Below are the first 10 and last 10 pages of uncorrected machineread text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapterrepresentative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 221
A Boundary Integral Approach
nntive Variables for Free Surface Flows
C. Casciola (I.N.S.E.A.N., Italy)
R. Piva (Universita di Roma, Italy)
ABSTRACT
The boundary integral formulation, very efficient
for free surface potential flows, has been considered
in the present paper for its possible extension to ro
tational flows either inviscid or viscous. We analyze
first a general formulation for unsteady Navier Stokes
equations in primitive variables, which reduces to a
representation for the Euler equations in the limiting
case of Reynolds infinity.
A first simplified model for rotational flows, ob
tained by decoupling kinematics and dynamics, re
duces the integral equations to a known kinematical
form whose mathematical and numerical properties
have been studied. The dynamics equations to com
plete the model are obtained for the free surface and
the wake. A simple and efficient scheme for the study
of the non linear evolution of the wave system and its
interaction with the body wake is presented. A steady
state version for the calculation of the wave resistance
is else reported. A second model has been proposed
for the simulation of rotational separated regions, by
coupling the integral equations in velocity with an in
tegral equation for the vorticity at the body boundary.
The same procedure may be extended to include the
diffusion of the vorticity in the flowfield. The vortex
shedding from a cylindrical body in unsteady motion
is discussed, as a first application of the model.
INTRODUCTION
One of the most successful approaches for the
analysis of free surface flow problems is given by the
boundary integral equation method. In particular,
if the governing equations are linear, as it is for the
potential flow approximation, this approach reduces
by one the space dimensions of the computational
domain. Moreover, it provides a description of the
boundary conditions (which are usually non linear
and unsteady) more accurate than any other compu
22
tational model. As a matter of fact, the boundary
integral equation method, together with some specific
techniques introduced to linearize and discretize the
kinematic and the dynamic boundary conditions at
the free surface, leads to an extremely efficient com
putational methodology for the evaluation of the wave
resistance and of the overall potential flow field, e.g.
see A.
However more realistic flows always contain re
gions of vorticitY different from zero. which after being
generated at the body wall, usually remains confined
in a narrow region close to the body itself and its
wake, provided the Reynolds number is sufficiently
high. The relevance of the rotational flow may be en
hanced by large separated regions about bluff bodies,
by the interaction of the wake with the free surface
or with other solid bodies (e.g. the propeller), or by
a larger effect of the diffusion for moderate values of
the Reynolds number.
In all these conditions, where either confined or
large vertical regions appear, the inability to intro
duce the velocity potential prevents from using the
very efficient model previously mentioned. In the for
mer case (i.e. confined vertical regions) the overall
picture of the flow field does not change much with
respect to the potential one, so that the classical sin
gular perturbation approach, that is a boundary layer
external solution interaction model, may give suffi
ciently accurate results. In the latter case (i.e. large
vertical regions) any kind of external flow field cor
rection becomes inefficient and the direct field dis
cretization of the Navier Stokes equations (or a sim
plified version of them, e.g. parabolized) seems to be
the only available approach for practical applications
A. Anyhow in both cases we have to deal with more
complicated techniques, which require a larger com
putational effort. In particular, extending the effect
of viscosity to the entire flow field, we may even spoil
from a numerical point of view the wave pattern sim
ulation at the free surface, with respect to the much
OCR for page 221
simpler potential flow model.
It is reasonable at this point to try to answer the
following questions: in which condition is it possible
and convenient the extension of the boundary integral
method to rotational flows? How large is the loss of
efficiency due to the presence of non linear terms in
the equations? In fact, these terms give rise to field
integrals and cancelling one of the main advantages of
the boundary integral formulation for potential flow
(that is the only presence of boundary unknowns).
Purpose of this paper is to give some prelimi
nary answers to these questions by illustrating a gen
eral boundary integral formulation for viscous flows
in primitive variables (velocity components and pres
sure) which reduces to a representation for inviscid
flows in the limiting case of Reynolds infinity. The
theoretical analysis, developed by the authors in pre
vious papers t3, 4, 5] and briefly summarized for the
reader's convenience in Section 2, is successively ap
plied to generate a set of computational models which
are increasingly complicated as long as they become
suitable to deal with flows presenting more relevant
vertical regions.
A first group of models is obtained by decoupling
the kinematics from the dynamics in the integral rep
resentation which holds in the limit of zero diffusion.
By doing so, we recover the purely kinematical inte
gral representation for the velocity vector, known as
Poincare formula, valid also for rotational flows A.
The dynamical part of the equations, still in differ
ential form, gives rise to auxiliary conditions for the
free surface and for the wake. The generation of vor
ticity and its release from the body, in the classical
case of sharp trailing edge, is assured by the enforce
ment of a "Kuttatype" condition, which accounts for
the local viscous phenomena, as explained in details in
Section 3. A further kinematical equation is required
to account for the unknown position of the field dis
continuity given by the free surface or by the wake.
The above assumptions provide a very simple
and efficient model, able to analyze rotational un
steady flows and well equipped to treat the non linear
free surface behaviour. Several computational results
obtained by this method t3] are reported in Section 3.
Also a steady state linearized version of the model
which resembles, in terms of velocity, a classical po
tential flow model t11 largely used for the wave resis
tance calculation, is reported in Section 3, in order to
show the versatility of the present approach and its
capability to reproduce the most interesting positive
features of the existing methods.
A further step in the direction of a complete sim
ulation of the rotational flow, is attempted by includ
ing the diffusion phenomena, neglected in the previ
ous group of models, as a first order effect acting over
the inviscid solution. In fact, without introducing the
boundary layer equations, the viscous effects are re
covered by the boundary integral formulation for the
vorticity transport equation, given in Section 2. As
suming, as in the first order boundary layer theory,
that the pressure does not change normally to the
body wall, we combine its value from the inviscid son
lution is combined in order to obtain an integral equa
tion in the wall vorticity. This procedure, described
in Section 4, allows to detect, within the limits of the
approximation, the separation point along the wall,
hence the position and the intensity of the issuing
vortex layers for the simulation of the rotational wake
region.
