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OCR for page 685

Theoretical Prediction of Midchord and Face Unsteady
Propeller Sheet Cavitation
S. A. Kinnas and N. E. Fine
Massachusetts Institute of Technology
Cambridge, USA
Abstract
In Ells worl;, first the linearized hydrofoil problem
with arbitrary ca.vit~ deta.cl~ment points is formulated
in terms of unl;no~`n source and vorticity distributions
Tl~e corresponding integral equations are inverted analyt-
ica.lly and the results are expressed in terms of integrals
of qua.r~tities which depend only on the l~yclrofoil shape.
Then, the cavitating hydrofoil problem is solved nu-
merically bar cliscretizing Else problem into point source
and vortex distributions and lay applying the boundary
conditions at appropriately selected collocation points.
I;`inally, the disrete vortex and source method is ex-
tendecl to predict unsteady propeller sleet cavitation with
arbitrary miclchord a,ncl/or face deta,cl~ment.
~ Introcluction
Helmholtz and I~irchoff t:3] more tl~a.n a. century ago.
The analysis of cavitating flows at non-zero cavitation
numbers created a lot of dixer.sitv on the cavity termina,-
tion models, i.e. Else Riabo~chinsl;y model t_S], the reen-
tra.nt jet model t7], t94], the spiral vortex models t394~ etc.
A complete description of the different cavity termination
models can be found in t394 a,ncl t3S]. The difficulty of
the hodogra,ph technique to treat general shaped bound-
aries necessitates the introduction of tl~e li~,ea.rized cavity
theory.
Linear theory was first applied by rI'uliI1 t304 to the
problem of a. superca.vita,ting symmetric section at zero
incidence and zero cavitation number. It was then a.p-
plied to general camber mea,~lines at zero cavitation num-
ber t33], and to a. sl~pelca.vita.ting flat plate at incidence
and a.rbitraly cavitation numbers (314.
Linear theory Divas slll~seq~lentl:! ext.elldecl to supercav-
itating hydrofoils of general shape at non-zero cavitation
numbers [39], (11], t9~. [S], [~>9~.
~ . . . .. .
Cavitation has always been a great concern in the de-
sign of marine propellers. A successful propeller design ~ he partially cavitating hydrofoil problem has also
is one which precludes cavitation at design conditions. been addressed in linear theory and analytical results
In recent times, however, with an increasing demand for have been produced for some s~ec.i,~.l h ~ cl, Sail ~~m~i ~ ins
higher propeller loadings and higher c~ciencies. the r~ro-
peller cavitation is fiery often unavoiclable. The tasI; of
the hyd~odyna.micist is, tl~ercfore, to predict,and con-
trol the propeller cavitation and its undesirable side ef-
fects. An analysis method for the prediction of unsteady
propeller cavitation is, therefore, an indispensable de-
si,gn tool. Furthermore. tlli.~; 1~ro~ll~l rnlrit~timn ~1
[1], [13] 7 [19], [14], [37].
The problem of a superca.~ril;atiIlg he dlofoil with arl~i-
trary cavity detachment was first formulated by Fab~lla
[S], who also gave results for ~ flat plate with different
detachment points.
Hanaol;a [15] formulated the linearized partial and su-
~ ~ _ ~ via Amp 1~~ An l~lUl1 all<(,l- percavita.ting hydrofoil problem faith al bitI ply cavit ~ de-
ysis method should lee able to treat cavities ~ hich start tachment. He also gave series representations for the cav-
on the suction side behind the leading edge towards the ita,tion number and the h~drody!na~nic coefficients when
blade midchord and/or on the pressure side, "face", of the hydrofoil slope could lee express d in therms of poly-
the propeller in front of the blade trailing e

