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OCR for page 469

A Boundary Integral Formulation for Free Surface Viscous
and Inviscid Flows about Submerged Bodies
C. M. Casciola
Istituto Nazionale di Studi ed Esperienze di Architettura Naval
Roma, Italy
R. Piva
University di Roma "La Sapienza"
Roma, Italy
Abstract
The possibility to generate numerical models based
on a boundary integral formulation for rotational free
surface flows, either viscous or inviscid, is explored
with the purpose to maintain some of the computa-
tional efficiency proper of potential flow approaches.
A simple method for the simulation of the unsteady
nonlinear behavior of the free surface has been de-
rived, starting from a mathematical model which de-
couples the kinematical and the dynamical aspects of
the flow field, without introducing the potential ap-
proximation. The strict relationship with the coupled
models based on the integral formulation for viscous
and inviscid flows, is discussed in details to extend
the present method to these more general conditions.
The mathematical and numerical aspects of the re-
lated integral equations are analyzed and an efficient
numerical solution is proposed for the solution of the
flow field generated by a moving submerged cylinder.
~ Introduction
Among the several methodologies adopted for the anal-
ysis of the free surface flow problems usually non-
linear and unsteady a special role is played by the
potential flow approximation. This actually leads to
a very appealing model consisting of a linear steady
equation for the field and a nonlinear unsteady bound-
ary condition on the free surface. The boundary inte-
gral equation method is particularly appropriate for
the numerical solution of such problem for two main
reasons. First, when applied to linear equations it
reduces by one the space dimensions of the computa-
tional domain, second it provides a description of the
complicated nonlinear boundary conditions more ac-
curate than any other computational approach. This
technique has been recently used for the numerical
simulation of the wave pattern generated by the mo-
469
tion of a submerged cylinder with both linear and non
linear free surface boundary conditions to], yielding
in a more efficient way the same results previously
obtained by finite difference in curvilinear coodinates
[51.
On the other hand, the viscous flow field about
a submerged body and the interaction between the
viscous phenomena and the free surface wave gener-
ation have been usually investigated by finite differ-
ence schemes of the Navier-Stokes equations in their
differential form. The proper numerical approxima-
tion of the boundary conditions of the free surface
7
togheter with the need to confine the computational
domain are the main difficulties connected with the
use OI only becnnlque. In spite of these drawbacks,
some interesting results have been presented recently
for the unsteady flow about a submerged body, even
in the case of breaking wave conditions A.
The main purpose of the present work is to analyze
the possibility to generate numerical models based on
a boundary integral formulation for the simulation of
rotational free surface flows either viscous or inviscid,
which maintain some of the capabilities briefly dis-
cussed for the potential approximation. In this case
the field equations either Navier-Stokes or Euler
are both non linear and unsteady thus preventing
the explicit confinement of the related terms into the
free surface boundary conditions, as for the potential
flow. In the past few years we investigated the in-
tegral equation approach for large Reynolds number
flows about streamlinead bodies [3] . We like here to
extend the formulation to viscous free surface flows
for the applications to ship hydrodynamics.
Following this idea, in section 2, the Navier-Stokes
equations are written in integral form by using the
fundamental solution for the unsteady Stokes operator
and a representation formula for the velocity vector in
the field in terms of the velocities and tractions at the
boundary is obtained. The numerical solution of the

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relevant integral equation presents some difficulties at
large Reynolds numbers, mainly related to the ten-
dential singularity of the kernel. The limiting case of
zero viscosity, that is the integral formulation for the
Euler equation, is discussed in section 3 to better ob-
serve the critical behavior of these equations and to
device the proper way to recover simplified models for
a quite general, but still sufficiently efficient, compu-
tational procedure. To this purpose we like to point
out that the great advantage of the potential models
in the framework of the free surface Bows, is not as
much connected to the introduction of the potential
itself as it is to the decoupling between the kinematics
(Laplace Eq.) and the dynamics (Bernoulli Eq.) of
the problem. The Bernoulli equation which is non lin-
ear and unsteady has to be solved at the free surface
(if the pressure is not required) to give the so called
dynamical boundary condition. A further kinematical
equation is required to describe, in its Lagrangian or
Eulerian form, the motion of the free surface in time.