Finally the boundary integral equations and the
computational procedure for a complete model are
briefly outlined in Section 5. By a complete model
we mean a model in which dynamics and kinematics
are fully coupled and the same viscous fundamental
solutions together with the related integral represen
tations are considered in the entire flow field. More
specifically the concept of interacting external and in
ternal solutions, which is typical of perturbation tech
nioues. is not adopted here. The close relationship
between this model and the previous ones may be
of great help to overcome the numerical difficulties
mainly due to the kernel of the integral equations,
which becomes highly singular for increasing values
of the Reynolds numbers. The numerical results con
cerning this model are still in progress and are going
to be presented in a further paper.
A GENERAL FORMULATION OF FLOW
PROBLEMS IN TERMS OF BOUNDARY
INTEGRAL EQUATIONS
The integral formulations of the Navier Stokes
equations have been mainly used for studying the
mathematical aspects of viscous flows. A detailed de
scription of the method is given in the book of La
dyzhenskaya t8], where the integral representation for
the steady state problem is also presented. More re
cently integral formulations have received new interest
for the numerical simulation of viscous flows. In par
ticular the authors investigated the flow about stream
lined bodies when no massive separation occurs A.
The analysis of the boundary integral equation
given in full details in a previous paper t10], is briefly
summarized here for completeness. Besides, an inter
gral representation for the vorticity is proposed for
its relevance to the solution procedure in the case of
rotational separated flows.
a. The velocity representation for viscous flows
We consider the case of the undisturbed fluid in
uniform translation with constant velocity UOO with
respect to the body frame of reference. The absolute
222
OCR for page 221
velocity is expressed as u = uOO + u, where u is a per
turbation velocity. We also introduce a perturbation
stagnation pressure P
+ ~ pU2pOO~ pU2 with
so that at infinity P = 0. Introducing these quantities
the governing equations become
V u=0 (1)
,uV2u  VP _ p flu = X (2)
where the righthand side term X contains the nonlin F
ear terms and the gravity force
X = pu x (+ pg
Notice that with the above expression of the v2E
equations, the non linear terms, which give rise to
a field integral, have to be accounted for only in the
regions where the vorticity is not negligible.
The integral representation of the velocity as a
solution of the system (1, 2) is given by t4]
Uk(X*,t*) = ~ ~ (Ujt; )tjU; )) dSdt
To Jan PVaUjUj dSdt (3y
+ ~ ~ Xjujk)dVdt~ pusujk)dVdt~
where the stress vector t is modified to include the
dynamic pressure. As shown by (3) the field veloc
ity is given in terms of surface as well as volume in
tegrals. The first surface integral gives the effect of
the boundary values of the velocity and the modified
traction. The second surface integral gives the effect
of the momentum flux due to the boundary motion.
In free surface flows, this is a non linear contribution,
because the boundary normal velocity component to
is strictly dependent on the fluid velocity field. This
source of non linearity is localized on the free surface
as in the case of potential flow. The first volume in
tegral, related to the term X, accounts for the body lotion become t5)
force and for the vorticity effects in the fluid. This
source of non linearity is within the field equations,
and in particular is connected to the rotational flow
region, which may be more or less confined, depending
on the flow field. Finally the second volume integral
gives the effect of the initial conditions. The funda
mental solutions uji) and tjk) are given by t4]
u(k) = [jkF `' ~(4)
p(k) = _ 49G bit*  t)
tj ) = p( )ni + ~ ( ~ + ~) ni
where for the twodimensional case
G =
lnr
2,r
1 '2/4 (t t)
47rp 1 (4~(t, t) )
and E1 is the exponential integral.
b. The inviscid flow as limiting case
(6)
We consider now the limiting case of the previ
ous theoretical formulation as the Reynolds number
goes to infinity, i.e. as ~ goes to zero. In fact the pa
rameter L,t could be more appropriate as we can see
from the expressions of the two functions F and E ap
pearing in the fundamental solutions. We can easily
see that their distributional limit is given by
F =  H(t*t)~(x*x)
E =  H(t*t) G(x*x)
and the equation V2E = F reduces to
V G = b(x*x)
(7)
Therefore the expressions of the fundamental so
u~k) = _(t*t) (0Cii ~z`02` Ba jade)
t(k) = LOG bit*t)nj (9)
Combining them with (3) we obtain the repre
sentation valid for rotational inviscid flows, which in
vector notation reads
223
OCR for page 221
( ' ) Man ( )
+ lo Jan p [(n V)VGnV2G] dSdt
Jordan va [(u V)VGuV2G] dSdt (10)
+ lo In (V* x V*G x X) dVdt
In(V* XV*GxuO)dv
where the vector identity
(b ~ V)VGbV2G = V* x V*G x b (11)
has been used in the expressions of the volume inte
grals, and uO is the initial velocity.
In order to keep the vector notation, when study
ing a two dimensional flow, we consider a cylindrical
body in a threedimensional space. In particular we
assume a local set of orthonormal coordinates defined
at each point of the boundary by the unit tangent
vector ~ (anticlockwise), the unit normal vector n
(external to the fluid domain) in the cross section and
k = ~ x n.
The integral representation (10) corresponds to
the differential model for inviscid flows given by the
Euler equations.
c. Analysis of the boundary integral equation
The integral representation (3) for viscous flow
gives, for x* going to the boundary, an integral equa
tion which is a constraint between the values assumed
by the velocity and the traction at the boundary. In
the limit a factor c (= 2 for smooth boundaries) ap
pears at the lefthand side to account for the jump
properties of the double layer kernel t(k). If either the
velocity or the traction is assigned as boundary con
dition, we obtain an integral equation of first kind for
the unknown traction or an integral equation of sec
ond kind for the unknown velocity respectively. Usu
ally for free surface flows about submerged bodies we
have a mixedtype boundary condition, that is the
traction is assigned at the free surface and the velocity
at the body wall. This exactly resembles the poten
tial flow formulation that for the same physical case
requires Neumann and Dirichlet boundary conditions
for the body and the free surface, respectively.
At increasing values of the Reynolds number the
kernel of the integral equation tends to become sin
gular, in any of the described cases, as shown by the
expressions (4) and (5) of the fundamental solution
u(k) and t(k). Actually, the functions F and E appear
ing in these expressions, become sharper and sharper
as the kinematical viscosity goes to zero. The main
difficulty in solving directly the integral equations for
large Reynolds number flows is essentially related to
the crucial behaviour of these functions rather than
to the presence of the volume integrals.
A deeper insight on the properties of these equa
tions for large Reynolds numbers, is provided by the
analysis of the limiting case of zero diffusion. It ap
pears from (8) that the kernel u(k) has a hypersingu
lar behaviour when the collocation point approaches
the boundary. It follows an interesting comparison
with the viscous case, showing the computational dif
ficulties to be expected asymptotically for increasing
values of the Reynolds number.
