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OCR for page 670
Modeling 3D unsteady sheet cavities using
a coupled UnRANS-BEM code
by
Chahine and Cha~Tsung Hsiao
Georges L.
DYNAFLOW, INC.
7210 Pindell School Road
Fulton, MD 20759
e-mail: info@dynaflowinc.com
http ://www. dynaflow-inc . com
ABSTRACT
The flow field of a propeller blade subjected to
sheet and cloud cavitation includes several complex
and strongly interacting features: flow separation,
turbulence, presence of vertical structures, deforming
and moving free surfaces, free surface instability, and
cavity break-up. To best describe this flow field we
are developing a numerical scheme combining a
viscous Navier Stokes code (UnRANS) and a
potential code (BEM) combining the capabilities of
each model to address portions of the problem and to
achieve a description level that is not possible with
each of the methods alone. The UnRANS code is
used to describe the turbulent viscous flow around
the blade, while the BEM code is used to describe the
non-linear cavity free surface deformations.
In this paper we apply the developed method
to study sheet cavitation dynamics on a straight and a
twisted elliptical hydrofoil. We show the results
obtained and discuss issues and future development
efforts. Cases presented consider the influence of the
cavitation number, the incidence angle, the
oscillation of the foil, and the Reynolds number on
the results. Also the influence on the cavity
dynamics of a perturbation in the inflow field, such
as in a wake is considered.
INTRODUCTION
The periodic shedding from a sheet cavity of a
bubble cloud and its subsequent convection
downstream followed by collapse leads to deleterious
effects such as noise, erosion and vibrations which
can strongly affect the expected performance of
marine propellers (Bark & van Berlekom, 1978, Shen
& Peterson, 1978, Brennen, 19944. Even though, this
phenomenon have been observed and documented for
many years, the processes by which cloud cavitation
inception occurs have not been elucidated. Numerical
modeling of the phenomenon remains one of the
frontier problems. The flow involves several complex
features --- transition zones, turbulence, presence of
vertical structures; deforming and moving free
surfaces; free surface instability and breakup;
detachment and flow of a bubbly medium. The lack
of fundamental knowledge of the basic physics at
play in the problem has made simulations using
conventional assumptions questionable until these
assumptions have been confronted.
At this time several numerical methods have
been developed. The more recent involve non-linear
three-dimensional modeling of partially cavitation
(Kinnas et al 1993, 1999, Pellone et al 1998, Dan and
Kuiper 1999 a,b, Lange, 19964. Other approaches
such as by Kubota et al. (1989) and Reboud &
Delannoy (1994) consider a two-phase flow or a level
set model to describe the periodic cloud shedding.
Some experimental studies were also conducted to
understand the mechanism of unsteady attached
cavitation (Franc & Michel 1988, Tassin Leger et al
1998 a,b, Katz et al 19994. The dynamics of bubble
clouds was studied among others by Brennen et al
(1994), Wang and Brennen (1999), Chahine et al
(1983,19924. We considered a potential flow and
computed by a three-dimensional boundary element
method (BEM), which efficiently and accurately
describes moving boundary flows (Brebbia et al
1989, Becker 1992) the bubble dynamics. The
advantages of the BEM lie in the economy of
unknowns only sought on the discretized liquid
domain boundaries, and in its accuracy in handling
boundary deformation. Moreover the movement of
the boundary is easily followed in a Lagrangian
fashion using the local velocity.
In order to study sheet cavitation instability and
cloud inception, we couple the BEM and the
UnRANS to use the best features of each of these
approaches. More particularly we modified and
coupled the UnRANS code, UNc~E, developed by
Mississippi State University (MSU) with
DYNAFLOW'S Boundary Element Method code,
3DYNAFS. UNc~E is used to accurately describe the
basic physics of the viscous flow around the propeller
OCR for page 671
blade, while 3DYNAFS accurately describes the non-
linear free surface dynamics of the cavities and their
interaction with the viscous flow.
Overview of the method
As described in the next section our
approach is justified through the use of the Helmholtz
decomposition where the flow velocity is
decomposed into a potential part and a vertical part.
The potential part is solved using the BEM code,
which is modified to account for the vertical
components, both in the time stepping procedure and
in the boundary conditions. The Helmholtz
decomposition is carried into the momentum
equation, which results in a modified Bernoulli
equation involving the vertical and viscous terms.
These are obtained from the Navier-Stokes
computations. The two-way coupling is furler
accomplished with the BEM cavity model providing
the free surface position and its normal speed to the
Navier-Stokes solver.
Solution of the UnRANS problem provides
the velocity, vorticity, and pressure fields in the
whole computational domain. Using a cavitation
criterion, say pressure lower than a critical pressure,
regions where cavities should be present are
delineated. This applies to both the region of the fluid
in contact with the blade and to the bulk fluid. A fine
surface discretization of these cavities / free surfaces
is then automatically generated for treatment by the
BEM. Initially, a Dual Reciprocity Boundary
Element Method (DRBEM) (Chahine et al 1997) was
used to solve a Poisson equation resulting from the
Helmholtz decomposition, and which describes the
potential component of the flow taking the viscous
vertical effects into account. A method for direct
evaluation of the df /aft terms needed on the free
:
_
~ :~_ _
: ~ ~ ~~
.~:~.~ =~ ,:~;~s,ylsy WP ~ ~
~ ~~ ~ (~? ~~>s>~-s,e-: ~ ~
:,,,:.
..,, ::
:. : ~
._ ~
~._.