A brief review of the set of boundary conditions
to be applied at the free surface for different approx-
imations of the flow model, starting from the more
general interface between two viscous fluids, is car-
ried out in section 4 to understand the role of each of
these conditions with regard especially to the coupling
or decoupling of the field equations. More specifically,
we like to understand how the free surface boundary
conditions reduce for flow models still rotational, but
simplified through a decoupling procedure analogous
to the one for potential flows. We analyze this subject
in section 5, where we derive directly from the Euler
representation a decoupled model which recovers for
the velocity the Poincare kinematical formula. The
role of dynamics in deriving the proper boundary con-
ditions is discussed in connection with this point. The
proposed model resulting from the above procedure,
is consistent and well equipped to treat the nonlinear
free surface behavior. It has a value in itself, as the nu-
merical results shown in section 6 for the submerged
cylinder may indicate. Furtherly, starting from this
model, for which we have studied the mathematical
and numerical aspects of the solution, we intend to
recover, through a backward procedure, the coupling
aspects and the diffusion phenomena neglected at this
moment. The direct connection between this model
and the more general representations for the Euler
and the Navier-Stokes equations, which has been de-
scribed in the present paper, indicates the main lines
to follow in the further investigations.
2 Integral formulation for vis-
cous flows
As mentioned before, in this section the Navier Stokes
equations are recasted into an integral form. It is well
known that the integral formulations for the Navier
Stokes equations have been mostly used for study-
ing the matematical aspects of viscous flows. A typi-
cal example is the theory of hydrodynamic potentials
which provides many important results for the lin-
earized Navier Stokes equations. A detailed descrip-
tion of the method, introduced by Odqvist, is given
in [1], where the direct integral representation for the
steady state problem is also presented. The aim of
Ladyzhenskaja's analysis, is devoted to the existence
and uniqueness of solution for the Navier Stokes equa-
tions. It is on these theoretical topics that the inte-
gral formulation for viscous flow has found their best
application, due to the simple analysis of the proper
boundary conditions.
More recently, due to the developments of the bound-
ary element methods, integral formulations have re-
ceived new interest for the numerical simulation of
viscous Bows. Although these methods have their best
application in linear problems, the advantages of inte-
gral formulations can be largely recovered in the nu-
merical simulation of viscous flows about streamlined
bodies when no massive separation occurs [3~. Actu-
ally in this case the nonlinear source term related to
the vorticity in the field, is confined to a small region
in the boundary layer and the wake, thus allowing for
an efficient discretization procedure in terms of finite
elements.
In this section the formulation, which has been
previously described in [2~ in full details, will be briefly
reviewed in order to present an extension to free sur-
face problems. In order to introduce the integral for-
mulation, for the sake of clarity, the Navier Stokes
equations are given here:
2
+V2 +X =
V.u = 0
— P +uV2u—Vie (1)
In equation (1) the term 11 = go is the poten-
tial energy per unit mass related to the gravity force.
The direct integral formulation for the system (1),
as obtained in [2], gives the velocity of the fluid in
terms of convolutions of the proper fundamental solu-
tions which will be denoted in the following by u(k)
and p(k) with the related traction given by tjk)
_p(k)nj + R(~; + ~j )
=
Uk(X ,t ) = | | (ujtj ) —tjuj )) dsdt— (2)
It in p~OujUj )dsdt +
| | Xjuji) dvdt—| uju(k)dv ltO
470

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In deriving the integral representation formula (2)
from the Navier Stokes equations in differential form,
the term X =—u x ~ (` = V x u) has been assumed as
a known forcing term; furthermore the traction vector
t(k) is modified in the form
i I (~::i + {1:rj) (3)
where:
p=P+tu2+0 (4)
is the Bernoulli group. For the sake of clarity it is
to be remarked the difference between the traction
vector tj involved in the boundary conditions and the
modified traction vector tj (tj = tj + 2u2 + II).
As shown by (2) it appears that the solution u is
given in terms of surface as well as volume integrals.
The first surface integral gives the effect of the bound-
ary values of the velocity and of the modified traction.
Nonlinear terms are present in this integral through
the modified tractions tj.