In particular u(k) is composed of two terms which
are both singular. The first one is the well known hy
persingular term which appears in the double layer
representation of the velocity potential for the Neu
mann problem t11~. The second one is a Dirac delta
function on the boundary itself. By combining the two
terms together with a few vectorial identities and the
Stokes theorem for a closed surface t11,12], the second
surface integral may be reexpressed in the form
(VP x n) x VGdS (~12)
an
which corresponds to a vortex layer of density By =
(VP x n) with P = lot* Pp aft. Moreover the kernel does
not show now the same singularity as in the original
form.
By similar manipulation through known vecto
rial identities the third surface integral may be reset
in the form
V Xi ~ 2va(V X u)GdSdt
* 0 an (13)
of* ~
v* x I J VatU X VG)dSdt
0 an
which is not presenting particularly attractive features
with respect to the original form. The volume integral
containing the non linear convective term, through an
integration by parts may be written as
rt* ~
/ / V* x V*G x XdVdt =
so an
(14)
V* x ~ G x (V x X)dV  V* x ~ Gn x XdS
with X = Jo* Xdt.
By the same procedure, the volume integral con
taining the initial velocity term gives
~ (V* x V*G x us) dV =
V* x ~ G`~odV + V* x ~ Gn x UodS
n an
(15)
Notice that in this form the initial term integral
is not extended to the entire flow field, but it gives a
224
OCR for page 221
contribution only in the rotational region and along
the boundary.
It is worth to notice that the representation (10)
of the velocity vector, for x* approaching the bound
ary, gives only one integral equation, when considering
its normal projection. This fact is strictly related to
the lower order of the corresponding differential equa
tion for the inviscid case. Hence, only one integral
constraint is expected among the two scalar quanti
ties (u n) and P. as the collocation point approaches
the boundary. Namely, we obtain a Fredholm integral
equation of second kind if P is assigned and a first
kind Cauchythype integral equation if the boundary
condition prescribes the normal velocity component.
Also in this case, for the problem of the free sur
face flow about a submerged body we have a mixed
type boundary condition and the resulting system of
discretized equations has to be carefully analyzed in
order to determine the mathematical properties of the
corresponding integral operator A.
d. The integral representation for vorticity
In some of the models we are going to discuss in
the next sections, we consider also the integral rep
resentation for vorticity in regions close to the body.
Let us write the vorticity transport equation in the
form
\7xx (16)
where the nonlinear terms are taken as a source at the
righthand side of the equation. The integral repro
sensation for the solution of (14) is given, in the case
of a fixed fluid domain, by
((x*,t*)  Jo /onL,(~8nF3~)dSdt
/o /n (V x x) FdVdt/n (oFdV
(17)
where F is the fundamental solution of the diffusion
equation given by (6) for the twodimensional case.
We notice that this representation is not independent
from the representation (3) for velocity. Actually it
can be obtained by performing the curl of (3) and by
accounting also for the expression of the stress vector
t in terms of its normal and tangential components
t15]
t = pn+,u~xn (18)
which is valid for a traslating rigid body.
For the collocation point x* approaching the
boundary we obtain an integral equation which is a
constraint between the values of ~ and ~ at the bound
ary.
No simple boundary conditions are available for
~ or ~ and the latter is usually approximated by using
the intensity of the vortex layer at the body boundary
and by assuming a reasonable model for the diffusion
of vorticity t13, 14~. For ~ assigned as known bound
ary condition, a second kind integral equation for ~ is
obtained. The kernel &oF still presents some computa
tional difficulties for large Reynolds numbers.
THE DECOUPLING OF KINEMATICS AND DY
NAMICS: A MODEL FOR ROTATIONAL INVIS
CID FLOWS
Let us consider the representation of velocity
(10) combined with the new expressions (12) and (13)
of the surface integrals, for the analysis of inviscid flow
fields. The representation contains in its terms both
the kinematical and the dynamical aspect of the phys
ical model, as clearly shown by the two unknowns,
namely the normal velocity and the stagnation pres
sure.
A coupled integral formulation like (10) allows
for a straightforward enforcement of the boundary
condition in terms of the boundary velocity and pres
sure. In addition their discretization does not imply
any further difficulty as in the numerical models based
on differential equations. Therefore a coupled integral
formulation would be ideal for the simulation of free
surface flow fields, where a particular attention has
to be paid for a simple and accurate application of
the boundary conditions. However, the presence of
some computational difficulties (mainly related to the
calculation of the surface integral (13) and the vol
ume integral (14~) suggested to still follow the classical
path of decoupling the kinematics from the dynamics,
as successfully experienced by all the potential flow
models.
a. The Poincare formula for kinematics
A purely kinematical integral representation is
obtained by eliminating the dynamical variables
through a back substitution of the Euler equation and
of the vorticity transport equation (in their differential
form) into the integral representation (10) for inviscid
flows.
For the sake of simplicity, we apply first the
above procedure to a fluid domain fixed in time, so
that the integrals which account for the free surface
motion are dropped out. We will see later that their
inclusion, is not going to modify the final result, al
though it complicates significantly the overall pro
cedure. After introducing the new integral expres
sions (12), (14), (15) and combining with the vorticity
transport equation for inviscid flow integrated in time
225
OCR for page 221
5^S^o = If X X (19)
2 ur + ion UT[,n.dS = (24)
the integral representation (10) becomes
ut *,t* = VP x n x VGdS
) fan ( )
+J&.n (u n) VGdSV* x ~ G`dV (20)
+V* x / An x (usX) dS
time
Combining now the Euler equation integrated in
(u  us) ~ X = VP (21)
we obtain a kinematical representation for the velocity
vector
u(x*,t*)  [ (u x n) x VGdS+
+ },.,ntu n)VGdSV* x ~ G`dV
(22)
where on the righthand side both u and I, as function
of time, are given for tt*. The velocity representa
tion (22) is the well known Poincare formula usually
written as [61
U(:l:*) = V* t/n(V · u)GdV/ (u n)GdSlJ
V* X t/n(V X u)GdV + / (u x n)GdS)
(23)
which is a velocity integral representation satisfying
only the kinematical equations
V x u = s. and V u = Q
In the present case Q is identically equal to zero.