I' ~.~ '~pS>~'5 ~~(X5~=
Figure 1. Sketch of the coupled UnRANS / BEM
surfaces is however presently preferred. Using the
velocities and vorticities computed by the UnRANS
code, df /aft is obtained using a modified Bernoulli
equation, and is used to update the values of f for
the next time step. The BEM is then solved; a new
cavity shape found and the cavities' grids are
interpolated back to the UnRANS Eulerian grids to
enable computation of the following time step. The
procedure is sketched in Figure 1 and can be
summarized as follows:
a. The UnRANS code, using a precise
discretization of the blade geometry, is used to
describe the turbulent viscous flow around the blades.
b. The UnRANS equations are solved with
adequate boundary conditions: no-slip velocity on
solid boundaries and stress balance (i.e. zero shear
and balance of normal stresses) on the cavity
surfaces. This leads to knowledge of velocity,
vorticity, and pressure fields in the whole
computational domain.
c. In order to account for the vorticity field, a
Dual Reciprocity BEM is used, which results in a
Poisson equation in which the right hand side, or
"source term", contains the vorticity contribution.
d. Using the velocity and vorticity field
computed by the UnRANS code, the "source term" in
the Poisson Equation is obtained, and is assumed
invariant for the following half time step.
e. An expansion of the "source term" field in
terms of a selected basis function is conducted
resulting in a transformation of the Poisson equation
into a Laplace equation, which is then efficiently
solved using the 3DYNAFS BEM method.
f. Alternatively, to replace c-e, a Dual-BEM
method is solved which directly computes
df /aft using the above described source terms
computed in the UnRANS procedure.
g. The BEM equations are solved using the
same surface grid as the UnRANS on the blade
surfaces and a fine discretization on the surface of the
cavities. This results in a description of the free
surface nodes velocities, which enables one to update
the shape of the cavities for the next time step.
h. The cavity fine grids are interpolated back to
the UnRANS grid to enable computation at the
following time step, and to iterate by going back to
Step a.
MATHEMATICAL FORMULATION
Let us consider a propeller blade immersed
in an incompressible liquid of density r . Noting the
vector velocity u, the time t, and the pressure p, the
continuity and the momentum equations can be
written:
OCR for page 672
v n=O,
p[3t+(u Vln]=V t+g, (1)
where t is the Newtonian shess tensor, ad g is a
body force such as She acceleration due to g a ity
In She frame of reference of She blade, the flow is
subject to a no-slip condition on She rigid surfae:
u = 0 (2)
Under some flow conditions cavitation occurs in the
liquid, a d cavities form over the blade so thee ad in
the liquid At these free surfae, neglecting mass
transfer Dross She interfaces, the liquid satisfies a
kinematic ad a dy amic free so thee condition he
kinematic boundary condition on each surfae of
equation Sj (x, rl = 0, can be w itten:
DSi = 0, (3)
while the dy amics condition expresses the
contimmity of stresses at the free are faces if w
neglect gas motion ad viscous forces on the gaseous
side of the free m fade, only the normal components
of the stresses remans, while the tangential liquid
stresses are mall in this case She normal shess
balm e equation can be written as:
~ (I n)=p Cy, (4)
where p' is She pressure on the f ee su fade side, C
the local surfae curvature, Andy is the surfae
tension he zero shess tangential components can
be written as:
t ~ [u (I n)] n=0
Cavity Model
(5)
he precise condition mder which a cavity
forms on a solid bo mdary is not yet w 11 mdzstood
Recently in a careful experiment Morch ad Song
(1992) have shown that a perfect contact between a
solid boundary ad She liquid carmot exist Ed that
naoscopic an cavities remam at anv wetted solid
surfae, Thus forming potential cavitation nudei We
will use 6 is to ju tify The use of the following model
that post tales inception of a cavity on a solid surfae
when The pressure at the surfae dkops below some
critical pressure, pa comparable to the liquid valor
pressure, Pv. a d acc o Ming for a gas pressure, pgO
P. = Pv + PA (6)
If the pressure at anv region is below The pressured,,
we make That pat of the surfae free to move with
the pressure Thereafter following a polyt opic gas
compression law of con tat, ~ he spade vol me
between The initial body so thee ad the freely
moving free surfae fomms the cavity of vol me V
that grows ad collar ses on The body
he pressure at The bubble/cavity surfae is given by:
+ ( V ) Cy (a)
he cavity suffice moves with The local fluidu
subject to The condition that no cavity point can
penetrate the physical solid m fade of the body
Durmg collapse a free surfae point that apron X.
the solid surfae is made a solid node again
Mooed Approach
In order to be ale to tudy with some
~ C! the physics involved with the highly
nonlinear dy comics of the cavities, especially in the
boundary layer of The blade, a mixed U RANS REM
Euleria / Lagr mgia Repro ah was selected his is
justifi d by The fat that The acurate description of
the dy amics of anv free su fames in the flow domain
will require a high level of discretization which
nnotbe achieved in the full computational spade
using a UnFtANS Euleria g id he two codes are
coupled mtimately th ough use of half pieud.
timesteps (one for each of the methods), ad th ough
a g id "overl y" of The results of one medhod onto the
other
UnRANS Approach for the Liquid Behavior
In turbulent flows a wide spech m of eddy
sizes wish a corresponding spectrum of fluctuation
frequencies exist he largest eddies have sizes on the
same order of magnitude as the flow domain, have
low frequencies, ad are affected by The boundaries
ad the mea flow he smallest eddies, on the ocher
h Ed, are detemmined by the viscosity of the fluid ad
have high frequency fluct ations As the Rey olds
m mber of a given flow increases, The width of the
spectr m, or the difference between The largest ad
smallest eddies more As
he large eddies exhort kinetic energy from
the mea motion Ed feed it to the large scale
turbulent motion Energy is passed dow The cascade
to smaller ad smaller eddies mtil viscosity cases
the dissipation of the eddies he rate of energy
dissipated is determined by the large Dale motion
although dissipation occurs at the smallest scales it is
impo tat to note that viscosity does not detemmine
the amo mt of dissipated energy, but only The scale at
which dissipation occurs, ad that away f om
boundaries much of the physics of t rbulence is
effectively mviscid
Direct mmmerical simulations ENS), which
involve n merically solving The full msteady N
OCR for page 673
equations ad resolving all re l es ant length ad time
scales are currently limited to only simple flow
geometries ad low Rey olds mmmbers, as the
computational resources required to resold all length
ad time scales of t rbulence is prohibitiy Iy
expensive Alternatiy is, the Unsteady R ynolds
iv raged Nayie~Stokes UnFtANS) equations,
obtained from averaging the msteady NS equations,
require much less computational resources ad have