The second surface integral accounts for the mo-
tion of the boundary, which moves with velocity ZJa
(normal velocity component of a geometric point be-
longing to the boundary), and gives the effect of the
momentum flux due to the boundary motion.
This integral gives a nonlinear contribution in free
surface flows for the dependence of the boundary ve-
locity u~ on the fluid motion. The source of this non-
linearity are always confined to the boundary as the
ones acting in potential flows.
Different is the case of the first volume integral
which is related to the term X and accounts for the
rotational effects in the fluid. This source of nonlin-
earity, which directly comes from the field equation,
is confined to the rotational flow region. In the gen-
eral case it would require a large computational effort
and it would penalize the computational procedure
in comparison with other computational approaches.
However for the flow about a submerged streamlined
body, either attached or without a massive separa-
tion, this term can be efficiently treated by finite ele-
ments tecniques retaining in this way a great deal of
the computational efficiency of the boundary element
method. In order to complete the description of (2)
the last volume integral carries the information on the
initial conditions.
The integral representation gives a boundary inte-
gral equation for the collocation point, at which the
velocity is evaluated, approaching the boundary. To
account for the jump properties of the double layer
kernel t(k) a factor c (= 2' for smooth boundaries)
appears in the limit at the left hand side . The ob-
tained integral equation is a constraint between the
471
values assumed by the velocity and the traction at
the boundary. If we know one of them, for a given
configuration of the domain representation (2) gives
a Fredholm equation for the remaining one (either a
first kind for the unknown traction or a second kind
for the unknown velocity). The fundamental solutions
u(k) and p(k) satisfy the equations:
V . up) = 0 `5'
TV U(k) _ Vp(k) _ p_ = §*e(k)
which must to be considered in the sense of distribu-
tion theory with fix—x*) and bit—t*) Dirac delta
functions in space and time respectively. The explicit
solution of (5) for the free space problem in two di-
· —
menslons gives
?1( ) = biiiF— t7D id (6)
pot) = _ adG bit* - t)
where:
E =
G = 2 Intr) (7)
1 * e—r2/4V(t*—t)
F = —petit —t) At*—ty
47rp ~ (4L'(e. 2~)
V2E = F
where El is the exponential integral.
Looking at the fundamental solutions it appears
that all the terms in the integral representation that
contain a derivative with respect to ok can be recasted
in the form of a gradient flak = —a3*) by taking
the derivative with respect to ok out of the integrals.
These terms give an irrotational contribution to the
velocity field, while the rotational effect is related to
the function F. which is not reducible to a gradient
form. The function F is the fundamental solution for
the heat transfer equation and it becomes sharper and
sharper as the kinematical viscosity ~ goes to zero.
The function E has the same behaviour as it ap-
pears from the last equation in system (7~. The main
difficulty in solving these equations for large Reynolds
number flows is related to the crucial behaviour of the
functions F and E, rather than to the evaluation of
the volume integrals.
In order to gain a better insight on the proper-
ties of representation (2) in the case of large Reynolds
numbers, we analyze the limiting case of Re infin-

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ity. An integral formulation related to the differential
model for inviscid flows (that is the Euler equations)
is obtained in the next section.
3 The limiting case for inviscid
flows
The Euler equations for an incompressible fluid are
obtained from the Navier Stokes equations for the
Reynolds number infinity. Consequently one of the
boundary conditions for viscous flow should be re-
laxed, due to the lowering of the order of the partial
differential equation. For this limiting case a bound-
ary integral formulation is directly derived from the
one valid for Navier Stokes equations, rather than
from the differential model. Some introductory con-
siderations on the proper boundary conditions to be
applied is presented while the general discussion on
the free surface boundary conditions for both viscous
and inviscid models is given in the next section.
As a first step we obtain the limiting expressions
for the fundamental solutions. As it appears from
(6) the pressure fundamental solution p(k) doesn't de-
pend on the kinematic viscosity a, so that it retains
the same expression as for the viscous case. This is
not surprising since the pressure fundamental solution
is related to the compressibility effects in the fluid,
which in both cases is assumed of constant density.