Let use underline the fact that the splitting be
tween kinematics and dynamics, inherent to the clas
sical velocity potential formulation, is here recovered
by recombining the Euler and the vorticity transport
equations with the original coupled formulation. In
order to verify the equivalence with the Poincare for
mula also in the case of a moving boundary, as is the
case for the free surface, it is more convenient to op
erate in a reverse way, that is to differentiate in time
equation (22), accounting for the variation in time of
part of the boundary an. A brief note on these cal
culation is reported in Appendix A.
Finally we deduce the boundary integral equa
tions which follow from (22) by performing the tan
gential and the normal projections for the collocation
point 2:* approaching the boundary (assumed smooth)
226
; unGdS+I~T
2 an + [n un en* dS (~25~)
+ 2, * ,/; urGdS + Itn
where I`' and I`n give the contribution of the volume
integrals in the two projections, respectively.
Equation (24) is a second kind Fredholm equa
tion for the unknown or or a first kind integral equa
tion with a Cauchy type integral (the kernel a8G is
singulars for the unknown an. The opposite is valid
for equation (25~. The two integral equations are com
pletely equivalent in the sense that if you solve the first
one, the solution will satisfy also the second one. The
choice may depend on the assigned boundary condi
tion and on the preference about the numerical tech
nique to be used.
b. The dynamics of the free surface
For a solid wall the value of un is assigned and
u, is the unknown. Instead, for a free boundary an
is the unknown, and the dynamical part of the model
should provide the boundary value or. The procedure
parallels exactly the one used for potential flows where
~ is the unknown and the values of the potential ~ are
evaluated by means of the Bernoulli equation. In the
case of rotational flows, the Euler equations must be
used to relate the tangential velocity component to the
pressure distribution assigned as boundary condition.
Notice that the coupled representation (10) contains
in it the pressure and no additional dynamic condition
would be required.
We write the Euler equation for a point of the
free surface, labeled by the Lagrangian variable I.
The tangent projection of the Euler equation on the
free surface at point ~ gives
Du ,__! ~P_93'7 (~26)
where ~ has the usual meaning of free surface elevation
and If is equal to zero for the boundary condition of
assigned constant pressure. The unit tangent vector
at point ~ changes in time for the boundary motion.
Therefore
Du D(u ~ Dr
_ . ~ = u
Dt Dt Dt
and, for r. DD' identically equal to zero, we finally
obtain
OCR for page 221
DOT DT Brl
_ = unn  9
The non linear evolution equation (27) for the
tangential velocity component gives at each time the
required boundary value Or for each of the two equa
tions (24) and (25~. We finally need to complete the
formulation by means of a Lagrangian description in
time of the fluid interface. Denoting by Xf((,t) the
position of the geometrical point, labeled by I, of the
free surface the new interface geometry is determined
by solving the initial value problem
``' = Of X'(6,O) = Xo(~) ~ ~ D' (28)
where DO is the set of values of the Lagrangian pa
rameter ~ and U' is assumed to coincide with the local
fluid velocity u.
c. The dynamics of the wake
The kinematical representation (22) expresses
the field velocity as function of its boundary value
and of the field vorticity both at the present time t*.
Hence, we have to determine the distribution of vor
ticity by adding a dynamic equation. For instance, the
vorticity transport equation for incompressible invis
cid flows in 2D provides the very simple result of con
stant vorticity along the motion. We consider in this
. . . . . . . . . . .
section the physical case of very large Reynolds num
ber flows about streamlined bodies with sharp trailing
edge, which do not experience any boundary layer sep
aration [16~. We may simulate these conditions by the
zero diffusion model with a vertical wake downstream
of the body. These wakes (or free vortex sheets) are
given by surfaces of discontinuity characterized by the
fact that both pressure and normal fluid velocity are
continuous across them, while the tangential compo
nents of velocity may admit a jump, that is a concen
trated vorticity
Ok = [u] x n _ [trek (29)
where ~ ~ is the symbol for the jump across the discon
tinuity surface. The volume integral It appearing in
(22), if the field vorticity is only concentrated on the
wake, may be expressed as
It =V* x / [Ur]Gds (30)
0.
where on is the wake surface.
Similarly to what has been done for the free sur
face, we introduce a dynamic equation to study the
evolution of furl, which is given by t17~.
DW (Jluri) = 0 (31)
where w is the velocity of a point ~ of the wake, DO is
r271 the material derivative along the wake motion and
J = ~ is the Jacobian of the trasformation
xu, = xw(6,t) which gives at each time the position of
the point ~ belonging to the wake. Equation (31) is
equivalent to state that
JO Judd = const. (32)
along the motion, which has the physical meaning of
conservation of concentrated vorticity for a portion
(~! < ~ < 62) of the wake.
The initial value ATE of the vortex layer intensity
at the trailing edge is taken to be the limit
ATE = km I.UT(X+) + Ur(X_~l (33)
I+TO
I_ _TE
where :z:+ are points on the upper and lower side of the
body and uric+) the corresponding tangential compo
nents of the velocity. In a sudden start the value of
ATE decreases in time asymptotically to zero when the
body reaches the steady state. The equation (31) plus
the initial condition (33) is called a Kuttatype con
dition, because at steady state t satisfies the classical
Kutta condition of zero vorticity at the trailing edge.
To describe the evolution of the geometrical con
figuration of the wake we adopt a Lagrangian model
completely analogous to (28)
where w = (u+ +u_~/2.
Ago = w (28)
d. Comparison with potential flow models
We describe now the solution procedure for the
model consisting of the integral equations (24) or (25)
plus the dynamics contribution for the free surface
(eq. 27) and for the wake (eqs. (31) or (32~) to which
the evolution equation (28) has to be added. A com
plete theoretical analysis for the solution of the two
equations (24) and (25) has been recently performed
[10] for the case of flow past bodies, that is for as
signed normal velocity component on the boundary.
The same would be for assigned tangential component
all over the boundary. The contemporary presence of
free surface and body complicates the analysis, be
cause we have a mixedtype boundary condition, that
is un assigned at the body wall and a' at the free
surface. Let us recall first the main findings of the
previous analysis for the case of uniform boundary
condition.
We consider here only the second kind Fredholm
integral equation (i.e. the tangential component (24)
for the unknown ur or the normal component (25)
227
OCR for page 221
for the unknown an). This formulation exactly corre
sponds to a Neumann internal problem for potential
flow with a simple layer representation. A complete
equivalence with regard to existence and uniqueness
of the solution and compatibility conditions, holds.