been shown to be a successful altematiy Wilson et
al 1998; Gorski, 1998),
Reynolds Averaging
UnFtANS equations have been show to be
successful in addkessing propeller ad ship flow
probl ms, (Gorski, 1998, Wilson et al, 1998, Hsiao
ad Parley 1999) They express a relationship
between the memo y locity field ad the memo
pressure field Howey I, Hey contain as additional
urJmow >, the averaged products of the fluctuatmg
y locity components, or She R y olds stresses, I:
P[at+V (uu)]= pVp+Dvn V I' (8)
In order to sold These equations, additional equations
relating the Rey olds stresses to the mea y locity
ad pressure field, are necessary (closure model) For
the propeller problem, w are especially interested m
a hey olds shess closure model chat e hibits, at least
to the low st order, the correct asymptotic behavior
in the near-wall region, ad which yields satisfactory
m. rall predictions in chat region Of herwise, the
pressures at the smfae of the propeller will be poorly
modeled, ad the cavitation sheet will be predicted
incorrectly
We used a y rsion of UNCLE den loped at
the Mississippi State Uniy rsity (Taylor 1991, Sheng
1995) chat includes both a algebrac, Baldwin
Lomax model Baldwin ad Lom ax, 1978), where the
eddy viscosity is set to be proportional to She
modulus of the local mea y locity y ctor, ad a ~ a
two-equations turbulence model Here we used a
simple tw3layer Baldwir~Lomax model successfully
used by MSU for propulsor, ship, ad free m face
problems The R y olds stresses are expressed as a
function of He average y locity field, with a
multiplying factor called the eddy viscosity, :,
=~ · · (9)
: at 0~ :s s,~Ow,
where s is the distance normal to He solid surface
This eddy s-iscosiries~ mug ~ ~ are in turn
written m terms of average flow characteristics
i yolvmg the local vorticity ad the law of He wall
coordinates, y+
The UNCLE code is based on the artificial
compressibility method m which a time derivatiy of
the pressure is added to She contimmity equation to
couple it with the moment m equations As a
consequence, a hyperbolic system of equations is
formed md cm be sold d using a timemarching
scheme The method can be marched in pseud.ntime
to reach a steady state solution when a din regency
free y lociy field is obtained To obtain a t me
acurate solution, a sub-iteratiy procedure with
pseudo-time steps is performed at each physical time
step
UnRANS Free Surface Conditions
When coupling the NarienStokes
computation with the bo mdary element code
3DYNAFS, He Nayier-Stokes flow sold r mu t be
ale to handle gas liquid mte face ad moving
bo mdaries We have implemented in UNCLE a
generali:osd free surface bo mdary conditions that
e force zero shear stress ad normal shess balm e
on He bubble surface For He curvilinear coordinate
system with orthogonal bo mdaries, the bo mdary
condition can be w itten as follows:
· V 733tg .. · g h ii a'
· V 733tg ·h g x i
p. pa. I.. Cg
2 :. W. ~ g . U L - V g · W
Pa; ~ V 2 t ·x ·h V i;
where U. VW are He contrayariat y locities, ad
x,h ad V are the curvilinear coordinates with W
ad V in the normal direction of the su fame The
cone ayariat metric sgs a dgg are defmed as follows:
(10)
Grid Generation
To conduct the Nayier-Stokes comput lion
with moving bo mdaries, a efficient V id generation
scheme, which can automatically generate a
apropriate V id based on new bo mdaries at each
time step, must be inteV Ted with the Nar-ienStokes
sold r Here we have implemented a grid generation
scheme combining both algebraic ad elliptic V id
generation techniques as described m Hsia md
OCR for page 674
Pauley, 1998) which creates a good quality grid at
each time step. This uses trans-finite grid
interpolation on most blocks, combined with an
elliptic grid generation technique for the block
including the free surface of the cavity. In this
approach equations such as:
2X; Pj ~ h)V
i 1,2,3, (12)
are solved subject to given boundary conditions br
the nodes on the block boundaries; i.e. known node
locations at the boundary or gridlines perpendicular
to the boundary. Given a problem, the strategy is to
take the region on which the problem is posed, and
map it onto a simple region (a cube) and calculate the
metric coefficients needed to transform the
differential equations and boundary conditions
governing the phenomena to this simpler domain.
The transformed problem is then solved using a finite
Figure 2. Multi-block "ridding used in the modified
UnRANS code UNCLE for the sheet cavitation
problem treatment.
difference method on the simple computational
domain. Any changes to the geometry of the physical
domain are performed dynamically by changing the
mapping, and proceeding with the solution d: the
physical problem on the computational domain. We
also use a multi-block method that allows only some
blocks to vary discretization with times, while the
other blocks remain invariant. This is very
compatible with the UNCLE code, which uses multi-
blocks and a multigrid approach. A typical multi
block grid for the problem here is shown in Figure 2.
A time varying grid configuration (presence and
absence of cavity) is shown in Figure 3.
Treatment of the Cavity
In order to proceed with the BEM/UnRANS
mixed approach we use the fact that any fluid
velocity field u can be expressed via the Helmholtz
decomposition as the sum of the gradient of a scalar
potential f and the curl of a vector potential A:
u up UV, up f, u v A (13)
Since the flow is considered incompressible, we have
2f I. (14)
By applying Green's identity one can determine f at
any point x in the fluid domain oncef and its normal
derivative are known at the boundary of the domain,
S. which includes submerged bodies, cavities and
free surfaces:
f p'
fat) n G(~c~c
S n f pi) GO )t dSxt (15)
Here is the solid angle subtended by the fluid at
the point x, n is the local outward normal at the
surface, and is the gradient operator in the primed
variable. G is the free space Green's function,
Gaff ~ 1 (16)
Ix x~
Taking the curl of (13) we see that A is directly
related to the vorticity v by
2A v. (17)
In our mixed approach the vorticity v is obtained
from the UnRANS 'half step" computation.
Then, the momentum equation can be rewritten by
decomposing u into its components:
Up UV v u l lul2
t t 2
P ~
n Mu.
r
the following modified Bernoulli equation is obtained:
OCR for page 675
UV u v n 2U, (19)
f 1lU12 P.
t 2 r
(20)
This is an exact expression of the modified Bernoulli
equation earlier derived in Chahine (1989, 1993).
By taking the divergence of (19) we obtain the
following Poisson equation
2 R lVI2 U
On the boundary, satisfies
v. (21)
n n Uv u v n 2U . (22)
The presence of the term R in the right hand side in
(21) gives rise to a volume term if we were to apply
the Green's identity. Therefore, (21) cannot be solved
using information only at the boundary and the
regular BEM. This can be overcome by applying the
"dual reciprocity" ho undary element method
(Partridge et al 1991) as summarized below (Chahine
et al. 19987). Once we obtain , and the pressure p
on the bubble surface from the dynamic boundary
condition, (20) provides an expression for f / t
needed to update f at the following time step.
