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OCR for page 819
24th Symposium on Naval Hydrodynamics
FuLuoka, JAPAN, 8-13 July 2002
A BEM Technique for the Modeling of Supercavitating
and Surface-Piercing Propeller Flows
Yin Lu Young
Spyros A. Kinnas
(The University of Texas at Austin, USA)
ABSTRACT
A three dimensional (3-D) boundary element
method (BEM) is presented for the numerical mod-
eling of supercavitating and surface-piercing pro-
pellers. The method has been developed in the
past for the prediction of unsteady sheet cavitation
for conventional fully submerged propellers. To al-
low for the treatment of supercavitating propellers,
the method is extended to model the separated
flow behind trailing edges with non-zero thickness.
For surface-piercing propellers, the negative image
method is used to account for the effect of the free
surface. The current method is able to predict com-
plex types of cavitation patterns on both sides of the
blade surface with searched cavity detachments. The
method is shown to converge quickly with grid size
and time step size. The predicted cavity planforms
and propeller loadings also compare well with exper-
imental observations and measurements. Finally, a
2-D study to investigate the effects of jet sprays at
the moment of blade entry is presented.
INTRODUCTION
Supercavitating propellers are often believed to
be the most fuel efficient propulsive device for high
speed vessels. The term supercavities refer to cavities
that are longer than the blade. They tend to have
smaller volume change and produce bubbles which
collapse downstream of the blade trailing edge, which
results in reduced noise and blade surface erosion.
However, they are also difficult to model due to the
unknown size of and the pressure in the separated
region behind thick blade trailing edges, which are
characteristic of supercavitating propellers.
A surface-piercing (also called partially sub-
merged ~ propeller is a special type of supercavitating
propeller which operates at partially submerged con-
ditions. Surface-piercing propellers are more efficient
than submerged supercavitating propellers because:
Reduction of appendage drag due to shafts,
struts, propeller hub, etc.;
2. Larger propeller diameter since its size is not
limited by the blade tip clearance from the hull
or the maximum vessel draft;
3. Reduction of blade surface friction and erosion
since cavitation is replaced by ventilation .
In the past, the design of surface-piercing pro-
pellers often involved trial-and-error procedure us-
ing the measured performance of test models in free-
surface tunnels or towing tanks. However, most of
the trial-and-error approaches do not provide infor-
mation about the dynamic blades loads nor the aver-
age propeller forces (Olofsson 1996~. Model tests are
extremely expensive, and are often hampered by scal-
ing effects (Shell 1975) (Scherer 1977) and influenced
by test techniques (Morgan 1966) (Suhrbier & Lecof-
fre 1986~. Numerical methods, on the other hand,
were not able to model the real phenomena. Difficul-
ties in modeling surface-piercing propellers include:
Insufficient understanding of the physical phe-
nomena at the blade's entry to, and exit from,
the free surface.
Insufficient understanding of the dynamic loads
accompanying a propeller piercing the water at
high speed.
· The modeling of very thick and very long venti-
lated cavities, which are also interrupted by the
free surface.
· The modeling of water jets and associated
change in the free surface elevations at the time
of the blade's entry to, and exit from, the free
surface.
· The effect of blade vibrations due to the cyclic
loading and unloading of the blades associated
with the blade's entry to, and exit from, the free
surface.
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Supercavitating Propellers
The development of numerical methods for the
analysis and design of supercavitating propellers has
been slow compared to conventional propellers. The
main difficult arises from the unknown physics in
the highly turbulent region behind the blunt trailing
edge, which is characteristic of supercavitating pro-
peller sections. The first theoretical design method
was developed by (Tachnimdji & Morgan 1958), and
followed by (Turin 1962, yenning & Haberman 1962,
Cox 1968, Barr 1970, Yim & Higgins 1975, Yim
1976~. However, these methods were based on 2-D
studies, and required many approximations and em-
pirical corrections. Recently, more rigorous methods
were developed by (Kamiirisa & Aoki 1994, Kikuchi
et al 1994, Vorus & Mitchell 1994, Ukon et al 1995~.
Nevertheless, these methods were still based on the
optimization of 2-D cavitating blade sections to yield
minimal drag for a given lift and cavitation number.
A 3-D vortex-lattice method was developed by
(Kudo & Ukon 1994) to predict the steady perfor-
mance of supercavitating propellers. Their model
assumed the pressure over the separated zone to be
constant and equal to the vapor pressure. A variable
length separated zone model using a similar vortex-
lattice method was presented in (Kudo & Kinnas
1995~. It was found that the length of the separated
zone had no effect on the results if all the blade sec-
tions were covered by the supercavity. However, the
length of the separated zone did have an effect on
the pressure and cavity length near the blade trailing
edge under fully wetted and partially cavitating con-
ditions. Finally, (Kinnas et al 1999) further extended
their method to search for midchord cavitation, and
coupled it with an optimization method (Mishima &
Kinnas 1997) for the design of supercavitating pro-
pellers.
However, all of the above mentioned lifting sur-
face methods cannot capture accurately the flow de-
tails at the blade leading and trailing edge due to the
breakdown of linear cavity theory. In addition, the
applicability of the thickness-loading coupling intro-
duced by (Kinnas 1992) in the analysis of supercav-
itating propellers is still under investigation.
Surface-Piercing Propellers
The first effort to model partially submerged
propeller was carried out by (Oberembt 1968~.
He used a lifting line approach to calculate the
characteristics of partially submerged propellers.
(Oberembt 1968) assumed that the propeller is
lightly loaded such that no natural ventilation of the
propeller and its vortex wake occur.
A lifting-line approach which includes the ef-
fect of propeller ventilation was developed by Furuya
in (Furuya 1984, Furuya 1985~. He used linearized
boundary conditions and applied the image method
to account for free surface effects. He also assumed
the face portion of the blades to be fully wetted and
the back portion of the blades to be fully ventilated
starting from the blade leading edge. The blades
were reduced to a series of lifting lines, and method
was combined with a 2-D water entry-and-exit theory
developed by (Wang 1977, Wang 1979) to determine
thrust and torque coefficients. Furuya compared the
predicted mean thrust and torque coefficients with
experimental measurements obtained by (Hadler &
Hecker 1968~. In general, the predicted thrust co-
efficients are within acceptable range compared to
measured values. However, there were significant dis-
crepancies with torque coefficients. Furuya (Furuya
1984, Furuya 1985) attributed the discrepancies to
the effects of nonlinearity, absence of the blade and
cavity thickness representation in the induced veloc-
ity calculation, and uncertainties in interpreting the
experimental data. He also stated that the applica-
tion of lifting-line theory is limited due to the relative
large induced velocities at low advance coefficients.