Namely, let us consider for instance equation (24),
the compatibility condition for the righthand side
An*  '9r. ./an UnGdS + Itr] dS* = 0 (34)
is identically satisfied, giving the existence of the solu
tion, for any assigned normal velocity at the boundary
and for any distribution of vorticity in the domain
Q (notice that It gives the velocity induced by the
field vorticity whose circulation is identically zero be
ing the vorticity only external to an). Moreover, it is
known from the potential theory that the solution is
not unique and it may be expressed in the form
Ur = Ur + OKUT
(35)
where uP is a particular solution and up is the eigen
solution, that satisfies the homogeneous equation for
the Neumann problem. The solution for the homoge
neous problem is given by the simple layer density for
the related Robin potential, for which the property
J;n us dS ~ O holds.
By imposing the further condition of conserva
tion of total vorticity, we find the circulation ~ around
the body, as far as we know by (32) the vorticity is
sued at each time on the wake. Finally from the value
of I' we can find the constant c' appearing in (35) and
determine uniquely the solution or.
As we said before, we do not need to consider
here the singular integral equations of first kind which
may require more sophisticated numerical techniques
t12~. In fact, a mixed approach is used, that is we
assume the normal component (25) to hold on the
free surface and the tangential component (24) on
the other boundaries. Altough a deeper theoretical
analysis would be required to study the mathematical
properties of the corresponding integral operator, we
experience a nice behaviour of the discretized equa
tion from a numerical point of view.
The above illustrated solution procedure is sim
ple enough to be comparable, for both theoretical and
numerical aspects, to the classical one for potential
flow. By the present procedure we may not only re
produce the results obtained by potential flow mod
els, but also obtain an immediate extension to rota
tional inviscid flows, with vorticity confined in wakes
or blubs. Moreover with regard to the free surface, the
present model provides a very efficient unsteady tech
nique to study the non linear evolution of the waves.
Few sample numerical results are reported. The
transient wave system due to the motion of a sub
merged circular cylinder, previously discussed in t3] is
here reported as a test case for the accuracy of the
model. The numerical results shown in Fig. 1, com
pare favourably with other classical results 18,19.
,' ~ ~ ~ ~ ~ . . .
....
Figure 1: Free surface configurations after a sudden
start of a submerged cylinder. Submergence h = 2,
Fr = .566, D = 1, t = 4.5,6,7.5,9.
solution, ~ Ref. 18, ~ Ref. 19
 present
A more interesting case is given by the unsteady
motion of a slightly submerged lifting airfoil. After a
sudden start of the airfoil the vortex layer shedding
from the sharp trailing edge, interacts with the free
surface giving rise to a wave pattern which feels the
influence of the wake vorticity. The airfoil at three
different values of the angle of attack is shown in Fig.
2.
228
OCR for page 221
a" = ~ ~(38)
Figure 2: Free surface and wake configurations af
ter a sudden start of a lifting airfoil. Submergence
h.75 * chord, Fr = 1 (with respect to the chord).
Angle of attack 0°, 10°, 20°. t = 3
e. A steady linearized free surface condition
To complete the comparison with potential flow
models, we briefly discuss now the application to
steady linearized boundary conditions at the free sur
face, frequently used for an efficient calculation of the
wave resistance. Equation (27) which describes the
dynamics of the free surface, for steady state and a
zeroth order linearization reduces to
UOO ~ r =  9,90 (36)
considering that the unit tangent vector r in this ap
proximation is constantly aligned with the undisturbed
velocity UOO The fluid interface motion given by the
Lagrangian description (28) is here conveniently ex
pressed through its linerized Eulerian form
UOO Bt7 = u" (37)
Combining (36) and (37) implies the well known
NeumannKelvin condition
usually written in terms of the velocity potential. In
troducing (38) into the right hand side integral of the
equation (24) and integrating by parts we obtain
Uoo ~ / BurGdS = _Uoo ~ at/ Ur3GdS
9 Br* an' B~ 9 Br* an' B~
(39)
with ~Q = aQb + aQf where b and f stay for body
and free surface respectively. Combining with (39),
the integral equation (24) becomes
2 r + Jan u, fin* dS + 9 Or. fan us ~, dS =
~ DIG dS + I (40)
This integral equation gives, for assigned normal ve
locity at the body boundary oaf, the tangential ve
locity component on the entire boundary ~Q. Once
u, is known, we easily obtain the free surface eleva
tion ?7 by integrating equation (36~. The tangential
derivative of the free surface integral in (40) is dis
cretized by the upwind finite difference scheme used
in the potential flow model proposed by Dowson A.
The steady linearized version of the present
model exactly reproduces the numerical results pb
tained by the above mentioned potential model. For
instance, the wave pattern generated by a submerged
lifting airfoil is shown in Fig. 3, for a zero contribu
tion of the term I`T. However, the present tecnique
may include some rotational effects relevant to steady
state conditions.
ODD
~D
~4
Figure 3: Wave elevation in a steady problem for
a NACA0012 airfoil. Submergence h = 1 * chord,
Fr = .5 (with respect to chord). Angle of attack 10°.