Dual Reciprocity BEM
To update the nodal values of f at the free
surface at successive time steps we need to solve in
an efficient way the Poisson equation for . To do
so, we expand the right hand side term R as a sum of
radial functions as follows:
N N
it(x) = ~,Rifd(x - xi ) = ~,RiV2g (x - xi ). (23)
i=1 i=1
This transforms the Poisson Equation into a Laplace
equation as follows:
V2~=v2:n -iRig(X-Xi)1=0 (24)
Now we can use a BEM scheme similar to
that for f, and obtain the field . We now have to
solve the two Laplace equations, V2f = 0 and
V2~ = 0, with the following boundary conditions:
Of
=-u n;
On solid
DO Il-llU12-P+U-Vf (25,
Dt 2 r
—=n (- w +ux? +~V2u)->R
an at i=1 an
Figure 3. Grids used in the modified UNCLE code at
two different steps, prior to and after sheet cavitation.
Dual BEM Approach
The computations of the function are very
sensitive numerically to the choice of the radial basis
functions. In addition, they require finite
differencing of quantities close to the body (see
Figure 5 showing the large values concentrated at the
boundary and prone to large errors). These
procedures, at least as we have implementing them,
turned out to be not too robust. We selected to
replace this scheme with a direct computation of
f / t. If 2f 0, then we also have:
v2(aflat) =o. (26)
The Boundary Element Method can thus be applied
to f / t with the appropriate boundary conditions;
i.e. zero normal derivative of f / t on the solid
boundary and in addition we account for the results
obtained with the UnRANS code. On the solid
boundaries we use:
P Puncle ~ (27)
to determine cavitation criterion, and on the cavity:
any at )=-3n(2~U~ + r )+
no.- w +uxv + d`72u 1.
~ at ~
(28)
The use of the pressures obtained by the UnRANS
code is an important step that insures a perfect match
between the potential flow and the Navier Stokes
pressure distributions.
Discretization
The free surfaces are discretized into
triangular elements, to generate a set of linear
OCR for page 676
Source
1 .6 E+07
1 .4E+O
1 .2 E+O
4E+O
~ E+07 ~
~1 _ '-5
8E+06 ~ O i=11
~ ~ i=15
6 E+06 ,
2 E+06 ~
0 0.0005 0.001 0.0015 0.002
Normal Distance
o
4000,
3500
30001
i=5
i=8
i=1 1
i=15
2500
2000
1 500
1 000
500
Module of Vorticity
~ ~ 1
)~ ~
) _i~'\
) _~\\\
, 1~1
J 0.0005 0.001 0.0015 0.002
Normal Distance
Figure 4. Concentration near the boundary of the term R in the Poisson equation. i represent the row numbers
equations relating f and f / n (and and
/ n ~ at the nodes,
Ajj USE ~ —~ ,B~ aid f x 0. (29)
The functions f and f / n (and and / n ~
are linearly interpolated inside each panel using their
values at the vertices of the triangle. Aid and Bij are
geometry dependent matrices relating the influence
of the ah node at the ith node. They are obtained by
integrating the surface integral in (15) analytically
over each panel separately. Since the problem has
both free surface and solid boundaries, f is known
on the free surface and f / n is known on the solid
boundaries. The equations are then solved for
f / n on the free surface and for f on the solid
boundaries (Chahine et al, 1988, 89, 94 ~ . (The
formulationis similar for and / n ).
Time Integration
The time evolution of the flow is governed
by the boundary conditions. We may update the
values of the velocity potential f using:
Df _ u f -IUI2 P U f, (30)
Dt t 2 r
where the material derivative is computed as the
surface moves with the fluid. Then, the obtained
normal and tangential velocities are used to
determine the motion of the cavities' surface nodes.
These integrations are performed with an explicit
Euler scheme. An adaptive time stepping selects a
time step proportional to the ratio of the smallest
internodal distance to the largest nodal velocity.
Wake and Tip Vortex Modeling in the BEM
In order to
properly describe the
pressure and the lift
_ <~ .~
distribution on the foil
it is necessary with an
inviscid method, to
introduce additional
singularity
distributions to
account for real fluid
effects that are not
directly recovered by
the irrotational model.
This includes, for
instance a proper
description of the lift of the blade, and means to
describe the presence of a wake behind the foil and of
a tip vortex in the blade tip area. This is a classical
problem which is solved effectively by enforcing the
Kutta condition at the trailing edge of the foil. This
condition insures pressure equality at the trailing
edge between the suction and the pressure side of the
Figure 5 Foil and wake
example discretization for
the boundary element
method.
OCR for page 677
blade. In order to achieve this we need to add a
vortex sheet behind the trailing edge of the foil,
which enables one through proper choice of the
intensity of the vortex sheet to satisfy the following
conditions:
a) Continuity of pressure across the sheet,
b) Continuity of normal velocites across the
sheet,
c) Existence at each node i of a jump, f i, in
the velocity potential across the sheet, in order to
accommodate the presence of shear or tangential
. . . .
velocity d~sconnnu~ty.
In addition, through application of Bernoulli
condition along two paths each leading to one side of
the sheet, we can write that:
D fit ~ luuPP~l2 l~bw~l2 (31)
where the us s are the tangential velocities on each
side of the sheet.
In order to implement this concept we discretize the
wake as an additional surface of the domain, (See
Figure 5), and modify Green's Identity (15) to
become:
fern G(~c~c
. . . . .
f P' dS l2)
fcil n f pa) G('c tic x
W eke
2
_ f upper p3E f by
W eke
fn. Good d: for x Boll
fpc)n G(~y
dS
fcil n f pa) G('c tic x
fn. Good dS Lo r x Wake
A special treatment is required for the junction line
between the wake and the trailing edge of the foil.
Triple points are used along this line, such as
illustrated in Figure 6 by nodes 1, N. and N+1. These
'. ~
.' 'it.
I. W~ ~~ .~
I: ~ ~:
· ~~ ~~d
Figure 6. Illustration of the special treatment of
the junction area between the wake and the foil.
are needed to enable the presence of a velocity
potential jump between nodes 1 and N. and of a
dipole distribution at N+1 of intensity,
fN ~ fN f:. The modified BEM procedure is
then to obtain at each time step: a) on the foil f
knowing f / n, and b) on the wake
f upper f b WE knowing f
The wake / vorticity sheet could be allowed
to evolve in time, and wrap up helicoidally behind
the foil tip into a tip vortex. However, this is CPU
time consuming and does not address the problem at
hand here. In addition, since the actual wake should
be recovered quite accurately by the UrRANS code,
the objective here is to have in the BEM half timestep
a description of the foil surface that is not too much
different from the
real fluid one.