An unsteady lifting surface method was em-
ployed by (Wang et al 1990a) for the analysis of 3-D
fully ventilated thin foils entering into initially calm
water. The method was later extended by (Wang
et al 1990b) and (Wang et al 1992) to predict the
performance of fully ventilated partially submerged
propellers with its shaft above the water surface.
Similar to (Furuya 1984, Furuya 1985), the method
assumed the flow to separate from both the lead-
ing edge and trailing edge of the the blade, forming
on the suction side a cavity that vents to the at-
mosphere. Discrete line vortices and sources were
placed on the face portion of the blade to simulate
the effect of blade loading and cavity thickness, re-
spectively. Line sources were also placed on the cav-
ity surface behind the trailing edge of the blade to
represent the cavity thickness in the wake. A heli-
cal surface with constant radius and pitch were used
to construct the trailing vortex sheets. The nega-
tive image method was used to account for the effect
of the free surface. The effect of the blade thickness
was neglected in the computation. Comparisons were
presented with both experimental measurements by
(Hadler & Hecker 1968) and numerical predictions by
(Furuya 1984, Furuya 1985~. The predictions were
within reasonable agreement with experimental val-
ues for a propeller with limited data range. However,
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Representative terms from entire chapter:
supercavitating propellers
substantial discrepancies were observed for another
propeller with both experimental values and numer-
ical predictions by (Furuya 1984, Furuya 1985~.
The 3-D vortex-lattice lifting surface method de-
veloped by (Kudo & Ukon 1994) and (Kudo & Kin-
nas 1995) for the analysis of supercavitating pro-
pellers has also been extended for the analysis of
surface-piercing propellers. However, the method
performs all the calculations assuming the propeller
is fully submerged, then multiplies the resulting
forces with the propeller submergence ratio. As a
result, only the mean forces can be predicted while
the complicated phenomena of blade's entry to, and
exit from, the water surface are completely ignored.
A 2-D time-marching BEM was developed by
(Savineau & Kinnas 1995) for the analysis of the flow
field around a fully ventilated partially submerged
hydrofoil. However, this method only accounts for
the hydrofoil's entry to, but not exit from, the water
surface. In addition, the negative image method was
used so the effects of water jets and change in free
surface elevation were ignored.
Objectives
The objective of this work is to extend an exist-
ing 3-D potential based boundary element method
to predict the performance of supercavitating and
surface-piercing propellers.
The method was first developed by (Kinnas &
Fine 1991) for the nonlinear analysis of flow around
partially and supercavitating 2-D hydrofoils. It was
then modified to treat partially cavitating 3-D hy-
drofoils (Fine & Kinnas 1993a). In (Kinnas & Fine
1992, Fine & Kinnas 1993b), the method was named
PROPCAV (PROPeller CAVitation) for its added
ability to analyze 3-D unsteady flow around cavi-
tating propellers. Later, (Mueller & Kinnas 1999)
modified the method to search for midchord cavita-
tion on either the back or the face of propeller blades.
Most recently, (Young & Kinnas 2001) extended the
method to predict alternating or simultaneous face
and back cavitation on conventional propeller blades
subjected to non-axisymmetric inflow. The bound-
ary element method inherently includes the effect of
non-linear thickness-loading coupling by discretizing
the blade surface instead of the mean camber sur-
face. Thus, it requires more CPU time and memory
than the lifting surface method. However, it offers a
better prediction of the flow details at the propeller
leading edge and tip than the lifting surface method.
non-axisymmetric midchordpari~al wake
effective inflow wake cavity on back side ~
An, ..,.
yor:(myz)b
- ~` ~ _ _..5
Ls-zxst
..-i~ 1 /d _
hub /
{supercavity
on face side l
supercavity on back side
Figure 1: Propeller subjected to a general inflow
wake. The blade fixed (x, y, z) and ship fixed
(xs~yS'zS) coordinate systems are shown.
FORMULATION
The general formulation for the prediction of
unsteady sheet cavitation on conventional fully sub-
merged propellers is presented in (Kinnas & Fine
1992, Fine October, 1992, Young & Kinnas 2001~.
It is summarized in this section for the sake of com-
pleteness.
Consider a cavitating propeller subject to a gen-
eral inflow wake, qw (AS ~ as ~ as ~ ~ as shown in Fig. 1.
The inflow wake is expressed in terms of the absolute
(ship fixed) system of coordinates (x,s,Ys,zs). The
inflow velocity, -tin, with respect to the blade fixed
coordinates (x, y, z), can be expressed as the sum of
the inflow wake velocity, qw, and the propeller's an-
gular velocity A, at a given location x:
qirl (x, y, Z. t) = qw (X, r, (JB—At) + W X X (1)
where r = x/~, cB = arctan~z/y), and x =
(x, y, z).
The inflow, qw, is assumed to be the effective
wake, i.e. it includes the interaction between the
vorticity in the inflow and the propeller (Choi 2000,
Kinnas et al 2000~. The resulting flow is assumed
to be incompressible, inviscid, and irrotational flow.
