. The distance
between the source and sink is chosen to be 0.1. The Froude
number based on depth is unity. The midpoint between the
source and sink is initially located at point (5,0,  1~. The
grid on Sf at t = 0 has 41 x 16 node points within 0 ~ x < 20
and O < y < 7.5. The spacing increases by 10 percent in the
y direction. The initial time increment in the time marching
is 0.2.
The potential ~ is expressed as a sum of 1) the source
sink disturbance pair at the distance h below the undis
turbed free surface, 2) the image disturbance above the
undisturbed free surface, and 3) a sum of Nf sources of
unknown strength in an array a distance La above the dis
turbed free surface using (16~.
420
OCR for page 417
The method is first applied to the waves generated by a
sufficiently small disturbance such that linear wave theory
is a good approximation. The results of the present method
using fully nonlinear free surface conditions are compared to
an "exact" solution computed from a timedependent Green
function for a Kelvin wave source that satisfies the linearized
free surface condition.~9 Fig. 2 shows the comparison of the
wave elevation along the symmetry plane at t = 10 com
puted by the present method to that computed by the linear
calculation for a weak disturbance (aO = 0.05~. The nonlin
ear and linear results agree very well. Independent compu
0.02
0.01
o.oo
O . 0 1
o . 02
1
l
 1~1onlinear
\/ ~ Line"
0 5 10
x
15 20
Fig. 2 Wave profiles along y = 0 (weak disturbance)
0.3
0.2
i: 0. 1
._
o.o
O. 1
i,
o . 2
^ Nonlinear
0 5 10
x
15 zo
Fig. 3 Wave profiles along y = 0 (strong disturbance)
tations using: a) a smaller computational domain (with the
same mesh spacing within 0 < x < 15 and 0 ~ y < 7.5), b)
finer mesh grids (81 x 16 and 41 x 31 with the same compu
tational domains, and c) doubling the time increment, result
in negligible difference for the nonlinear calculation. This in
dicates that even for the small disturbance example studied
here, the differences in Fig. 2 are primarily due to nonlinear
effects. Fig. 3 shows the results for a stronger disturbance
COO = 0.75), showing the larger nonlinear effects of the free
surface conditions, especially at the troughs.
6.3 Waves generated by a spheroid moving be
low a free surface
In this example, the length scale is chosen to make the
length of the spheroid l unity. The center of the spheroid is
initially located at (2,0,h), again with a moving compu
tational window. On the free surface, we use 61 x 16 nodes
within 0 < x < 7.5 and 0 < y < 1.875. The spacing in
the y direction increases by 10 percent for each row of nodes
further from the centerline. For comparisons to the results
presented in Doctors and Beck, we use the same submer
gence depths of the spheroid (h/L = 0.16 and 0.245~.
The potential ~ is expressed as 1) a sum of NO sources of
unknown strength inside the body, 2) the image of 1) above
the undisturbed free surface, and 3) a sum of Nf sources
of unknown strength above the disturbed free surface. The
desingularization distances of the sources above Sf are given
by (16~. To represent the body, the singularities (except
at the bow and stern) were distributed on a spheroid of
smaller minor axis inside the body. After some preliminary
calculations, the ratio of the minor axes of the two spheriods
was fixed at 0.3 in our calculations.
Fig. 4 shows a threedimensional view and contour lines
of the wave pattern caused by the spheroid for h/L = 0.245
and Fr = 0.6 at t = 25. A smooth startup (,ll = 2) and
Nb = 2N6' = 16 are used. We compare the waves pro
duced by the spheroid and those made by the relevant sim
ple sourcesink pair disturbance in Fig. 4 and Fig. 5. The
strength of the pair and the distance between the source
and sink are determined to give a Rankine oval having the
same length and midsectional area as the spheroid moving
in an infinite fluid. The comparison becomes meaningless if
the disturbance is too close to the free surface because the
simple sourcesink no longer represents the body well. The
same depth of submergence, location of the center and mo
tion of the disturbance as those for the spheroid are used
for a direct comparison. Comparison of Fig. 4 and Fig. 5
shows that the wave patterns of the spheroid and the rel
evant sourcesink are very similar except that the spheroid
generates steeper waves near the stern.