Trailing edge position x = 4.5, y =
229
OCR for page 221
A VISCOUSINVISCID INTERACTION MODEL
FOR SEPARATED FLOWS
The model described in the previous section is
appropriate for the simulation of inviscid rotational
flows with either a field vorticity whose initial posi
tion and intensity are assigned or a vortex sheet is
suing from a sharp trailing edge whose intensity is
determined by a Kuttatype conditions. Therefore,
the vorticity generated at the body wall may be ac
counted for, always in the limit of Reynolds number
going to infinity, but it is restricted to flows where
no separation occurs. Actually, the diffusion phenom
ena were neglected everywhere in that model if we
exclude, in a sense, the unsteady generation of the
wake by the Kuttatype condition which provides the
limiting behaviour of a viscous fluid. A further step
in the direction of a complete simulation of rotational
flows (i.e. with separated regions) is attempted by
including an internal viscous solution which is going
to modify the external inviscid solution. In fact, we
do not introduce the boundary layer approximation,
that would be inappropriate to the present aim, but
we try to recover the viscosity effects in a more con
sisting way through a boundary integral formulation
for the full vorticity transport equation, which is valid
in principle in the entire flow field. However, as in the
first order boundary layer theory, we introduce sev
eral drastic assumptions to simplify as much as pos
sible the numerical solution of the boundary integral
equation resulting from the representation (17) for x*
appoaching the boundary (assumed smooth)
LIP ~ {u2N
= __ _
B ~B~ 2
MU' (43)
It follows the boundary condition (42) for the nor
mal derivative of vorticity. The two volume integrals
of the non linear and the initial terms, appearing in
eq. (41), are confined to very narrow layers (in the
limit of Re ~ oo that we consider here) where the
vorticity is different from zero. Consistingly with this
approximation, the fundamental solution F is taken
constant across these layers. Hence, we may compute
the integral across the layer of the non linear term
V X X = V X Huron2lnS~) ~
~ (V x Den = k / [fir (ur`)
~ funny] dn
In particular, for a layer close to the body boundary,
whose thickness is i, taking into account that us"0
at the body, ~ 0 at the outer edge of the layer and
that within the layer ~ =~ we have:
~ (V x Den = _ ~.721 k (44)
Instead, for a wake of thickness 264 with a jump
furl ~ and a tangential velocity defined as wr
u, +u,
2
2~(2*~t*) = lo dnV (fun F37~) dSdt  (V x Dan =kin 2'1 =k~9r(WrY) (45)
I* (41) By the same reasoning, we obtain respectively
+ lo In (V x X) FdVdt   ~oFdV
This is a Fredholm integral equation of the second
kind if ~ is assigned as boundary condition. From
the differential Navier Stokes equation on the body
boundary we obtain the general local identity which
gives in the tw~dimensional case, i.e. for I= ok,
Vp x n =p(V x <3 x n = wok (42)
In the framework of a first order theory we as
sume that the pressure doesn't change normally to the
body wall within the viscous layer close to the body
itself, and its value is given by the inviscid rotational
model of Section 3. As a matter of fact, from that
model we find us, and the pressure follows from the
Euler equation at the wall
/ ~Odn = urology (46)
o
/_`S^Odn = ~0 (47)
By combining the approximate values of the vol
ume integrals (44), (45), (46), (47), with the integral
equation (41), we have a relatively simple model to
determine the value of the wall vorticity.
As a further crucial feature of the model, we
place the separation point at the wall position where
S. is changing its sign. A vortex layer is issuing from
this separation point and its intensity is determined
by a Kuttatype condition completely analogous to
the one introduced in Section 3 for the wake at the
sharp trailing edge of a streamlined body t20;.
230
OCR for page 221
(
\
/
Figure 4: Wake evolution for flow past a cylin
der. Separation point at fixed positions ax =+ 108°.
t= 1,2,3,4
The numerical procedure alternates the solution of
equations (24,25) for the external inviscid flow, with
the solution of eq. (41) for the internal viscous layer.
They are connected through the condition (42) which
in a first order approximation relates the external
pressure gradients with the generation of vorticity at
the wall t21~. The simplifying hypotesis for the cal
culation of the volume integrals may be released to
obtain better approximations.
The model has been applied to a cylindrical body.
We didn't consider here the presence of the free sur
face to focuse our attention on the generation scheme
for the separated regions. In fact the model of gen
eration still requires a deeper understanding and it
seams resonable to select a test case for which a large
experience is available. Several computational mod
els, using a Kuttatype condition for the calculation
of separated flows in the framework of vortex methods
have been recently presented 23,24,25.
First we discuss some numerical results with two
fixed separation points in symmetric position on the
cylinder boundary (c' =+ 108° from the front stagna
tion point). A symmetric rear separated region, like
the one shown in Fig. 4, is obtained if no pertura
bation is introduced in the solution procedure. By
retarding of one time step the lower side of the cylin
der with respect to the other, we introduce a large
oscillation in the two vortex layers. By advancing in
time, they assume a configuration (see the sequence
in Fig. 5) which resembles the initial displacement
of the vortices in the classical Karman street. For a
more stable behaviour of the vortex layer we adopted
a desingularization technique t22] in order to elimi
nate the singularity of the kernel for x ~ x*.
231
OCR for page 221
,~/,
l.''>
/(
Figure 5: Wake evolution for flow past a cylinder with
an initial perturbation. Separation points at fixed po
sitions c' =+ 108°. t = 3, 5, 7, 8
Finally the complete procedure including the in
teraction with the internal viscous solution, has been
applied to the case of Re = 104. At each time step
the Kuttatype condition is applied in a new position
corresponding to the zero value of vorticity which is
determined from the solution of the integral equation
(41~. The sequence in Fig. 6 shows the motion of the
separation points towards the rear part of the cylin
der, starting from the initial position of 90°, which
232
OCR for page 221
\


\

\
~7
corresponds to zero tangential derivative of pressure
(i.e. zero vorticity for no volume integrals in the first
inviscid solution). The wiggles appearing in the wake
configurations in the present case, ace given to the
motion of the separation point (i. e. the source of the
vortex shedding) and not to a vortex layer instability.
The convergence to a steady state solution is uncer
/
me/
)
'¢,
/
Figure 6: Wake evolution for flow past a cylinder.
Separation point at variable positions determined by
the solution of the vorticity equation. t = 1, 2, 3
fain in the sense that a steady state is almost reached,
but is not mantained, as if it were an unstable solu
tion. We don't have at the moment sufficient data to
understand if this is just a numerical instability or it
simulates the ineherent physical instability present at
those values of the Reynolds number.
233
OCR for page 221
CONCLUDING REMARKS AND PERSPECTIVES
In the present paper we tried to answer several
questions about the possibility and the convenience
to extend boundary integral methods (usually very
efficient for potential flows) to rotational free surface
flows, either Inviscid or viscous.
Starting from an integral formulation in primi
tive variables for unsteady viscous flows, we deduced a
set of simplified models strictly connected, the one to
the other, through their relevant mathematical struc
ture. Actually the basic integral equations are very
similar, so the experience may be transferred from
simpler to increasingly complicated models.
The analysis of the limiting case for viscosity go
ing to zero, has been of great help to understand the
behaviour of the integral equation for tendentially sin
gular kernels. The decoupling of kinematics and dy
namics has been another crucial feature to reduce the
integral representation to a form (the Poincare for
mula) easy to be treated from a computational point
of view. A further point to be stressed is the coupling
of the original equations in primitive varaibles with
the integral equation for vorticity, which leads to a
more convenient approach for viscous flow solutions.