Therefore, the time dependent perturbation poten-
tial, ¢(x, y, z, t), can be expressed as follows:
(X, y, Z' t) = qin(X'y, Z't) + VO(X'y, Z. t) (2)
where ~ satisfies the Laplace's equation(V2
The perturbation potential, gb, at every point p
on the combined wetted blade and cavity surface,
SWB(t) U SC(t)' must satisfy Green's third identity:
2~¢ptt) =
~JrSWB(~)USC(~) [ q() ~nq(~) —G(p;q)~3 q(~]dS
Y ~ ~ ~
+I is Bird eq t) ~~,¢P¢;~) dS
for p ~ (sweat) Ascot)) (3)
where the subscript q corresponds to the variable
point in the integration. G(p; q) = 1/R(p; q) is the
Green's function with R(p; q) being the distance be-
tween points p and q. nq is the unit vector normal
to the integration surface, with the positive direction
pointing into the fluid domain. /~
OCR for page 823
Note that if s, v, and rz were located on the "ex-
act" cavity surface, then the total normal velocity,
On, would be zero. However, this is not the case
since the cavity surface is approximated with the
blade surface beneath the cavity and the wake sur-
face overlapped by the cavity. Although Vn may not
be exactly zero on the approximated cavity surface,
it is small enough to be neglected in the dynamic
boundary condition (Fine October, 1992~.
Equations 5 and 6 can be integrated to form a
quadratic equation in terms of the unknown chord-
wise perturbation velocity A. By selecting the root
which corresponds to the cavity velocity vectors that
point downstream, the following expression can be
derived:
INS = --tin S + Vv COST + sin ¢ ~/~qc ~2 - Vv2 (8)
where ~ is the angle between s and v directions, as
shown in Fig. 2.
Equation 8 can then integrated to form a Dirich-
let type boundary condition for ¢:
,}: '[ --tin S + Vv COS ~ + sin ¢~/~qc ~2 _ VV2 ~ ds
where s = sO corresponds to the cavity leading edge.
The terms ~ and ~ inside A and Vv on the right-
hand-side of Eqn. 9 are also unknowns and are deter-
mined in an iterative manner. The value of otsO, v, t)
is determined via cubic extrapolation of the unknown
potentials on the wetted blade surface immediately
upstream of the cavity.
On the cavitating wake surface, the coordinate
s is assumed to follow the streamline. It was found
that the crossflow term (rev) in the cavitating wake
region had a very small effect on the solution (Fine
October, 1992, Fine & Kinnas 1993b). Thus, the to-
tal cross flow velocity is assumed to be small, which
renders the following expression for ~ on the cavitat-
ing wake surface:
as
o+(s,v,t) = 0 (ST' V, t) + / {—qin s + Iqcl} ds
ST
(10)
where ST iS the s-direction curvilinear coordinate at
the blade trailing edge. o+ and o- represent the
potential on the upper and lower wake surface, re-
spectively. The value of ¢+ (ST, v, t) can be obtained
by applying Eqn. 9 at the blade trailing edge.
· Kinematic Boundary Condition on Cavitating Sur-
faces
The kinematic boundary condition requires that
the total velocity normal to the cavity surface to be
zero:
DO (n—has, v, t)) = (11)
[ 63 + qua, y, z, t) Vie (n—kits, v, t)) = 0
where n and h are the curvilinear coordinate and
cavity thickness normal to the blade surface, respec-
tively.
Substituting Eqn. 6 into Eqn. 11 yields the fol-
lowing partial differential equation for h on the blade
(Kinnas & Fine 1992~:
,, As - cos¢Vv] + ,:' tVv - cos¢Vs] (12)
sin2¢ (Vn—8t ~
Assuming again that the spanwise crossbow ve-
locity on the wake surface is small, the kinematic
boundary condition reduces to the following equa-
¢(s, v, t) = ¢(sO, v, t)+ (9) tion for the cavity thickness (hw) in the wake:
[ on on ~ ~~ = em As (13)
Note that hw in Eqn. 13 is defined normal to the
wake surface. In addition, the quantity hw at the
blade trailing edge is determined by interpolating the
upper and/or lower cavity surface over the blade and
computing its normal offset from the wake sheet.
The extent of the unsteady cavity is unknown
and has to be determined as part of the solution.
The cavity length at each radius r and time t is given
by the function lfr, t). For a given cavitation num-
ber, an, the cavity planform, lfr, t), must satisfy the
following condition:
~ (I (r, t); r, cony _ h (I(r, t), r, t) = 0 (14)
where ~ is the cavity height at the trailing edge of the
cavity. Equation 14 requires that the cavity closes at
its trailing edge. This requirement is the basis of
an iterative solution method that is used to find the
cavity planform (Kinnas & Fine 1993~.
Solution Algorithm
The unsteady cavity problem is solved by in-
verting Eqn. 3 subject to conditions 4, 9, 10, and
14. An iterative pressure Kutta condition (Kinnas
& Hsin 1992) is applied to ensure equality of pres-
sures at both sides of the trailing edge everywhere
on the blade. The numerical implementation is de-
scribed in detail in (Kinnas & Fine 1992~. In brief,
for a given cavity planform, Green's formula is solved
with respect to unknowns S) on the wetted blade and
hub surfaces, and unknowns ~ on the cavity sur-
face. The cavity heights on the blade and the wake
are then computed by differentiating Eqns. 12 and 13
with a second order central finite difference method.
Finally, the correct cavity planform for each time
step is obtained iteratively using a Newton-Raphson
technique which requires the cavity closure condition
(Eqn. 14) to be satisfied. It should be noted that
the split-panel method developed by (Kinnas & Fine
1993) is used to treat blade and wake panels that are
intersected by the free surface. In addition, at each
time step, the solution is only obtained for the key
blade. The influence of each of the other blades is
accounted for in a progressive manner by using the
solution from an earlier time step when the key blade
was in the position of that blade.
· Cavity Detachment Search Criterion
The cavity detachment location is determined
via an iterative algorithm. First, the initial detach-
ment lines at each time step (or blade angle) are
obtained based on the fully wetted pressure distribu-
tions. The search algorithm at each spanwise strip
begins at the section leading edge and travels down-
stream to the section trailing edge. The initial de-
tachment location for each strip is given as the first
point along the chordwise direction where the wetted
pressure is less than or equal to the vapor pressure
(i.e.,—Cp > Any. The cavity detachment locations
at each radial strip are then adjusted iteratively at
every revolution until the following smooth detach-
ment condition is satisfied:
1. The cavity has non-negative thickness at its
leading edge, and
2. The pressure on the wetted portion of the blade
upstream of the cavity should be greater than
the vapor pressure.