Therefore, we use a
fixed wake, and
account for its
wrapping into a tip
vortex by adding a
free to deform vortex
line composed of a
succession of linear
segments, and ending
with a semi-infinite Figure 7. Sketch of the line
vortex line as vortices added to the blade
sketched in Figure 7. "ridding
The velocity induced
by all of these line vortices of length, dry, at a field
point, x, is obtained using the Biot-Savart law:
on
o.s
0.7
0.6
0.5
0.4
0.3
0.2
0.1
o
-0.1
-0.2
-0.3
-0.4
-0.5
_~
~4
, _
U NCLE
----------------------------- 3DynaFS without Kutta condition
3DynaFS with Kutta condition
-0.25
o
X
0.25
Figure 8. Pressure coefficient distribution on
both sides of the blade computed with the Navier
Stokes code and with the inviscid code with and
without the Kutta condition applied.
OCR for page 678
( t) ~ Or Ax ? j~k(~x ?; ~ )
~0 V TE Root Section
The potential due to the line vortices is added to the
incoming flow velocity potential in order to properly
compute the velocity and pressure fields.
n1
1 1
-0.1
no
Figure 9. Pressure distribution and velocity contours
near the trailing edge showing separation in the
region where the cp coefficient of the BEM did not
match the RANS computations in Figure 9.
With the addition of these two vorticity
distributions, the BEM method is capable of
recovering the pressure distribution quite accurately
everywhere but in areas where separation occurs. For
instance Figure 8 shows the pressure distribution at
mid-span on both sides of the foil discussed bdow in
the results section for an angle of attack of 12
degrees. The figure compares the results of the
Navier Stokes code with those of the BEM code
when the Kutta condition is and is not accounted for.
Near the trailing edge, a separation area exists for this
high incidence angle, that is captured by the RANS
code (see Figure 9), but obviously not with the
simple wake model BEM code. Again, the objective
here of using the coupled codes is to compensate for
the lack of precision in some areas of the BEM,but it
is advantageous for the numerical simulations to
minimize the differences between the potential and
viscous code so that the vertical part is more easily
computed.
y
Non-Twisted I ,, Twisted
1
it,
Figure 10. NACA 16-020 foil used for code
development. Straight and twisted versions.
12
1 1
10
9
8
c' 7
~ 6
85 5
. - 4
3
2
O ~
0 0.5 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 .5
y
Figure 11. Pitch angle selected for the NACA foil shown in
Figure 10.
RESULTS AND DISCUSSIONS
To test the developed numerical methods we
selected to use a 3D elliptical hydrofoil with a NACA
16-020 cross-section, which was extensively tested
experimentally in the hydrodynamic tunnel of the
University of Grenoble (Boulon, 1996). A rendering
of the shape of the foil is seen in Figure 10. Also
shown in Figure 10 is a twisted version of this foil,
with the pitch angle was varied along the span as
shown in Figure 11 to minimize tip vortex cavitation.
The chord of the foil is equal to 12 cm and its half-
span length is 18 cm. Figure 12 is a picture taken
from Boulon and Chahine (1998) and shows the
development of a sheet cavity on the foil when the
OCR for page 679
Figure 12. Sheet and cloud cavitation on an
oscillating NACA 16-020 foil. Pictures taken in
Grenoble. V=10 m/s, s =1.5, and the incidence
angle is 11.4 and 12 .
inflow velocity is 10 m/s, the cavitation number is
1.5, and the incidence angle is 11.4° and 12°. We can
see under these particular flow conditions the
attached sheet cavity and the shedding of a part of the
cavity into a cloud convected downstream into the
flow where it collapses. One disadvantage, however,
of this foil shape is the noticeable presence of a tip
vortex cavity that interacts with the sheet cavity.
This is in fact more important in the numerical
modeling that in the experiment, because the
description of the tip vortex cavity requires special
treatment that we have not implemented yet, because
they are beyond the scope of this study. Numerical
results were then obtained using the BEM code
3DYNAFS alone, the modified UnRANS code UNCLE
alone and the BEM/UnRANS coupled code that is
still being developed.
The mesh of the foil was chosen to be at the
beginning of the calculation denser close to the
leading edge, where the cavitation inception takes
place, and coarser near the trailing edge. A
regridding scheme was then implemented to
redistribute the nodes properly once a sheet cavity
develops and tends with the 3DYNAFS grid moving
1.5
0.5
-05
0.12
n 1
_..
0.08
0.06
0.02
O
-0.02 _
— 1
~ b' ~
-0.15 -0.1 -0.05 0
X
Figure 13. Crosscut of an unacceptable UnRANS
grid that results when the reentrant jet starts
developing
with the liquid velocity to increase the node density
near the trailing edge of the cavity. This procedure
and its continuous improvement and adaptation are
essential for a proper operation of the developed
code. In fact while regridding and adding panels
and nodes, may be relatively easy for a 2D BEM, it
is more challenging for the 3D code developed here,
and becomes extremely difficult with the UnRANS
code using regular grids as is the case for the
modified version of UNCLE presently used here. In
order to be able to pursue the computations shown
below to a significant length of physical time, we
1 1 1 1
-0.5 0 0.5 1 1.5
Figure 14. Time evolution of the sheet cavity and
development of a reentrant jet. NACA 16-020,
V=12m/s, a = 10 ~ s = 0.9
OCR for page 680
2.5
1.5
ns
-0.5
id'
1
1.5
2
2.5 3
Figure 15. Snapshot at a given time of the sheet cavity
along the foil span showing the 3D nature of the
reentrant jet. NACA 16-020, V=12m/s, a = 10, s = 0.9
had to compromise in this respect by imposing a
"rounding" of the cavity near its trailing edge thus
preventing the beginning of the formation of the
reentrant jet. Otherwise as shown in Figure 13, the
"ridding scheme fails afterwards and the Navier
Stokes code cannot continue the computations.