Fig. 6 shows the drag, lift and moment acting on the
spheroid as a function of time. As seen, the solution is
close to the steady state after the body has moved 10 to 15
body lengths. Fig. 7 shows an influence of the two different
startups of the body (p = 2 and oo) on the hydrodynamic
forces. Although both eventually merge to the same values,
the body experiences very different forces during the transi
tion. We notice that the body experiences a negative drag
for a short time soon after an impulsive startup.
421
OCR for page 417
1 875, ~~~:,
! / ~
/ ''
Ir t
~ \ ~
1 8751 _ 1_
O.0000
_ _ ~ 1 _ _ _ _ _ _ _ _ _ I J _ _ _
1.875 3.750
Fig. 4 Wave pattern (by spheroid)
5.625 7 500
0.008
(elevation contours are 0.02 apart) ~
.~, 0 006
0. 004
0. 002
° ~ °°°
Fig. 9 shows the convergence of the drag, lift and moment
on the body as a function of node number Ni using Nb =
2N~ = 8,12,16 and 20. For all these cases, the free surface
grid (61 x 16) adequately resolves the waves.
The comparison of the hydrodynamic coefficients to lin
ear theories at different Froude numbers is shown in Fig. 10.
Our results for h/L = 0.245 (solid triangles) compare well
with linear calculations. Our computations used a finer free
surface grid (71 x 16) for the smaller Froude numbers to re
solve the waves. For h/L = 0.16, the body is too close to the
free surface. For all attempted Froude numbers, the free sur
face is sucked down and touches the body surface which in
turn stops the computation. The linear calculations are not
affected by this because the free surface boundary conditions
o.o~o
0.002
o .004
o .006
\\\/ 
"_...
0 5
CM
0 15 20 25
t
Fig. 6 Hydrodynamic coefficients vs. time
0.020
Fig. 5 Wave pattern (by sourcesink pair) 0.015
(elevation contours are 0.02 apart)
Fig. 8 compares the pressure on the spheroid using the
present method and the NeumannKelvin calculation at t =
25 for the same conditions as those for Fig. 4. The compari
son is made for the pressure along the body centerlines of the
top, bottom and side. In the NeumannKelvin calculation,
the body surface is divided into flat panels with distributions
of constant source strength. The strengths are determined
by satisfying the body boundary condition at the centers of
the panels. The pressure is calculated at the center points.
The pressure elsewhere is obtained by interpolation. The
differences between the linear and the nonlinear results are
noticeable.
a
,~ 0. 010
o
~ o . 005
;^
o
:r
o .005
0.010
0 1
Gradual startup
 Imoulsive startup
2 3
t
4 5
Fig. 7 Influence of the startup of the spheroid
422
OCR for page 417
are satisfied at z = 0. (Note: To compare to Doctors and
Becki8, the moments for this figure are taken about the ver
tical projection of the spheroid center onto the undisturbed
free surface. The moments shown in the other figures are
taken about the centroid of the spheroid).
The computationswere carried out on a Cray YMP. The
results shown in Fig. 4 took approximately 4.3 CPU seconds
to solve the boundary value problem (69) which required
~ .o
0.8
~ 0.4
0.2
o.o
o . 2
Fr=0.6
t = 25  Nonlinear
0.6 L I ~ Linear
Bottom center line
~1 1 ~:
0.4 l l l l
Bow
X
Stern
Fig. 8 Comparison of the pressure on the spheroid
.° 8.0
B.0 .