It can be shown that the above criterion is equiva-
lent to the Villat-Brillouin condition (Brillouin 1911,
Villat 1914).
SUPERCAVITATING PROPELLERS
Experimental evidence shows that the separated
zone behind the thick blade trailing edge forms a
closed cavity that separates from the practically ideal
irrotational flow around a supercavitating blade sec-
tion (Russel 1958~. Furthermore, the pressure within
the separated zone (also called the base pressure)
can be assumed to be uniform (Raibouchinsky 1926,
Tulin 1953~. However, in high Reynolds number
flows, the mean base pressure depends on the me-
chanics of the wake turbulence (RoshLo 1955~. This
implies that a turbulent dissipation model, such as
the one used in (Vows & Chen 1987), is necessary to
determine the mean base pressure and the extent of
the separated zone. However, the use of such models
in the prediction of unsteady 3-D cavitating propeller
flows is not practical for engineering purposes.
To simplify the physics, (Kudo & Ukon 1994) as-
sumed the supercavitating blade section to be base
ventilated (i.e. the mean base pressure equal to the
vapor pressure), and solve the steady cavitating pro-
peller problem using a 3-D vortex-lattice lifting sur-
face method. Later, (Kudo & Kinnas 1995) modified
the method to allow for a variable length separated
zone model which determines the mean base pres-
sure. However, the length of the separated zone is
arbitrarily specified by the user, and has found to
affect the pressure and cavity length near the blade
trailing edge under fully wetted and partially cavitat-
ing conditions. Furthermore, the method of (Kudo
& Kinnas 1995) cannot be applied in unsteady cavi-
tating analysis since the length of the separated zone
changes with blade angle.
In the present method, the assumption of (Kudo
& Ukon 1994) is used for the analysis of supercav-
itating propellers subjected to steady and unsteady
inflow. The base pressure is assumed to be constant
and equal to the vapor pressure, and the extent of
the separated zone at each time step is determined
iteratively like a cavity problem. The logic behind
this assumption are:
1. The base pressure should equal to the vapor
pressure in the case of supercavitation,.
2. The separated zone has to communicate with
the supercavity in the span-wise direction in the
case of mixed cavitation (i.e. one part of the
blade is wetted or partially cavitating while an-
other part is supercavitating).
3. Most supercavitating propellers operate in su-
percavitating conditions.
Hence, the present method solves for the sepa-
rated zone like an additional cavitation bubble. How-
ever, the "openness" at the blade trailing edge poses
a problem for the panel method. Thus, a small clos-
ing zone, shown in Fig. 3, is introduced. The precise
geometry of the closing zone is not important, as long
as it is inside the separated region and its trailing
edge lies on the aligned wake sheet. ~
The method is
modified so that it treats the original blade and the
closing zone as one solid body. Thus, the integral
surface in Green's formula (Eqn. 3) must now in-
cludes the blade and hub surfaces, the closing zones,
and the wake surfaces. Moreover, Eqns. 9 and 12
should also be applied over the closing zone. The
solution method is the same as that for conventional
fully submerged propellers. However, additional care
is needed to ensure the potential to be continuous
between the wetted portions of the blade, the cav-
ity surfaces, and the closing zones. Furthermore, an
additional condition which requires the cavities to
detach prior to the actual blade trailing edge is also
needed. In the force calculation, the pressure acting
on the thick blade trailing edge (shown in Fig. 4)
must also be included. This is accomplished by mul-
tiplying the separated region pressure acting normal
to the blade trailing edge with the trailing edge area.
Details of the numerical algorithm and numerical val-
idations of the method are presented in (Young &
Kinnas 2002, Young 2002~.
Fully wetted
G-~
~ blade section
_ ~ ~ r~
/ """"my
Amp., =T
Partially cavitating
cavity blade section
(P= _ ~
Supercavitating
can ty I ,\
= p\:) N\\ __
- closing zone
wake
- s~par
I
Figure 6: Discretized geometry of propeller SRI.
0.8 L 1°4
0.35
0.3
0.25
0.2
0.15
experiment 0.1
-am PROPCAV
0' ' ' I I, I, I,, 005
0.8 0.9 1 1.1 1 .2 1.3 1 .4
0.7 t
0.6
C 0.5
0.4
To
0.1
~ ~ ~ ~ ~ G
C
/.
10*K
- am. me. ~ ~ ~ o o
63 ~ 4E~-~WO
~Ceec
0.3 -
0.2
a
JA
l (
Figure 7: Comparison of the predicted and versus
measured IT, KQ, and lip for different advance co-
efficients.
SURFACE-PIERCING PROPELLERS
Since surface-piercing propellers are partially
submerged, the computational boundary must also
include the free surface. Hence, the perturbation po-
tential, Op. at every point p on the combined wet-
ted blade surface SWB(t)' ventilated cavity surface
Scot (t)USc2(t) USC3(t), and free surface SF (t), must
satisfy Green's third identity:
2~op~t)=J~is(~) [¢)q~t) anq(~) (P;q) ~nq(~)]
(15)
where S(t) _ Swatt) U Sc~(t) U SC2(t) U SC3(t) U
SF (t) is the combined surfaced as defined in the blade
section example shown on Fig. 10. nq is the unit
vector normal to the integration surface, with the
'be,'\\
Figure 8: Predicted cavity shape and cavitating pres-
sures for propeller SRI at JA = 1.3. 50x20 panels.
Uniform inflow.
I'
50X20: KT=0.1372, 1OKQ=O.4072
W-''''N'N'
70X30: K'O.1362, 1OKLQ=O.4048
\\'
70X20: KT=0.1366, KQ=0.4050
!_
80XSO: KT=0.1353, 1OKQ=O.4031
Figure 9: Convergence of cavity shape and force coef-
ficients with number of panels for JA = 1.3. Uniform
inflow.
positive direction pointing into the fluid domain.
As in the case of fully submerged propellers, the
"exact" ventilated cavity surfaces, Scat (`t) U SC2 (t) U
SC3(t), are unknown and have to be determined as
part of the solution. Thus, the ventilated cavity sur-
faces are approximated with the blade surface under-
neath the cavity, SC2(t) ~ SCatt), and the portion
of the wake surface which is overlapped by the cavity,
Scat (t) U Sc3(t) ~ Sow (t). The definition of SOB (t)
and Smart) are shown in Fig. 10.