Figure 13 illustrates the distortion of the grids as the
reentrant jet begins to initiate. Increasing grid
density at the expense of significant computation
time could improve a little the picture, but a
different "ridding scheme, such as an overset grid
scheme in the reentrant jet region is definitely
needed to pursue the computations with the
UnRANS beyond this point.
Figures 14 and 15 illustrate, using the BEM
code 3DYNAFS alone, for an incoming flow velocity
of 12m/s, and an angle of incidence a of 12°, and a
cavitation number s of 0.9 the time evolution of the
cavity along the foil surface. Figure 15 is a crosscut
of the foil and cavity surface at 55% of the span of
the NACA foil. The line behind the foil is the
crosscut of the imposed vortex sheet wake. Starting
from the fully wetted foil shape, the sheet cavity is
seen to develop over time, form a long wavy
surface, then a reentrant jet which advances under
the cavity along the foil surface. At this time the
code does not pursue computations after the
reentrant jet hits another part of the cavity and the
computation domain becomes multi-connected.
However, we will be able to pursue such
computation in the future, such as we have done for
van
30
ut ) 10 20 30
Time (ms)
— oc=1 2°
---- oc=1 0°
oc=8°
Figure 16. Sheet cavity volume versus time at
different incidence angles obtained with the
modified UnRANS code
reentrant jet in collapsing bubbles (Zhang et al,
199*, Chahine et al, 19964. Figure 15 shows for the
same flow and foil conditions a snapshot at a given
time of the sheet cavity shape at various sections
along the span of the foil. This figure illustrates the
3D nature of the reentrant jet which has here a
funnel filament shape that touches back the top of
the sheet at one end close to the tip while it is still
advancing under the sheet cavity at other locations.
The volume of the cavity versus time can be
seen in Figure 17. This figure also shows the
influence of the angle of attack of the foil on the
cavity volume variations with time. Asexpected the
volume increases with the incidence angle, with the
transient portion of the computation being also
longer for the higher incidence. One can also
observe some periodic volume oscillations whose
relative value is larger for smaller incidenceangle.
This is more obvious when observing animations of
the 3D results which show larger amplitude wave
motion for the smaller angle of attack. Including the
viscous effects in the computations result in a
significant reduction of the computed sheet cauty.
This is illustrated in Figure 17, which shows cavity
volume versus time using both approaches for an
incidence angle of 12°, an incoming speed of 12 m/s
and a cavitation number of 0.9. The volumes are
about 50% smaller when viscous effects are taken
into account. Similarly, volume fluctuations are
damped out, and as seen in Figure 17, the surface of
the cavity is much smoother when viscous effects
are included.
OCR for page 681
70L
Got
o -
50
co
40
30
20
1n
!
s
s
s
20
o=o.s oc=12°
UnRANS Solution
10 20
~',1,, 1,,,,1,,,,' ~
v Ail
30 40 ~ ~ ~ 1 __
E ~ , ~-
Figure 17. Comparison of the volume versus time
results of the potential flow solver alone and the
viscous code results.
This smoothing effect of viscosity is stronger
for lower Reynolds numbers. Figure 19 shows for
the same foil and for s 09 the volume time
dependence for three values of the Reynolds
number, 105, 106, and 107. We can see from the
figure that the Reynolds number has some effect on
the absolute value of the cavity volume, with the
tendency being as seen in Figure 17, i.e. increase
volume with increased Reynolds number. In
addition the volume fluctuations are seen to also
increase with the Reynolds number. The same is
true concerning the cavity free surface shapewhich
become rougher with the increase of the Reynolds
number. Let's restate here that stronger oscillations
than those shown in the above results actually exist,
since we have prevented here the development of the
35
An
~ ~ ~ $~ I:. ~
Figure 18. Shape of the sheet cavity at a
given time obtained with the inviscid code
alone (top) and with the UnRANS code
(bottom). The back blue surface is the
wake. Full volume history is in Figure 18.
reentrant jet to be able to pursue the computations
over a long time period.
Figure 20 shows the influence of the cavitation
number on the cavity volume. Three values of s are
considered 0.9, 1.0 and 1.1. Here too as expected
lowering the cavitation number results in increased
cavity volume and increase volume oscillations.
Figure 21 shows how the lift coefficient
distribution, cp, along the foil is modified by the
development of the sheet cavity. As expected G., is
i~-
,! ~ ~ ~ ~
/ ~ ~~ a< ~ On
25
20
15
10
.E
not I I I I I I I I I I I I I I I I I I I
~o 10 20 30
Time (ms)
Figure 19. Influence of the Reynolds number on
the sheet cavity volume time variations.
— Re=1 05
Re=1 o6 0
--------------------- Re=107 ~
oc=12° Re=1 .44x1 o6
20
a,
~ 1R
O-
0 10 20 30
Time (ms)
~=o.s
o=1 .0
o=1 .1
Figure 20. Influence of the cavitation number on
the sheet cavity volume time variations.
OCR for page 682
very much modified on the suction side taking the
value - s at the location of the cavity. Here, we also
see that the pressure side pressures are also affected
along the full length of the foil and drop by more
than 10% further resulting in loss of lift.
6 = 0.9 0c = 12° Re=1.44x106 at Y=0
0.8
0.6
0.4
nor
o
-0.2
-0.4
c
(A -0.6
-n ~
-1 ~
-1 4
-1
-1
T = 0ms
.............. T=40ms
2 it ~ ~ ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ 1
-0.5 -0.25 0 0.25 0.5
X
Figure 21. Modification on the lift coefficient
on the NACA16-020 foil with the development
of the sheet cavity.
.5
.4
.3
.2
.1
1
O.'
0.E
0.,
0.E
O.'
0.i
O.'
O.~
0.1
c) Non-uniform wake-like inflow conditions.
We have already seen in Figure 10 a rendering of the
shape of the twisted NACA 16-020. Figure 11
showed the values of the imposed pitch angle. Figure
22 compares the resulting pressure coefficient
distribution over the blade surface between the
straight and the twisted foils. With the pitch angle
being reduced near the tip and the root we have
aimed at generating sheet cavitation only over the
mid-span of the blade. The actual shapes of the
cavities generated on the twisted and untwisted foils
can be seen in Figure 23. The cavitation in the
twisted case for this configuration appears to be away
from the root, where the incidence angle is zero. The
change in the pressure distribution as the sheet cavity
develops can be seen in Figure 24. Finally Figure 25
shows the difference in the cavity shape and in the
flow field pressure distribution at a selected time and
at various span locations over the foil.