4.0 _
2.0 _
0 50 100 150
Ni
200 250
Fig. 9 Sensitivity of body mesh
Slender body
Haveloclc
Doctors and Beck
x x x x Farell
~ ~ ~ ~present method (h/L = 0.245)
0.02
0.016
0.012
CD
BOOR
0.004
O
_
_ ~
1 i///'~16
0.2 0.3 on 05
l
0.6 0.7 0.8
Fr
003
0.025
0.02]
0.015 
CL
0.01
0.005
01
0.005
0.01
0.002
01
0.002
0.004
C,~
0.006
O.OQ8
0.01 4
0.012
0.014 ~.
0.2 0.3
16
~ \
,= 02450
~16
/
H/L = 0.2450
OR
l
02 0.3 0.4 0.5 0.6 Q7 0.8
Fr
Fig. 10 Comparison of hydrodynamic coefficients
(modified from Doctors & Beck)
423
OCR for page 417
solving each subsystem approximately 5 times. About 10
percent of this time was required for matrix setup. The
CPU for the entire time simulation was 45 minutes. The
computations for the simple disturbance in Fig. 5 took ap
proximately 3.9 CPU seconds to solve (69) and 40 minutes
for the entire time simulation. The difference in the CPU
time for the two computations is small because the splitting
procedure for the body problem oniv r~r~llir~ the ~'l~l;+;~l
evaluation of multiple righthand sides.
0.02
c 0.01
._
0.00
0.01
.J I.. I 3~"ll~;~ b11= "UUlblOll~
0.02
02 4
X
,' Id = 1.0, 2.0, 3.0~4.0
Id = 0.5
~ 8
Fig. 11 Effect of disingularization factor Id
° 30 . 0
25.0
20.0
5.0
0.0
s.o
o.o
1 _
l ~ Work
// ray
, . . ,
o
2 4 ~8 10
t
Fig. 12 Balance of wave energy and work done by spheroid
424
A faster iterative matrix solver can be used effectively
in the simple disturbance example.7 About onefifth of the
CPU time was required using the General Minimal Residual
Algorithm (GMRES) as compared to using LU decompos
titon. Although GMRES could be used for the spheroid
example, it would require special preconditioning. We tried
solving the entire system for the spheroid example using
both LU decomposition and GMRES without precondition
ing. The LU algorithm gave inaccurate solutions while GM
RES did not converge.
Fig. 11 shows the eRect on the wave computations of
five different values of desingularization Old = 0.5, 1.0, 2.0, 3.0
and 4.03. The results are not significantly affected except
for la = 0.5, which is too small for the integration by the
summation (or equivalently one point Gauss quadrature).
We also performed an energy conservation check for a
fixed control volume bounded by the free surface, the body
surface, a horizontal bottom and four vertical surfaces repre
senting the truncated farfield boundaries. The body starts
to move from rest at the center of the control volume. The
energy of the fluid in the control volume and the work done
by the body are shown in Fig. 12. The energy and the work
balance each other well for t < 6 before the waves and the
body reach the truncated boundary. Since the energy flux
is neglected, agreement is not to be expected after the body
or the waves reach the boundary.
7 Conclusions
Desingularization performs well for fully nonlinear free
surface problems without surface piercing bodies. Desingu
larization is not dispersive nor dissipative. Similar to those
methods requiring free surface discretization, our method fa
vors high Froude numbers since fewer nodes are required to
resolve the waves. Iteration between the free surface and the
body surface conditions is required. The waves produced by
the spheroid and the relevant sourcesink disturbance are
similar if they are sufficiently submerged. This indicates
that a simple disturbance (with a much simpler iterative
procedure and fewer unknowns) can be used if the surface
waves are the main interest.
Acknowledgment
This work is supported under the Program in Ship Hy
drodynamics at The University of Michigan, funded by The
University Research Initiative of the Office of Naval Re
search, Contract Number N00018486K0684. Computa
tions were made in part using a CRAY Grant at the Univer
sity Research and Development Program of the San Diego
Supercomputer Center. We acknowledge A. Magee for the
linear calculation of pressure shown in Fig. 8.
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