Boundary Conditions
· Dynamic Boundary Condition on the Free Surface
and Ventilated Cavity Surfaces
The dynamic boundary condition requires that
the pressure everywhere on the free surface and on
the ventilated cavity surface to be constant and
equal to the atmospheric pressure, Palm. Redefining
If
SF P = Patm P = Patm
1
1'1~
, /Sew
"/ ¢+ known (P = Patm)
—~ unknown D
~SWB
| ~ unknown
~~ known (=—qin n)
Figure 10: Definition of "exact" and approximated
flow boundaries around a surface-piercing blade sec-
tion.
an —(Po—Pa~m)/~n2D2) as the ventilated cavi-
tation number, the dynamic boundary condition re-
duces to Eqns 9 and 10 on the on SoB(t) and Smart)'
respectively.
· Kinematic Boundary Condition on the Ventilated
Cavity Surfaces
The kinematic boundary condition on the ven-
tilated cavity surfaces renders the same expression
as Eqns. 12 and 13 on SoB(t) and SCw~t)' respec-
tively. However, for partially submerged propellers,
the cross-flow velocities are also assumed to be small
on the blade surface (i.e. Vv ~ viscose on SCB(t)~.
This reduces the 0oh term in Eqn. 12 to zero. The
justification of this assumption can be found in (Fine
October, 1992), where it is shown that the cross-flow
term (evaluated iteratively) on the blade has a very
small effect on the predicted supercavity on either a
3-D hydrofoil or a propeller blade. In addition, the
8oh term is difficult to evaluate due to the interrup-
tion of the ventilated cavity by the free surface.
· linearized Free Surface Boundary Condition on the
Free Surface
As a first step to model partially submerged pro-
pellers in 3-D, the linearized free surface boundary
condition is applied:
6~J~°((x, y, z, t) + g~33v (x, y, z, t) = 0 (~16)
at Us =—R + h (i.e. free surface)
where h and R are the blade tip immersion and blade
radius, respectively, as defined in Fig. 11. ys is the
vertical ship-fixed coordinate, defined in Fig. 1.
' ,
. ...... .... ~
. . ~
~ S~ : - split panels
-
- ~ x~S
>4 z
I J '
Lo
. .
..
Figure 11: Definition of ship-fixed (xs~yS'ZS) and
blade-fixed (x,y,z) coordinate systems.
Assuming that the infinite Froude number con-
dition (i.e. Fr = rt2D/g ~ oo) applies, Eqn. 16
reduces to:
¢(x, y, z, t) = 0 at Us =—R + h (17)
The assumption that the Froude number grows with-
out bounds is valid because partially submerged pro-
pellers usually operate at very high speeds. Studies
by (Shiba 1953, Brandt 1973, Olofsson 1996) have
also shown that the effect of Froude number is neg-
ligible for Fn~ > 4 or Fr = ngD > 2 in the
fully ventilated regime.
The Negative Image Method
Equation 17 implies that the negative image
method can be used to account for the effect of the
free surface. Consequently, only vertical motions are
allowed on the free surface. This is accomplished by
distributing sources and dipoles of equal strengths
but with negative signs on the location of the mirror
image with respect to the free surface. A schematic
example of the negative image method on a blade
section is shown in Fig. 12.
Solution Algorithm
For surface-piercing propellers, Green's formula
is only solved for the total number of submerged pan-
els on the key blade and the cavitating portion of the
key wake. The values of the 0 and ~ are set equal
to zero on the blade and wake panels that are above
the free surface. Note that the current algorithm
does not re-panel the blades and wakes at every time
step, in order to maintain computation efficiency. As
a result, there are some panels that are partially cut
by the free surface. In the present algorithm, the
~~ 1
/1 ~
1 1 1
', .~ dipole (opposite normal)
~ 1 ~~
1
/ 1
sinkl,,',`~\~\ "in
1
1 1
I \ ,,
~ I let
Figure 12: Schematic example of the negative image
method on a partially submerged blade section.
strengths of the singularities are also set equal to
zero for the partially submerged panels. Neverthe-
less, a method similar to the split-panel technique
(Kinnas & Fine 1993) can be applied to account for
the effects of these panels.
The solution algorithm for partially submerged
propellers is similar to that explained earlier for fully
submerged supercavitating propellers. However, it-
erations to determine the correct cavity lengths are
no longer necessary since the ventilated cavities are
assumed to vent to the atmosphere, as observed in
experiments.
· Ventilated Cavity Detachment Search Algorithm
Depending on the flow conditions and the blade sec-
tion geometry, the ventilated cavities may detach aft
of the blade leading edge. Thus, the cavity detach-
ment locations on the suction side of the blade are
searched for iteratively at each time step until the
smooth detachment condition is satisfied. In addi-
tion, due to the interruption of the free surface, the
following detachment conditions must also be satis-
fied for partially submerged propellers:
· The ventilated cavities must detach at or prior
to the blade trailing edge; and
· During the exit phase (i.e. when part of the Figure 14: Axial velocity distribution at the pro-
blade is departing the free surface), the venti- pelter plane. Propeller model 841-B. h/D = 0.33.
lated cavities must detach at or aft of the inter- Copied from (Olofsson 1996~.
section between the blade section and the free
surface.
It should be noted that the ventilated cavities on
the pressure side of the blade are always assumed
to detach from the blade trailing edge. It is pos-
sible to also search for cavity detachment locations
on the pressure side. However, such occurrence is
unlikely due to the high-speed operation of partially
submerged propellers.
Validation with Experiments
In order to validate the partially submerged pro-
peller formulation of the method, numerical predic-
tions for propeller model 841-B are compared with
experimental measurements collected by (Olofsson
1996~. A photograph of the partially submerged pro-
peller and the corresponding BEM model are shown
in Fig. 13. The velocity distribution at the propeller
plane is shown in Fig. 14. The experiments were
conducted at the free-surface cavitation tunnel at
KaMeWa of Sweden. Details of the experiments are
given in (Olofsson 1996~.
am_ ~
- ~ ~
_ _~
~ ~ ez)
Figure 13: Photograph of propeller model 841-B
shown in (Olofsson 1996), with corresponding BEM
model on the right.
o
on
0.4
.....................