In order to check if the code is able to
handle properly incoming oscillations in the flow
field we considered a uniform velocity at infinity
oscillating between 6 and 12°. This resulted as
expected in strong oscillations of the volume as
shown in Figure 26.
Finally a wake-like non-uniform flow field
Non-Twisted
.4 4
3
.1
O l
-0.5 0 0.5
X
pi
0.10
0.03
-0.04
-0.11
-0.19
-0.26
-0.33
-0 .40
-0 .47
-0.54
-0.61
-0.69
-0 .76
-0.83
-0.90
1 .5
1 .4
1 .3
1 .2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
O.
1 4
1 .2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
n 1
No-Twisted With Wake Inflow
Figure 22. Comparison of the pressure distribution on the fully wetted foil under three conditions:
a) Non-twisted foil in a uniform flow field, b) Twisted foil in a uniform flow field, and
c) Non-Twisted foil in a wake-like flow field
Complex 3D simulations
In order to show the capabilities of the
developed codes to address complex 3D geometries
and flow conditions, we consider the following
effects on the sheet dynamics:
a)Presence of oscillations in the incoming
flow or in the foil incidence,
b)Presence of significant twist on the blade
shape, and
pi
0.10
0.03
-0.04
-O. 1 1
-0.19
-0.26
-0.33
-0.40
-0.47
-0.54
-0.61
-0.69
-0.76
-0.83
-0.90
was imposed at the entrance to the computation
domain. Figure 28 describes this flow field by
representing the resulting velocity contours at the
start of the computations where the foil is fully
wetted. We purposely selected a wake like sinusoidal
distribution in the leading edge to minimize or
eliminate the tip vortex cavitation. As expected, the
code operated properly under these conditions and
the resulting sheet cavity volume was significantly
smaller (see Figures 28 and 29) than that obtained
OCR for page 683
,>
1.5
1.4
1.3
1.2
1.1
1
o.s
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
n
~=~
Figure 23. Shape of the cavitation region over the
straight and twisted NAC16-020
Figure 24. Change in the pressure distribution with
the development of the sheet cavity.
with the uniform flow under the same condition, i.e.
V=12 His, a =12°, and s =0.9.
CONCLUSIONS
We have developed a 3D viscousfinviscid
scheme to describe sheet cavitation on propeller
blades. The code uses a BEM and an UnRANS
procedure to describe the cavity shape and time
c, = 0.9 oc=12° Re=1 .44x1 o6 Oscillation Prelod=40ms
25
an
`, 15 ~ 1
a,
~ - 1
10 _ ~
n I l l 1l l
: V ; ~ r :) b)
U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 50 100 150 200
Time (ms)
Figure 26. Time evolution of cavity volume on a
NACA16-020 foil oscillating between 6 and 10
degrees with an oscillation period of 40ms.
Figure 25. Cavity shape and pressure distribution in the
flow field at various span locations and at a selected time.
Left set is for un-twisted foil and right set is for twisted
NACA16-020 foil.
evolution up to the point of formation of a reentrant
jet which would result in the inception of cloud
cavitation. Beyond this point further development of
the codes is needed and is on-going. This
development involves mainly definition of adequate
grid generation procedures.
From our numerical results it appear that there
are three schemes for cavity shape oscillations at
various flow conditions. These may occur
simultaneously under some flow conditions:
c)
a) The oscillation of the cavity extent, i.e
length, width, and height,
The development of a wavy over the cavity
surface whose amplitude can be large
enough to cut the cavity into two parts, and
The formation of a reentrant jet, which
curls under the cavity and detaches a
portion of it into a cavity cloud.
The last two conditions may lead to cloud
cavitation formation.
We have also found that the Reynolds
number affects the results making the cavity
appearance more distorted and its volume larger at
larger cavitation numbers.
ACKNOWLEDGMENTS
This work was supported by the Office of Naval
Research under Contract No. N0014-97-C-0167
OCR for page 684
No-Twisted With Wake Inflow
.~.,
20
o
-5 -4 -3 -2
X
-1 0 1
Figure 27. Non-Uniform wave-like flow field imposed
around the NACA16-020 foil. Notice imposed strong
velocity deficit in the tip area.
it.
-
_
I:
~ .~
_,,..
Figure 29. Cavitation distribution over the NACA16-
020 foil in a uniform flow and in the wake-like non-
uniform flow field described in Figure 26.
under the monitoring of Dr. Edwin Rood. This
support is greatly appreciated.
REFERENCES
1. Taylor, L. K., "Unsteady Three-Dimensional
Incompressible Alogrithm Based on Artificial
Compressibility," Ph.D. Dissertation, Mississippi
State University, Mississippi, May 1991.
2. Sheng, C., "Development of a Multiblock
Multigrid Algorithm for He Three-Dimensional
Incompressible Navier-Stokes Equations," Ph.D.
Dissertation, Mississippi State University,
Mississippi, Dec. 1994.
3. Baldwin, B. S., Lomax, H., 'The Thin Layer
Approximation and Algebraic Method for Separated
Turbulent Flows", AIAA Paper 78-257, 1978.
6 = 0.9 a=12°
~—
f ~
Uniform Inflow
Wake Inflow
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
10 20 30
Time (ms)
Figure 28. Comparison of the cavity volume
generated on the non-twisted NACA16-020 foil
in auniform field and in a non-uniform field
4. Bark, G. & van Berlekom, W. B.,
"Experimental Investigations of Cavitation Noise,"In
Proc. of the 12th Symposium on Naval
Hydrodynamics, Washington, D. C., USA. National
Academy Press, 1978.
5. Becker, A. A., The Boundary Element
Method in Engineering, McGraw-Hill, New York,
1992.
6. Boulon, O. "Etude Experimentale de la
Cavitation de Torubillon Marginal -Effects
Instationnaires, de Germes et de Confinement,"
Thesis, Institute National Polytechnique de Grenoble,
1996.
7. Boulon, O. and Chahine, G. L., "Numerical
Simulation of Unsteady Sheet Cavitation on a 3D
Hydrofoil," Third International Symposium on
Cavitation, Grenoble France, April, 1998.
8. Brebbia, C. and Dominguez, J., Boundary
Elements an Introductory Course, McGraw-Hill
Book Co., New York, 1 989.