Looking at the limiting behavior of the function F in
(7) the following relationship holds:
Limo in f (X) F (X, X* ; t, t* )dV = (8)
-lI(t* - t~f~x*)
from (8) it directly follows the distributional limit for
the function F as:
F = —H(t*—telex—x*) (9)
The limiting value for the function E (note that V2G =
6(x—x*) is given by
V2E = —H(t*—telex—x*) (10)
Therefore we have:
E = —H(t*—t)G (11)
Finally the limiting expressions for the velocity fun-
damental solutions are
472
~ ~ ( j aziazi a~jozE )
with the corresponding traction vector given by:
t(k) = _~(t—t No (13)
The direct combination of (12) and of (13) into the
integral representation (2) readily gives the limiting
formulation which is given below in vector notations
with the explicit expressions of the surface integral
while the remaining volume integrals are denoted by
Iv
u = |`n(t*) (u n) VGds + (14)
o l2(t) [(n V)VG—nV2G] dsdt+
Ito lift) Ha [(U V) VG—uV2G] dsdt + Iv
It clearly appears from (14) that the kernels in
the second and third integrals have an hypersyngular
behaviour when the collocation point approaches the
boundary. The formulation (14) does not present a
direct computational interest, but it has great interest
for the comparison it provides with the viscous case
showing in particular the computational difficulties to
be expected for large Reynolds number. In this case
in fact the fundamental solutions do not present a real
hypersingular behavior, but they are very difficult to
be numerically evaluated in an accurate way for their
close relationship with the hypersingular ones.
Moreover, the following analitycal manipulations
to set eq. (14) in a better form from the compu-
tational point of view will give some usefull sugges-
tions for an accurate treatment of the viscous equa-
tions. Following the procedure suggested for potential
cows by Hsiao and Nedelec, eq. (14) can be rewritten
through some known vector identities, in the form
u = {n(~*) (u n) VGds+ (~15)
/~0 i;n(~) (VP x n) x VGdsdt
+V* x ~ ~ 21J~ (V x u) Gdsdt
to en(~)
—V* X ~ J. L,o (U X VG`) dsdt + Iv
to no)
where V* = en if* . In the present form the hypersin-
gular kernel doesn't appears any more and the con-
tribution of the modified pressure P is closely related
to that of a vortex layer of density by = VP x n. Fur-
ther manipulations may be performed on (15) in or-
der to show its close relationship with an uncoupled

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model based on the Poincare kinematic formula as it
is shown in the next section. At the present stage it is
of some interest to notice that eq. (15) could be used
in the numerical simulation of nonlinear free surface
flows with no difficulty other than the computation of
the volume integrals. It is also worthwhile to notice
that, for a two dimensional case, the velocity u at any
point in the field is given in terms of two scalar valued
functions at the boundary, instead of the two vector
valued functions appearing in (2~. This fact is strictly
related to the lower order attained by the differential
model for the limiting case. As a result only one inte-
gral constraint is expected among the two scalars as
the collocation point approaches the boundary. Ac-
tually the boundary integral equation is given by the
normal component of the vector formula (153. The
other component gives, even on the boundary, a rep-
resentation for the tangential velocity component in
terms of given normal component and the modified
pressure.
In the next section the proper boundary condi-
tions for the free surface will be reviewed for both
the viscous and the inviscid model. The kinematical
description of the interface will also be presented.
4 Free surface boundary condi-
tions for couplet! models
In both the integral formulations presented in the pre-
vious sections the kinematical and dynamical parts of
the flow problem are strictly coupled. The proper
boundary conditions for the free surface problem in
the coupled models is introduced here in some details
in order to show the differences with the correspond-
ing treatment in decoupled models (splitting of kine-
matics and dynamics) to be introduced in the next
section.
The two possible kinematical descriptions of the
free surface (Lagrangian and Eulerian) are also intro-
duced and discussed at the end of this section. The
boundary conditions which directly follow from the in-
tegral formulations are treated first. Let us consider
the general case of an interface between two different
fluids. Two different cases are considered. In the first
one both fluids are assumed to be viscous while in
the second one they are considered both inviscid. For
a given position of the interface the boundary condi-
tions are given by the dynamical balance of the free
surface and by the no slip property of the viscous flu-
ids. Neglecting surface tension effect, the boundary
conditions are
u = uu
t = to
(17)
where the subscript u stays for the upper fluid and
the values of both members of (16) and (17) are un-
known. From the integral representation (2) it follows
that the boundary conditions are appropriate for the
interface problem. Collocating the integral represen-
tation for the lower fluid at a point belonging to the
free surface, one integral constraint between u and t
is enforced. Analogously the other integral equation
is obtained for u';, and t". In this way at each point
of the interface we have four vector unknowns and
four independent vector equations that is two bound-
ary integral equations and two boundary conditions.