0.6 _
0.E
flat plate
~ free surface
It.,.,,,_,,,,,,,_ _ v 1
, . ~
................................................... — `
velocity distribution in propeller plane 'I,
l
. ~ —.
a. -
0.25 0.5
V'
x
0.75 1
The current method assumes the blade to be
rigid, the cavities to be fully ventilated, and the
Ffoude number to be infinite. Thus, the following
combination of test conditions were selected to mini-
mize the effect of Fioude number, cavitation number,
and blade vibration:
shaft yaw angle: ~ = 0°
shaft inclination angle: by= 0°
blade tip immersion: h/D = 0.33
advance coefficient:
Froze number:
cavitation number:
JA = Vie = 1.0—1.2
F'l D = N/~ = 6. O
~V= ~=0.25
Note that PO is the pressure far upstream on the shaft
axis.
Comparisons of the observed and predicted ven-
tilated cavity patterns are shown in Figs. 15 to 17.
Comparisons of the measured and predicted individ-
ual blade force and moment coefficients are shown
in Fig. 18. The solid lines and the symbols in
Fig. 18 represent the load coefficients predicted by
the present method and measured in experiments, re-
spectively. (,KFX, KEY, KFZ, KMX, KMY, KMZ ~
are the six components of the individual blade force
and moment coefficients defined in the coordinate
system shown in Fig. 1.
~ = oO7 gO°, ~ soo
Figure 15: Comparison of the observed (top) and
predicted (bottom) ventilated cavity patterns for
JA = 1.2. Propeller model 841-B. 4 Blades. h/D =
0.33. 60x20 panels. /\d = 6°.
As shown in Figs. 15 to 18, the predicted venti-
lated cavity patterns and blade forces agree well with
experiments for JS = 1.2. In addition, the method
also converged quickly with time step size and grid
size, as shown in Figs. 19 and 20. However, there
is more discrepancy between the predicted and mea-
sured individual blade forces for JS = 1.0, which is
shown in Fig. 21.
if_
~ = 30°, 120°, 210°
Figure 16: Comparison of the observed (top) and
predicted (bottom) ventilated cavity patterns for
JA = 1.2. Propeller model 841-B. 4 Blades. h/D =
0.33. 60x20 panels. /\d = 6°.
9=60° 150° 240°
Figure 17: Comparison of the observed (top) and
predicted (bottom) ventilated cavity patterns for
JA = 12 Propeller model 841-B. 4 Blades. h/D =
0.33. 60x20 panels. i\§ = 6°.
The authors believe that the discrepancies are
attributed to:
.
.
Inadequate simulation of the blade entry phe-
nomena. At the instant of impact, a very strong
jet is developed near the blade leading edge,
which results into very high slamming forces. In
the current formulation, the presence of the jet
cannot be captured due to the application of the
negative image method. In other words, a non-
linear free surface model should be applied to
capture the development of the jet, so that the
added hydrodynamic force can be directly eval-
uated.
Inability of the current method to capture the
increase in free surface elevation. The overall
free surface rises due to the cavity displacement
O.O5 1
-0.04
.0.03
= -0.02
Fox (P)
·1811~1~1~1~1~1 an (P)
~ Pox A)
O KPZ (E)
0. 0 1 . ... .. .
O. tic
0.01
0.02 1 1
0.015
0.01
0.005
-0.00
0.0025
0.0025
0.005 ~
-0.025
-0.02
~~~~~ ~~~~--~ ~ KF / ~ C) -0.015
blade angle 0.01 360 it, -0.01
~ -0.005
— KM,, (P)
·I.I.I.I.I.I.I KMZ (P)
'` KM,. (E)
0 KMZ (E)
_ .K /- I
90 1 80
blade angle
270
.02
0.01
_
~60
.I-~ ; KMy (P)
KMY (E) . . ~ . . . . . . . . . .
~ .
~0 ~ ~
0 005
0 90 180 270 360
blade angle
Figure 18: Comparison of predicted (P) and mea-
sured (E) blade forces for JA = 1.2. Propeller
model 841-B. 4 Blades. h/D = 0.33. 60x20 pan-
els. /\§ = 6°.
.
effect (Olofsson 1996~. As a result, the actual
immersion of the propeller increases, which in
turn adds to the hydrodynamic blade load. This
effect can be observed in the experimental data
shown in (Olofsson 1996) for low JA values.
Inability of the current method to model the ef-
fect of blade vibrations. Blade vibration is a
resonance phenomenon which affect the blade
shapes and loadings. The effect of which is evi-
dent via the "humps" (amplified fluctuations su-
perimposed on the basic load) observed in the
experimental data shown in Figure 21. It was
also observed during experiments that the fre-
quencies of these fluctuations modulate between
the blade's fundamental frequency in air and in
water (Olofsson 1996~. However, the current
model assumes rigid body motion. Thus, the
effects of blade vibrations cannot be captured.
2-D STUDY OF FREE SURFACE EFFECTS
In order to quantify the added hydrodynamic
forces associated with jet sprays generated at the
AD = 9O
AD = 6°
AD = 3°
o
0.005` ) 90 180 to
blade angle (degrees)
~ , , 1
_ 360
Figure 19: Convergence of thrust (KT) and torque
(KQ) coefficients (per blade) with time step size.
Propeller model 841-B. JA = 1.2. 70x30 panels. 6
propeller revolutions.
blade entry and exit phase, a systematic 2-D study
has been initiated. The objective of the 2-D study is
to find a simplified approach to quantify the added
hydrodynamic forces due to slamming and change in
free surface elevations. The progression of the pro-
posed 2-D study is shown in Fig. 22.