The theoretical analysis suggested a first model
for the study of free surface flows with regions of as
signed vorticity or vortex layers generated by the body
motion. The resulting numerical procedure, efficient
as the potential one, allows for an easy evaluation of
several rotational effects.
A second more refined model, suitable for the
investigation of separated flows with large rotational
regions, was provided by the interaction with a first
order viscous solution. The model implies a numeri
cal procedure, which still requires a further analysis.
However the numerical results are very promising in
spite of the simplicity of the model.
The latter may be considered as a first step to
wards the fully coupled and fully viscous model which
is the straightforward approach to study the genera
tion of vorticity and the creation, through separation,
of large vertical regions. The integral representation
(3) for the velocity vector and the scalar version of the
(41) for the vorticity (2D case) lead to boundary in
tegral equations in the same unknowns if we account
for the relations (18) and (42) valid at the body wall.
A mixed procedure which solves alternatively the nor
mal component of the first integral equation for the
dynamic pressure and the second integral equation for
the vorticity, is now in progress.
REFERENCES
1. Dowson C.W., PA practical computing method
for solving ship wave problem", 2nd Int. Confer
ence Numerical Ship Hydrodynamics, Berkeley,
1977.
2. Miyata H., Sato T., and Baba N., "Difference
Solution of a Viscous Flow with Free Surface
Wave about an Advancing Ship", J. Comput.
Phys., no. 72, 1987, p. 393.
3. Casciola C.M., and Piva R., PA Boundary In
tegral Formulation for Free Surface Viscous and
Inviscid Flows about Submerged Bodies",
Proceedings of the 5th International Conference
on Numerical Ship Hydrodynamics, Hiroshima
(Japan), Sept. 2529,1989.
4. Piva R., and Morino L., "Vector Green's Func
tion method for Unsteady Navier Stokes Equa
tions", Meccanica, vol. 22, 1987, pp. 7~85.
5. Piva R., Graziani G., and Morino L., UBoun
dary Integral Equation Method for Unsteady
Viscous and Inviscid Flows", Advanced Bound
ary Element Methods, Cruse T.A. (ed.), Spring
er Verlag, New York, USA, 1987.
6. Brard R., Vortex theories for bodies moving
in water", Proceedings of 9th Symp. on Naval
Hydrodynamics, Brard R., and Castera A.
(eds.), U.S. Gov. Printing Office, 1972, pp. 1187
1284.
7. Piva R., UThe Boundary Integral Equation Me
thod for Viscous and Inviscid Flows", Proce
edings ISCFD, Oshima K. fed.), Nagoya, Japan,
1989.
8 . L adyzhenskaj a, O. . A ., The Mathemat ic al Theory
of Viscous Incompressible Flows, Gordon &
Breach, New York, 1963.
9. Casciola C.M., Lancia M.R., and Piva R., "A
General Approach to Unsteady Flows in Aero
dynamics: Classical Results and Perspectives",
ISBEM 89, East Hartford, USA, 1989, Springer
Verlag, Berlin.
10. Bassanini P., Casciola C.M., Lancia M.R., and
Piva R., "A boundary integral formulation for
the kinetic field in aerodynamics. Part I: Math
ematical analysis. Part II: Applications to Un
steady 2D Flows", submitted to European J.
Mech., B/Fluids, 1990.
11. Nedelec J.C., Approximation des equation
integrates en mecanique et en physique", Lec
ture Notes, Centre de Mathematiques Applique
es, Ecole Polytechnique, Palaiseaux, France,
1977.
234
OCR for page 221
12. Hsiao G.C., "On boundary integral equations of
the first kind", ChinaUS Seminar on Boundary
Integral Equations and Boundary Element Met
hods in Physics and Engineering. Jan. 1988.
Xi'an, People's Republic of China.
13. Schmall R.A., and Kinney, R.B., Numerical
Study of Unsteady Viscous Flow Past a Lifting
Plate", AIAA Journal, Vol. 12, 1974, pp. 156
1673.
14. Wu J.C., "Numerical Boundary Conditions for
Viscous Flow Problems", AIAA Journal, Vol. 14,
1976, pp. 10401049.
15. Berker R., Integration des equations du Move
ment d'un Fluide Visquex Incompressible",
Encyclopedia of Physics, Flugge S. (ed.),
Vol. VIII/2, 1963.
16. Morino L., ~Helmoltz Decomposition Revisited:
Vorticity Generation and Trailing Edge Condi
tion", Computational Mechanics I.
17. Friedrichs K.O., Special Topics in Fluid Dyn
amics, Gordon & Breach, London, 1966.
18. Liu, P.LF., and Ligget J.A., Applications of
the Boundary Element Method to Problems of
Water Waves", Developments in Boundary
Element Methods  2, Banerjee P.K., and Shaw
R.P., Applied Science publishers, London, 1982.
19. Haussling, H.J., and Coleman R.M., ~Finite
Difference Computations Using Boundary Fit
ted Coordinate Systems for Free Surface Po
tential Flows Generated by Submerged Bodies",
Proceedings of the 2nd Inter. Conference on
Numerical Shin Hydrodynamics Wehausen J.V..
and Salvesen N. (eds.), 1977.
20. Sears, W.R., "Unsteady Motion to Airfoils anbd
Boundary Layer Separations, AIAA Journal,
Vol. 14, No. 2.
21. Lighthill, M.J., "Introduction. Boundary Layer
Theory, Laminar Boundary Layers, L. Rosen
head (ed.), Oxford at the Clarendon Press, 1963,
pp. 4~59.
22. Krasny, R., Computation of vortex sheet roll
up in the Trefftz plane", Journal of Fluid
Mechanics, Vol. 184, 1987, pp. 123156.

23. Katz, J., "A discrete vortex method for the non
steady separated flow over an airfoils, Journal
of Fluid Mechanics, Vol. 102, 1981, pp. 315328.
24. Kiya M., and Arie M., "A contribution to an in
viscid vortexshedding model for an inclined flat
plate in uniform flown, Journal of Fluid
Mechanics, Vol. 82, Part 2, 1977, pp. 223240.
25. Sarpkaya T., An inviscid model of twodimensi
onal vortex shedding for transient and asymp
totically steady separated flow over an inclined
plate", Journal of Fluid Mechanics, Vol. 68,
Part 1, 1975, pp. 109128.