the case of no motion of the fluid acting upon the
interface the problem is more simple and only one
boundary condition, the dynamical one, has to be im-
posed, in the form, for instance, of a known pressure
distribution:
t = —pan
(18)
In this case we have one unknown (the fluid ve-
locity at the free surface) and one integral equation
where the value specified by (18) for the traction vec-
tor is introduced.
A similar procedure leads to the boundary condi-
tions to be used for the formulation (15) for the second
case. Due to the inviscid character of the fluids it is
expected to get a discontinuity in the tangential ve-
locity component across the interface while only con-
ditions on the normal velocity component and on the
pressure may be used
In = Sun
P = Pu
(19)
(20)
We have now four scalar unknowns and four scalar
equations: the two boundary conditions and the two
integral equations relating the normal velocity com-
ponent to the pressure. As in the viscous case, for no
motion of the fluid acting upon the interface, a given
pressure distribution is assigned.
In order to complete the formulation, the kine-
matical description of the interface has to be briefly
recalled. As it is well known, for the geometrical anal-
ysis of a moving surface two possible decription are
available. The Lagrangian one, gives the position of
the geometrical points on the surface as a function
of a Lagr~ngian parameter I. Denoting by Xf this
function, once the velocity of the point labeled by ~
were known, the surface geometry may be determined
(16) solving the initial value problem:
473

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dXf
- = us
x(~, o)
(21)
= x0(~) (ED:
where Do is the set of values of the Lagrangian pa-
rameter ~ and Uf is assumed to coincide with the lo-
cal fluid velocity. Some problems are expected in the
case of the interface between two inviscid fluids due to
the discontinuity of the velocity across the interface.
Some intermediate value for the tangential component
should then be assumed to move the free surface.
In the Eulerian description the free surface config-
uration is defined by the implicit function f (x, t) = 0.
In the restrictive hypotesis in which a single valued
function is assumed for the free surface configuration,
the geometric description is given in the more usual
form z = If (x, t) . In this case the only normal compo-
nent of the surface velocity (~a) is defined. By equat-
ing its value to the normal fluid velocity the proper
equation for the kinematic evolution of the free sur-
face is attained.
The two geometrical formulations are equivalent.
In fact the Lagrangian description is slightly more
general than the Eulerian one, in which some reg-
ularity assumptions on the function f must be ac-
cepted. Moreover, whenever an Eulerian description
exists, the two formulations are equivalent. Only in
the computations some difference appears. Actually
the Eulerian description doesn't modify significantly
the dimensions of the boundary elements, while the
Lagrangian one concentrates the elements near the
crest of the waves where sharper velocity gradients ap-
pears and more computational accuracy is required.
For these reasons the Lagrangian description is pre-
ferred in the computational model for the nonlinear
free surface problem discussed in section 6.
From the previous analysis it clearly appears the
simple use of the boundary conditions for free surface
problems when using coupled formulations in integral
form. Coupled models allow to impose the boundary
conditions directly in terms of boundary velocity and
tractions. The integral form allows for a simple use
of the boundary conditions also in the discrete form,
with no additional difficulty as in the numerical mod-
els based on the differential equations.
In the next section a decoupled model is intro-
duced as a semplified version of the inviscid one. In
this case the definition of the boundary conditions is
not as direct as for the coupled models.
A general kinematical repre-
sentation and the dynamical
boundary condition
In this section a purely kinematical integral represen-
tation will be recovered as a semplified version of the
integral formulation for inviscid flows. It follows a
direct similarity between the two representations, in
the sense that boundary integral equations with sim-
ilar properties are attained in the two cases.