Previous Work
The problem of a 2-D rigid wedge entering the
water was first studied by (don Karman 1929) and
(Wagner 1932~. Both assumed that the velocity field
around the wetted part of the body can be approx-
imated with the flow field around an infinitely long
flat plate. The model in (don Karman 1929) as-
sumed that the free surface is flat, while the model
in (Wagner 1932) accounted for the deformation of
the free surface. However, the similarity method of
(Wagner 1932) reduced the unsteady problem to a
steady one. Since then, the slamming problem on a
2-D body has been extensively studied by (Makie
1969, Cox 1971, Yim 1974~. In particular, (Yim
1974) applied a linearized theory to study the wa-
ter entry and exit of a thin foil and a symmetric
wedge with ventilation. Later, (Wang 1977, Wang
1979) also applied linear theory to study the verti-
cal and oblique entry of a fully ventilated foil into
a horizontal layer of water with arbitrary thickness.
The method of (Wang 1977, Wang 1979) was later
extended by (F'uruya 1984, Furnya 1985) for the per-
formance prediction of surface-piercing propellers.
More recently, the 2-D wedge entry problem was
thoroughly investigated by (Zhao & Faltinsen 1993,
-0.025
-0.02
C) -0.01 5
cur
~0.005
-0.01
o
0.005
~f ~
I I ~ I I I I , , I
C) 90 180 270 360
blade angle (degrees)
Figure 20: Convergence of thrust (KT) and torque
(KQ) coefficients (per blade) with panel discretiza-
tion. Propeller model 841-B. JA = 1.2. /\§ = 6°. 6
propeller revolutions.
Zhao & Faltinsen 1998~. They applied a boundary
element method with constant source and dipole dis-
tributions. The exact nonlinear free surface bound-
ary condition was used. A special model was used
to treat the thin jet that develops at the intersection
between the free surface and the body. The method
was verified by comparisons with similarity solutions
by (Dobrovol'skaya 1969) and asymptotic analysis by
(Wagner 19321. Similar methods were also developed
by (tin & Ho 1994, Falch 1994, Fontaine & Cointe
19971. However, the focus of the previous investiga-
tions was on slamming loads on ship hulls with small
dead rise angles and no ventilation.
Formulation
The first step for the proposed 2-D analysis in-
volves analyzing the flow around a rigid hydrofoil
entering the water at an arbitrary angle of attack.
The formulation is similar that presented in (Zhao
& Faltinsen 1993), which was derived for the vertical
water entry of a symmetric wedge without ventila-
tion.
Consider a rigid, 2-D hydrofoil entering into
initially calm water of an unbounded domain at a
constant velocity V and angle attack or, as shown
in Fig. 23. For incompressible, inviscid, and ir-
rotational flow, the perturbation potential ¢, de-
fined with respect to the undisturbed free surface
coordinates (x, y) shown in Fig. 23, at any time
t satisfies Laplace's equation in the fluid domain:
(V24(x, y, t) = 0~.
Thus, the perturbation potential on the bound-
-o~o5
-0.04
50X10 -0.03
60X20
70X30 ~ 0.02
-0.01
O. _
0.02
n n1 ~
~0.01 _
0.005
O'
...... .. .......
-0.005
n 90 180 270 360
blade angle
.os-
0.04
o 03 ~
0.02--~=
0 01 ~
_~~~~ 1:' ~~_ {~ ~
0.0 1 ~ lo
blade angle
. fox (P) ..... A ~ .. .. .. .
~I.I.I.I.I.I.' KFZ(P) ^^ ~ ~ K1:Y
~ Kp7 (E'
KM,` (P) I
_ ~I~I~I~I~I~I~I KMZ (P)
K VIX (E)
KMZ (E)
~~~ KMX
-0.02—
0.01
F~ ,
0.0025
O
al~lalelelalai KMy (P)
~ KMY (E) ~~~- ~~ ~~~ ~~ ~ K
; . ~ . it. .
''. - - ' ' ''-'- - -- - -- 1—-- ' - - -' --—-- - — - --— - - -- - . - -
. ~~ ~
-0.0025
-0 005 . . , . . 1 . . . . . I
O 90 180 270 360
blade angle
Figure 21: Comparison of predicted (P) and mea-
sured (E) blade forces for JA = 1.2. Propeller
model 841-B. 4 Blades. h/D = 0.33. 60x20 pan-
els. /\§ = 6°
ary, S(t), of the computation domain, is represented
by Green's third identity:
o~x,y,t) = Jr [_ r1,t)~G(,;~71'~' (18)
+ ,94~;'71'~) G((, rid, tic dS((, r1, t)
where G = inn, r = :/(x—(12 + (Y _ 7112 and
S(t) = SWB (t) U SF (t) U SOO (defined in Fig. 23~.
Notice that SF(t) includes the free surface and the
ventilated surface as a whole. n is the unit vector
normal to the integration surface, which points into
the fluid domain. It should be noted that for this
problem, the perturbation potential (~) is the same
as the total potential (~) since the system is defined
with respect to the undisturbed free surface coordi-
nates (x,y).
· Kinematic Boundary Condition on SF: The kine-
matic free surface condition requires fluid particles
on the free surface and ventilated surface to remain
(1) (2)
A.
(3) , (4) n
(~ Scc ~
1 so
~ 1
Figure 22: Planned progression of the 2-D nonlin-
ear study for the water entry and exit problem of a
surface-piercing hydrofoil.
on the surface:
09 + 04 00 = 0° (19)
at ~xbx by
DO
Dt
where 71(x, t) iS the vertical coordinate of the fluid which can be written as:
particle, as shown in Fig. 23.
· Dynamic Boundary Condition on SF:
On the exact free surface and ventilated surface, where
the pressure should be constant and equal to the at-
mospheric pressure:
bt + 2 [( 0~ ) + ( i9y ) ] + 90 = 0 (20)
· Combined Kinematic and Dynamic Boundary Con-
dition on SF:
Equations 19 and 20 can be combined to form ((x, t) and 71(X, t) are the horizontal and vertical co-
a system of three equations using the definition of ordinates of the fluid particle, which at t = 0 was
Ox ~
(5) it------ /
r ~
v24=o
on SF:
_
Dt ax
it\_ ' ]~ So
D71
Dt By
Dt 2 [(0X) + (0Y) ] —99
//
Figure 23: Definition of coordinate system and con-
trol surface for the water entry problem of a 2-D rigid
hydrofoil.
substantial derivative, Dt = aft + V) · V:
D: So
=
Dt fix
DO So
_ =
Dt by
52 + (0o j2~ - (21>
DF G (22)
F = ~ 71
G =
~ 1
l 2 [(~¢,)2 + (~,)2~ J (23)
located on the undisturbed free surface at (x, t = 0),
as shown in Fig. 23.