9. Brennen, C. E., "Observation of Cavitating
Flows," In Proc. of the 20th Symposium on Naval
Hydrodynamics, Santa Barbara, California, USA.
National Academy Press, 1994
10. Chahine, G. L., 'Cloud Cavitation: Theory,"
Proceedings, 14th Symposium on Naval
Hydrodynamics, Ann Arbor, Michigan, 2527,
National Academy Press, pp. 165-195, Washington,
D.C., 1983.
11. Chahine, G.L. and Perdue, T.O.,
"Simulation of the Three-Dimensional Behavior of
an Unsteady Large Bubble Near a Structure," in
A.I.P. Conference Proceedings 197: Drops and
Bubbles, Third International Colloquium, ed. T.G.
Wang, American Institute of Physics, New York, pp.
188-199, 1989.
OCR for page 685
12 Chahine, G L, ad Duraswami, R.
IA amical h reracrions in a Multi Bubble Cloud,"
Journal of Fluid Engineenng, Vol. 114, pp 680-
686, 1991
13 Chahine,G L,D rcswamiR, -D amical
Interations in a Multi-Bubble Cloud," Journal of
Fluid Engineening,Vol 114,pp 680-686,1992
14 Chahine,G L,DuraswamiR, Kal muck,
K, "Bo mdary Element Method for Calculatmg 2-D
ad 3-D Underwater E plosion Bubble Loading on
Nearby Struct res, including Fluid Struct re
Interation Effects," Naval Surface Weapons Carter,
Dahlgen Division, Weapons R search ad
Technology D payment, Techmical Report
NSWCDD/TR-93/46, Sep 1996
15 Chahine G L, Sarkar, K, Duraswami R.
'Shong Bubble Flow h leranons ad Cavitation
Inception," DyNAFLow, INC Techmical R po t
94003-1, 1997
16 Dag, J. Kuiper, G. -he-E tract Jet
Modeling of Partial Cavity Flow on Two
Dimensional Hydkofoils," Journal of Fluid
Engineenng, Vol. 121, pp 773-780, 1999
17 Dag, J. Kuiper, G. -he-E tract Jet
Modeling of Partial Cavity Flow on Tlr en
Dimensional Hydkofoils," Journal of Fluid
Engineenng, Vol. 121, pp 781-787, 1999
18 Franc, J. P & Michel, J. M, -1 Mead
Attahed Cavitation on a Oscillating Hydkofoil,"
Journal of Fluid Me panics, 193, 171 189, 1988
19 Gorski, J. J. 'Pressure Field A alysis of a
Propeller with Unsteady Loading ad Sheet
Cavitation," Proceedings, A AA pager ?????
20 Gorski,J J,"TheE tropyEquationBudget
of Bo mded Turbulent Shear Flows," Physics of
Fluid, ?????
21 Hsiao, C -T. Parley, L L 'N merical
Study of fhe Steady-State Tip Vo tex Flow over a
Finite-Spa Hydkofoil," Journal of Fluid
Engineenng, Vol. 120, pp 345-353, 1998
22 Hsia, C -T. Paley, L L 'N merical
Computation of Tip Vo tex Flow Generated by a
Marine Propeller," Journal of Fluid Engineenng,
Vol. 121, pp 638-645, 1999
23 Kimmas, S A & Fine, N E, 'A Numerical
Nonlinear A alysis of the Flow aro md Two ad
Three-Dimensional Partially Cavitating Hydkofoils "
Journal of Fhid Me hanics, Vol. 254, pp 151 -181,
1993
24 Kimmas, S A, ' he Prediction of Unsteady
Sheet Cavitation," Third ~temational Symposi m on
Cavitation, Grenoble, Fra e April 1998
25 Kubota, A, Kato, H & Yunagmchi, H. 'A
New Modeling of Cavitating Flows: A N merical
Study of Unsteady Cavitation on a Hydkofoil
Section," Journal of Fluid Me hanics, Vol. 240, pp
59-96, 1989
26 Lage, DF D, "Observation ad
Modelmg of Cloud Formation Behind a Sh:et
Ca ty," Thesis Univ Twente, E schede, April,
27 Morch, K A, Song, J. P. 'Cavitation
Nuclei at SolidLiquid ~te faes," Cavitation:
Proceedmgs of the ~stitution of Mechaical
Engineers, pp 1-7, 1992
28 Patridge, P W. ad Brebbia, C A, ad
Wrobel, L C, ' he dual reciprocity bo mdary
element method," Computational mechaics
publications, Soubhampton, Elsevier, London, 1991
29 Pellone, C, Maibe, T. Bria omMarjollet,
L, 'Partially Cavitating Hydkofoils: E perimental
adN merical Analysis','Jou~nalofShipRerear h,
Vol. 44No 1,pp 40-58,2000
30 Wilson, R. Paterson, E, md Stern, F.
'ILsteady RANS CFD medhod for naval combatat
in waves", 22~ Symposi m on Naval
Hydkody amics, pp 183 197, Washington, D C
1998
31 Tassm L ger, A, C ccio, S L,
'Examination of the Flow near the L admg Edge of
Attahed Cavitation Part I D tahment of Two-
Dimensional md Axisymmetric Cavities, Jouxnal of
Fluid Me hanics, Vol. 376, pp 61-90, 1998
32 Tassin L ger, A Bernal, L P. Ceccio, S L,
'Examination of the Flow near the L admg Edge of
Attahed Cavitation Part 2 incipient Breakdown of
Two-Dimensional md A isymmetric Cavities,"
Journal of Fluid Me hanics, Vol. 376, pp 91 -113,
1998
33 Wag, Y-C Bremmen, C E, 'N merical
Computation of Shock Waves in a Spherical Cloud of
Cavitation Bubbles,"Jou~na/ of Fluid Engineenng,
Vol. 121, pp 872-880, 1999
34 Zhag S. D nca, J. H. Chahine, G L '~Ihe
final stage of the colla se of a cavitation bubble near
a rigid wall," Jou~nal of Fhid Me hanics, 257, pp
147-181, 1993
OCR for page 686
DISCUSSION
C. Song
University of Minnesota, USA
Assuming you can overcome the
simu ation of re-entraut jet, how wou d
t y to simu ate the very small cavities
generated in the cloud cavity? I thing
you wi I face a very difficu t problem of
geuerahng bounds y fitted god system
AUTHOR'S REPLY
Representative terms from entire chapter:
time step