The procedure which leads from the inviscid inte-
gral formulation (15) to the present one is based on
the idea of a back substitution of the Euler equations
in differential form into the inviscid integral repre-
sentation. In this way the dynamical variables are
dropped out thus reducing (15) to a purely kinemat-
ical formula. In order to apply in a simple form the
above procedure a quite drastic approximation is per-
formed by considering the fluid domain fI to be fixed
in time. A more rigorous approach should reach the
same conclusion through fairly more complex calcu-
lations without leading to a deeper understanding of
the analogy between (15) and the present model. Un-
der this assumption the surface integrals containing
the boundary velocity L,a no longer appear and the
representation (15) may reduce to a much simpler ex-
pression. Actually, by combining the Euler equations
and the vorticity transport equation
`~9~ + VP + X = 0 (22)
aB: + V x X = 0
(23)
the following representation is attained from ea.(151
by a simple integration in time
~ ~ _ ,
u = ~ (u · n) VGds— (~24~)
isntuxn~x VGds—
x VGdv
n
Representation (24) is the well known Poincarei
formula when the velocity field is assumed solenoidal.
This is a purely kinematical identity which can be
used to obtain boundary integral equations for the
kinematics of the flow. When the collocation point
approaches the boundary, two equations follow from
eq. (24) by performing the tangential and normal
. ~ .
projections.
474

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cu. + Isn upon* dS
con + |6n US 45n' dS
=
_ l; | u"Gds+( (25)
=
~ * | U?GdS + 17 (26)
where It stays for the contribution of the volume inte-
grals. Equation (25) is a second kind Fredholm equa-
tion for the unknown us or a first kind equation with
Cauchy kernel for the unknown a". Analogously for
eq. (26~. The properties of the Eredholm equation are
well known from the potential theory while matching
properties can be found to hold for integro differen-
tial one (see (~7~. As expected the equations (25) and
(26) are not independent, in the meaning that, what-
ever is the unknown, for a given data the solution
of the first one is also a solution of the second one.
When the boundary is a solid wall the value of an is
assigned and us is the unknown. Different is the case
of a free boundary, where an is the unknown. In this
case to determine the boundary data u, the dynamical
part of the model should be used. The procedure is
strictly similar to that used for potential flows where
the values of the potential ~ are evaluated by means of
the Bernoulli's equation. In the present case, due to
the vorticity in the field, which has been retained, no
potential function exists. The Euler equations must
be used to obtain the relation between the tangential
velocity component us and the known pressure distri-
bution. In order to write the required equation we
shall fix a point on the free surface, labeled by the
Lagrangian variable I. Due to the boundary motion,
the unit tangent vector A, at the boundary point ~
will change in time. By projecting the Euler equation
on this tangent vector we obtain
DUr [3~
—u ·—
Dt Bt
=
_o
(2r
In order to semplify eq. (27) the second term on the
left hand side is written as:
u bitt = UTAH Ott +unn~ Butt (28)
Noting that ~ · ~ = 0 the final expression is
DUE
— = Anne
. B~ _ 1 an _ 90?] (29)
Bt 2 B~ B~
where the dynamic boundary condition P = COnSt has
been used. This nonlinear evolution equation for the
tangential velocity component gives the free boundary
condition for the two boundary integral equation (25)
and (26~.
It is worthwhile to add few more comments on the
relationship between the present decoupled model and
the coupled inviscid one. The normal component eq.
(15) is very similar to (26~. In the case of the coupled
inviscid formulation the role of us is played by the
tangential derivative of the dynamic pressure. A part
from the integration in time the kernels are identical,
so that the numerical methods for the two equations
are expected to be essentially the same. The main
difference between the two formulations is due to the
fact that from (15) it follows only one integral equa-
tion, while in the kinematical decoupled model any
of the two equations (25) and (26) may be selected.
This feature of the kinematical model may be used to
avoid the solution of the Cauchy type equation which
is more difficult from the numerical point of view.
6 The numerical simulation for
a submerged body
Some numerical results obtained by the formulation
introduced in the previous section are presented for
the unsteady flow about a submerged body. The flow
field is assumed to be irrotational and the nonlinear
free surface configuration is followed in time by the
Lagrangian description. The evolution of the wave
pattern generated from rest by a moving cylinder is
simulated and the steady state configuration is reached
in the case in which no breaking wave occurs.