· Kinematic Boundary Condition on SWB:
The kinematic boundary condition requires the
following condition to be satisfied on the wetted body
surface:
(
2. Refine the panel distribution in highly curved
regions. Calculate Fi+1 t+i and Gi+~ ~ at new
panel midpoints.
Apply Eqns. 24 and 25 to determine the known
values of Pi+ i- i;+ L on the wetted body bound-
ary and infinite boundary, respectively.
4. Solve Green's formula, Eqn. 18, to obtain Hi,+
and ~ i+ ~ t+~ everywhere.
5. Calculate velocities, ~i+1 i+ and ~i+'- +'
at panel mid-points.
Calculate Gi+ 1 ~+~ on the free surface. ~ 6
· Pressure and Impact Force Calculation
The pressure at the wetted body surface is
calculated at panel mid-points of each time t via
Bernoulli's Equation:
;' 0.4
~loo.3
V=0.2
and
( P )i+2,t
let i+ 2 ,t 99i+:,t (31)
2 [(do) (0Y ) ~ it -
where ~i+1 t iS calculated as follows:
TV _ DO ~ ~ `9o: 2 ~ `9o) 2]
0t i+2,! Dt i+2,t L<5XJ ~ YJ ~ i+2,{
(32)
Dt in 2 ,t 2/\~ (hi+ 2 ,t+i—hi+ 2 ,t-~ ) (33)
Once the pressure has been computed, the im-
pact force can be calculated by integrating the pres-
sure over the wetted area on the solid body.
Preliminary Results
· Vertical Entry of a Symmetric Wedge
In order to validate the method, the predictions
for the 2-D wedge entry problem are first compared
with those presented in (Zhao & Faltinsen 19934.
Note that the formulation for the water entry prob-
lem of 2-D symmetric wedge is the same as that ex-
plained in above with the following exceptions:
· The wedge is symmetric with respect to the y-
axis.
· There is no ventilated cavity surface. Thus, SF
only includes the free surface.
For surface-piercing hydrofoil/propeller applica-
tions, the dead-rise angle is often very high (i.e.
a < 10°~. Thus, the case of or = 9° (highest dead-
rise angle presented in (Zhao & Faltinsen 1993~) is
selected for validation studies. The predicted free
surface elevation and pressure distribution on the
body are shown in Fig. 24. The current method com-
pares very well with predictions by (Zhao & Faltinsen
1993), which is also shown in Fig. 24.
n
-1 L
10C = 9°1
. . . . .
O 0.5 1
XIV:
present method I_
(Zhao & Faltinsen 1993) ~.
1 -0.5 0
ylvt
Figure 24: Predicted free surface elevation and pres-
sure distribution during water entry of a 2-D wedge.
00 = 9°.
· Oblique Entry of a Flat Plate
To further validate the method, numerical pre-
dictions for the oblique entry of a flat plate are also
presented. The predicted pressure distributions for
or = 5° and 8° are shown in Figs. 25 and 26, re-
spectively. Also shown in Figs. 25 and 26 are the
results obtained using the method of (Savineau &
Kinnas 1995), which applied the linearized free sur-
face boundary conditions. As expected, the current
method predicted higher forces and increased wetted
area compared to (Savineau & Kinnas 1995~.
CONCLUSIONS
A 3-D boundary element method has been ex-
tended to predict the performance of supercavitating
and surface-piercing propellers. The current method
is able to predict complex types of cavity patterns on
6 r
5
4
5,- 3
2
present method
(Savineau, 96) do,
la = sol
-
0
N\
~
present method
—— (Savineau, 96)
1 -0.8 -0.6 -0.4 -0.2 0 0.2
ylvt
1 -0.8 -0.6 -0.4 -0.2 0
ylvt
Figure 25: Pressure distribution on the wetted body Figure 26: Pressure distribution on the wetted body
surface predicted by the present method and by the surface predicted by the present method and by the
method of (Savineau 1993~. Flat plate. or = 5°.
both sides of the blade surface for conventional and
supercavitating propellers in steady and unsteady
flow. The method is also able to simulate the un-
steady separated region behind blade sections with
non-zero trailing edge thickness. For surface-piercing
propellers, the negative image method is used in the
3-D model to account for the effect of the free surface.
The method is also able to search for detachment
locations of ventilated cavities. In general, the pre-
dicted cavity planforms and propeller loadings com-
pare well with experimental measurements and ob-
servations. The method also appeared to converge
quickly with grid size and time step size.
A 2-D study using the exact free surface
boundary conditions has been initiated to quantify
the added hydrodynamic forces associated with jet
sprays during the entry phase. An overview of the
formulation and preliminary results for the 2-D study
was presented. Current efforts include the following:
· Determine the effect of prescribed pressure on
the separated zone behind non-zero trailing edge
sections in fully wetted and partially cavitating
conditions.
Complete the 2-D nonlinear study of surface-
piercing hydrofoils, and find a simplified ap-
proach to incorporate the results into the 3-D
model.
Couple the hydrodynamics with a structural
analysis model to include the effect of blade vi-
bration.
Validate the results with experimental measure-
ments, and perform convergence studies with
time and space discretizations.
.
method of (Savineau 1993~. Flat plate. of = 8°.
ACKNOWLEDGMENT
Support for this research was provided by Phase
III of the "Consortium on Cavitation Performance of
High Speed Propulsors" with the following members:
AB Volvo Penta, American Bureau of Shipping, El
Pardo Model Basin, Hyundai Maritime Research In-
stitute, John Crane-Lips Norway AS, Kamewa AB,
Michigan Wheel Corporation, Naval Surface Water
Center Carderock Division, Ulstein Propeller AS,
and VA Tech Escher Wyss GMBH.
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