APPENDIX A
We remove here the assumption of fixed fluid domain,
introduced in section 3a to simplify the calculations
for the equivalence between the kinematical represen
tation stemming from (10) and the Poincare formula
(23~. Hence, we have to consider in (10) also the sur
face integral accounting for the motion of the free sur
face, neglected in section 3a. An easy way to prove
this equivalence is to derive with respect to time the
Poincare formula. In particular the first surface inte
gral in (23) perfectly coincides with the first in (10)
therefore we focuse our attention on the second sur
face integral
V* x ~ (u x n)G dS =V* x k / urGdS
an(t) an(t)
(1A)
where now ~Q is a function of time, described by the
parametric equation x = x(t,,t) with ~ ~ D`. The
time derivative of this integral, expressed in terms of
the Lagrangian parameter ~ is
/. urG dS =
d t an(t)
dt /De Or [Xt6,t),t] GtX((,t),x*~1 J(: to do, =
iD dt~urJ)G+urJ dtVGd~ (2Ay
whereis a derivative for a given point ~ and J is
the Jacobian of the transformation. We introduce in
t2A~ the following identities
Dw
where n f
d Dwur dJ
dt(Ur]) = Dt J + or dt
DWU dr dJ
= Jo Dt · r+ U d t) + Or d t (3A)
Is the total derivative following the free
surface motion. Using the definition of J
J l~x
235
OCR for page 221
taking the time derivative of &~ ~ = TJ
d ax Bw d dr dJ
dt(~) Be, = dt(rJ) = Jdt +Tdt
its scalar product times u divided by J
dr 1 dJ dw
unit + u,Jdt = u. d
and combining these results with the expression (3A)
divided by J. leads to
.~d.~.(u~]) = dot ~ + us ~ (~4A)
The free surface velocity w has the normal com
ponent w" = a", while for the tangential component
we may assume wr = 2ur. The relationship between
the material derivative and D" is
Du DWU Bu
D! = Dt + (or Wr)o
which combined with(4A)gives
~7,d`(ur]) = Dt ·7 + ~r(2Un)
(5A)
where the last term results after some tediuos manip
ulation by
UT BU BW {1 1 2
+ U di9 = ~ ~ ( 2 Un)
Now from the Euler equation
Du _ ~ ( p ~ 9z) (6A)
from(5A)plus(6A)combined with(2A)we obtain
/ ()[~ (P+9z2u2)G]dS+
+Jr [Uris G+uTu"0G]dS (7A)
Integrating by parts the first integral and factorizing
, leads to
IOn(t) [[Jr (P +9Z2Un + BUT) + Urgent] dS
which corresponds exactly to the term
/ (PnjUkIJ U U(k)) dS (9A?
including the second and the third surface integrals of
equation (10). Actually by introducing
l~j = PnjL,oUj
and using the vector notations
~I&n`~) ( ) (1OA)
IlxVG= (~Ton Hoper)
with
Tin = _ _ u2 = P + us _ u" + go
JET = on or
by comparing we may verify the exact correspondence
between(lOA)and (1A)
In conclusion we see that the term(iOA)which
in the Euler equation is integrated in time, even in
this case of moving boundary, perfectly coincides with
the time derivative of the term(iA,of the Poincare
formula.
236
OCR for page 221
DISCUSSION
Gerard Van Oortmerssen
Marin Research Institute Netherlands, The Netherlands
The authors have given a rather fundamental analysis of freesurface
flows and showed results of boundary integral computations for some
basic cases involving simple geometries. Could you please elaborate
on the perspective for applying these methods for more realistic cases
of practical relevance.
AUTHORS' REPLY
I'd like to thank Dr. Van Oortmerssen for his question which gives
me the opportunity to discuss the perspectives of our work. We have
presented in this paper several models for the analysis of Cortical
flows at different levels of complexity. In particular, the simplest
one, for inviscid attached flows about bodies with a sharp trailing
edge, may be directly used for applications in ship hydrodynamics.
For instance, slightly submerged hydrofoils have been studied
accounting properly for the nonlinear effects due both to the free
surface and to the wake. The extension to the threedimensional case
has been completed from the theoretical point of view (see ref. 10)
while its numerical application is still in progress.
On the other hand, more complex models for flows about bluff bodies
still require a large amount of basic work, mainly about the vortex
shedding modelling, so that the application to real problems is far
ahead and presently the investigation is confined to twodimensional
test cases. In the framework of this application we have proposed
two possible approaches. In the first one we considered the integral
representation for the complete NavierStokes equations. A flow field
simulation by this model would require a very large computational
effort and only the simple case of two vortices is now under
investigation. The second simplified approach is essentially based on
the coupling of an external solution, obtained by the Poincare
identity, with an internal one for the detection of the separation point
locations. Even for this model, a deep investigation is required to
better understand its capability to represent the physical phenomenon
and its range of applicability.
DISCUSSION
Philippe G. Genoux
Bassin d'Essais des Carines, France
Is your solution able to simulate alternated vortices (Strouhal effects)
in the case of a flow past a cylinder?
AUTHORS' REPLY
In the present approach, the vorticity field is modeled by vortex
sheets issuing from the separation points on the cylinder. We find
that, after perturbing the system, the symmetric solution no longer
exists and is replaced by a flow which shows the developments of
alternate vortices. The solution is obtained with fixed (prescribed)
separation points. Moreover, we found that under these conditions
the flow doesn't evolve towards a periodic solution. Actually the
strength of the vortex layer, still oscillating, decreases in time. This
behavior may be explained by the fact that we keep the separation
points fixed. In order to develop a procedure with moving separation
points, we have introduced the interaction between the external and
the internal solution. Presently we have an increasing oscillatory
motion of the separation points; therefore, a comparison of the
Strouhal number seems to be premature.
DISCUSSION
J.M.R. Graham
Imperial College of London, United Kingdom
Have the authors compared their prediction of the separation points
on the circular cylinder in impulsively started flow with other
published results? For example, the analysis of the onset of
separation (Van Dommelen & Shen) and other similar work.
AUTHORS' REPLY
As I said in the previous answers, the model we propose for vortex
shedding after a bluff body is still under investigation and we
obtained only some preliminary results. We don't consider however,
the model suitable for studying the onset of separation at its very
initial stage. Let me stress again the point that our purpose here is
to devise a simplified model able to analyze recirculating or separated
regions without solving the complete Navier Stokes equations.
Presently, we are still in the stage to reproduce the physical
phenomenon and to understand the essential features to be included
in the model. From this point of view, we are very interested to
consider the suggestions in the paper you mention.
237
OCR for page 221