The numerical procedure adopted makes use of the
dynamical equation (29) end of the kinematical equa-
tion (21) to evalute us on the free surface and the
free surface configuration respectively. The following
notations are used: [FIB and [Q' denote the body
boundary and the free surface respectively. [LOO de-
note the fictitious boundary used to cut the computa-
tional domain at a finite distance from the submerged
body. When treating the flow in a channel, the fi-
nite depth bottom is denoted by [Qb while ~QOO is
given by an inflow part (Bali) and an outflow part
(AGO). As a boundary condition to be used in the
boundary integral equation, the normal inflow veloc-
ity, time dependent in general, is be assigned on [Eli.
The zero normal velocity condition is assigned on the
bottom boundary for both the channel flow or the infi-
nite depth case. At the outflow boundary the normal
velocity component is also given or a suitable radi-
ation condition is applied. On the free surface the
tangential velocity component is evaluated by the dy-
namic equation (29). The reference frame is assumed
connected with the body, which is moving at a costant
depth beneath the indisturbed free surface. The dy-
namic equation is used in the inertial frame connected
to the indisturbed fluid and the proper value of us
is evaluated by changing the reference frame to that
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fixed to the body.
As shown in the previous section any of the two
boundary integral equation either (26) or (25) can
be used to determine the unknown velocity compo-
nent. The most efficient choice, from the computa-
tional point of view, is the use of the second kind EYed-
holm equation, in order to avoid the variational tec-
niques required for an accurate solution of the Cauchy
type equation. To this purpose a mixed approach is
used here by assuming the normal component (26)
to hold on the free surface and the tangential com-
ponent (25) on the other boundaries. The equation
which follows from this mixed formulation does not
present a Cauchy type singularity and can be easily
solved by numerical methods. In the case of multi-
connected regions, some attention must be devoted
to the eingenvalue related to the circulation around
the body which in the present numerical study will
be assumed to be zero.
In the computational procedure standard finite el-
ements tecniques are used to discretize the equation.
In more details, piecewise linear shape functions are
used to describe the geometry while piecewise con-
stant functions are used for the unknown. In order to
assure the solution uniqueness the condition of zero
circulation around the body is imposed. The com-
putational procedure is splitted in two parts. The
boundary unknowns are evaluated first by solving the
boundary integral equation. With the velocity dis-
tribution on the free- surface completely known the
configuration of the domain and the values of us are
updated. The second order accurate Adams Bash-
forth scheme is used to perform this calculation. No
redistribution procedure is used for the free surface
panels which tends to concentrate near the crest of
the waves. Due to the mean motion, at each time
step a new panel is added at the inflow side and one
is canceled at the outflow, where a Sommerfeld radi-
ation condition is used.
By the described numerical tecnique the simula-
tion of the transient wave system due to the motion
of a circular cylinder has been performed. The ob-
tained results are compared with those presented by
Haussling and Coleman [5] and with those of Ligget
and Liu [6] for the potential flow equation. Haussling
and Coleman use a finite difference approach in gen-
eralized curvilinear coordinates and Ligget and Liu a
boundary element approach. Both use an Eulerian
description for the free surface. Although these ap-
proaches are completely different the three solutions
show a good agreement (fig. 13. It could be noted that
the present solution shows clearly the effects of non-
linearity given by the adopted Lagrangian description
of the free surface. In fact, with the present method
a steady state solution couldn't be reached due to the
steeping of the wave crest, which cause the solution
to break before the last instant showed by Ligget and
Liu. In the case, shown in fig. 3, although the Froude
number is higher than in the previous one, a steady
state solution has been attained due to the greater
submergence of the body. Fig. 4 shows the time his-
tory of the forces acting on the cylinder. Although the
free surface configuration appears to reach the steady
state, both lift and drag are oscillating around the
presumed steady value. In fig. 8 a simulation with
a smaller channel depth shows the evolution towards
the breaking of the wave, which is attained by an over-
turnig of the front face of the first crest behind the
body.
7 Concluding remarks
A computational method for unsteady free surface
flows has been derived from a mathematical model
which decouples the kinematical and the dynamical
aspect of the flow field. The method, valid in gen-
eral for rotational flows, has been shown to have the
same computational efficiency of the potential based
approaches. The strict relationship with the coupled
models based on the integral formulation for viscous
and inviscid flows, have been discussed. While the
numerical solution of the Fredholm integral equation
gives very satisfactory results, the numerical tecnique
for an efficient as well as accurate resolution of the
Cauchy type equation is to be completed. This is the
crucial point for the numerical implementation of the
limiting coupled formulation for inviscid flows. The
viscous model, should be treated in similar way by re-
casting the equation in terms of vortex layers as shown
for the inviscid case.
References
t1] Ladyzhenskaja, O.A., (1986~: The Mathematical
Theory of Viscous Incompressible Flows, Gordon
& Breach, New York, NY, USA.
t2] Piva, R. and Morino, L., (19873: "Vector Green'
Function Method for Unsteady Navier Stokes
Equations", Meccanica, Vol. 22, pp. 76-85.
t34 Piva, R., Graziani, G., and Morino, L., (1987~:
Boundary Integral Equation Method for Un-
steady Viscous and Inviscid Plows", Ed.: T.A.
Cruse, Advanced Boundary Element Methods,
Springer Verlag, New York, NY, USA.
[4] Miyata H., Sato T., Baba N. "Difference Solu-
tion of a Viscous Flow with Free Surface Wave
about an Advancing Ship", J. Comput. Phys.,
72 (1987), 393
476

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[5] Haussling, H.J., and Cole-
man, R.M., (1977) :~ Finite-Difference Computa-
tions Using Boundary Fitted Coordinate Sys-
tems for Free-Surface Potential FlowsGenerated
by Submerged Bodies~, Eds.: J.V. Wehausen and
N. Salvesen, The Procedings of the Second Inter-
national Conference on Numerical Ship Hydro-
dynamics.
[6] Liu, P.L-F. and Ligget J.A., (1982~: "Applica-
tions of the Boundary Element Method to Prob-
lem~ of Water Waves", Eds.: P.K. Banerjee and
R.P. Shaw, Developments in Boundary Element
Methods - 2, Applied Science Publishers, Lon-
don.
[7] Casciola, C.M., Lancia, M.R., and Piva, R.,
(1989~: "A General Approach to Unsteady Flows
in Aerodynamics: Classical Results and Perspec-
tives~, ISBEM 89, East Hartford Usa.
[8] Nedelec, J.C., (1977~: " Integral Equations with
Non Integrable Kernels, Integral Equations and
Operator Theory, Vol. 5, pp. 562-572.
[9] Hsiao, G.C., (1986~: " On the Stability of Integral
Equations of the First Kind with Logaritmic Ker-
nels", Archive for Rational Mechanics and Analy-
sis, Vol. 94,2 pp.179-192, Springer Verlag, Berlin.
t = 6.
t = 7.5
· · . ~0a .
t = 9
Fig. 1. Free - surface configurations after the start of
a submerge cylinder. Submergence h = 2.
Fr = .566 D = 1
_ present solution
° Ligget and Liu
· Haussling and Coleman
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0.150
0.100-
C. 050-
, , . . .
I.Io 1.90 9.00 9.10 t.20 t.30
Fig. 2. Steeping of the wave for the same case as in
Fig. 1. t = 10.
o
t= 15
t—30
t = 45
. ~
0
t=60
Fig. 4. Configurations of the computational domain
for the some case as in Fig. 3.
Fig. 3. Free surface configuration vs time
Fr = .95 D = 1
Submergence h = 3
Dt = .25
t= 75
0
t = 80
a
o
t = 85
o
t = 90
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/
1 1
,~\
- - ~
Fig. 7. Pressure distribution on the cylinder. Same
case as in Fig. 5. t = 85
l.tO-
Ill~ 8'll.
_' I! ~ ~~-
t. 000
5 11 15 28 25 30 15 40 10 20 ]' 41 51 11 T'
Fig. 5. Wave elevation. Fig. 6. Time History of the forces on the cylinder.
Fr = .95 Same case as in Fig. 5 (forces adimentionalized by
Submergence h = 3. pu2D)
Dt = .25 t = 85
~ _ _
o
Fig. 8. Wave overturning. D = 1
Fr = .95
Submergence h = 3
Channel depth ~ = 4
t= 14.25
Dt = .25
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