expressed in terms of singular integral equations of un-
l;nown source and vorticity distributions. Those integral
equations are inverted analytically and expressions for
the cavitation number, the source and vorticity distribu-
tions are given in terns of integrals of functions which
depend only on the geometry of the hydrofoil. Those
integrals are then computed numerically and the cavity
shapes are finallly computed t21], t904. The same tech-
nique has also been extended for pa.rtia.1 cavities with
arbitrary deta.chn~ent t93~. The leading edge correction
has also been implemented in the formulation of the cav-
itating hydrofoil problem to account for the non-linear
foil thickness effects t2.34.
In the present work, the technique used in t214 for
supercavitating hydrofoils is extended to treat superca`- and
ities with arbitrary detachment on either Else suction side
and/or the pressure side of the l~drofoil. The effect of
the detachment point on the cavity shapes and foil pres-
sure distributions is investigated. In the case where the
supercavity detaches on the pressure side in front of the
trailing edge, an equation for the chordwise location of
the cavity detachment point is given.
The cavitating hydrofoil problem with arbitrary suc-
tion and/or pressure cavity detachment is then solved by
employing a disrcete vortex and source method.
A numerical vortex and source lattice method has
been developed at WIT for the prediction of the unsteady
propeller sheet cavitation in spatially non-uniform wakes
[25i, t4], [IS]. The computer program which implements
this method is called PUF-3.
Finally, PUF-3 is modified to predict unsteady pro-
peller sheet cavitation with arbitrary detachment on ei-
ther the pressure or the suction side of the propeller. The
effect of the location of the cavity detachment on the time
history of the cavity voluble and the car itv shapes is in-
vestiga.ted.
2 The Cavitating Hyclrofoi! -The
Analytical Method
In this section, the linearized cavitating hydrofoil prol'-
lem is formulated in terms of unl;no~vn vorticity and
source distributions. For given cavity length and spec-
ified cavity detachment points, the involved singular in-
tegra.1 equations are inverted analytically. Expressions
are then found for the corresponding cavitation launder,
vorticity and source distributions in terms of integrals of
quantities which depend only on the foil geometry.
First, the superca.vitating hydrofoil problem for three
different cavity detachment situations is considered.
2.1 Leading Eclge Detachment
Consider a hydrofoil of chord length one, subject to a
uniform flow UOO and superca`;itating at a length x = I,
as shown in Figure 1. The cavity starts at the leading
edge x = 0 on the suction side and at the trailing x = 1
on the pressure side.
686
The corresponding cavitation nuttier ~ is defined as:
~ eU2 (1)
where pOO is the ambient pressure and PI the vapor pres-
sure inside the cavity.
In the context of the linearized cavity theory the cor-
responding Hilbert prol~lem can be formulated t91] in
terms of vorticit.y and source distril~tions Eye) and qtx)
respectively, located on the x axis as shown in Figure 1.
With the use of the definitions:
_ Add
i= tT
4(x) = At ), (3)
the complete boundary value problem becomes t914:
v
Ye ' Ax) LU
Ian
~ , ~ x
o
ll
A I
us = O2 up I
1 - ~ ~ ~
~.L~ u-=O2UOO
1 v-=u-0 ~
1
l\~7(x)
1
i
!
1 \
1 1
1 1
+1 1
1
1
l
1, x
1 ~ 1 ~~t 1 ~ ibex
i \ 1
Figure 1: Supercavitating hvUl ofoil

OCR for page 685

1. Cinematic Boullda.ry Condition for z > t, where:
-2+2lI)~ig=~(X)O

1~ · ( ~ ( S)~1 + 92)d~ 0250~
~ cry
where
+ (3l(2S) ~~ · [A -'`/;~1 (17)
zs = A; t = Am; r2 = ,~ (IS)
By observing the behavior of the vorticity distribu-
tion by varying s, we conclude that the correct detach-
ment point is the one for which the vortic.ity distribution
goes to zero at ~ with zero slope. This can be seen in
Figures 3 to 5, where the cavity shapes and the vortic-
ity distributions as predicted by the presented analytical
method, are shown for different detachment points. In
Figure 3, the circulation distribution is negative at the
trailing edge ~ A(s,l) ~ O ), thus violating the condi-
tion 15. In Figure 4 the vorticity distribution is positive
everywhere on the foil ~ A(s, 1) ~ O ), but the cavity inter-
sects the foil. The correct detachment point is somewhere
between these two, and the one which satisfies both con-
ditions is the one for which the vorticity has a zero slope
at s, which is shown in Figure 5. At this point, we should
have:
A(~, 1) = 0 (19)
A different approach of deriving equation 19, is given
in Appendix A.
The detachment point is determined lair solving equa-
tion 19 with respect to ~ numerically, utilizing a Newton
R.a.phson (secant) method A. A typical case requires
about five iterations, depending on the accuracy of the
initial guesses.
The effect of the location of the detachment point on
the cavitation number and the lift and drag coefficients
is shown in Table 1. The importance of the correct face
s . ~
.200 .0136
.542 .0247
.700 .0138
CD
.
~ 0010
.
.0018
.0017
SIGH
.1224
.1329
.~248
Table 1: Lift and drag coefficient and cavitation number
for the foils shown in Figures 3, 4 and 5.
688
0. 050
0.050~
0. 150 _
o.2s8. 0OO 0.200 0.400 0.600 0.800 1.000 1.200 1.4001.~5C
Figure 3: Cavity shape and vorticity distribution -
A(s, 1) < 0
.
0.150
O. 050
O. 050
o. 2s8. 000 ' 0.'200 ' 0.'400 ' 0.'600 0.'800 ' t.'000 · 1.-200 · 1. 4001. ~t
Figure 4: Cavity shape and vorticity distribution -
A(s, 1) > 0
0.250L . . . . . . . . . . . . . . .
0.150
n non
0.050
0. 150
Vor tights do - Ail om
~ = . 5~2
0 258. 000 0. 200 0. 400 0. 600 0. 800 1. 000 1. 200 ~ . 4001c
Figure 5: Car ity shape and vorticity distribution
A(8, 1) = 0
detachment point in the prediction of the cavity extent
and the forces on the foil is apparent.
Some further discussion on the determination of the
correct cavity detachment point is given in the Section 6.

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y
up ~ U+(~)= 2
_~1 ~ _ _ ~ ~ ~
l
O lo ~ l
:* = ~—9~l~+w—2) ; 0 < x < lo
A* = By; 10 < x < 1 (26)
02 — 01 ~ F (97)
Figure 6: LIidehorcl detachment on asupereavilatingfoil. Fall d f _ 1 ~ ~6 (28)
2.3 Midchord Detachment
For the ease where the superea.vity detaches aft of the
leading edge on the suction side of the foil, as shown in
Figure 6, the linearized boundary value problem can be
formulated as follows:
The dynamic boundary condition on the cavity:
9~) 1 ~ 'I = tt+~2 ~ = 1 10 < X < 1, y = 0
side:
where:
(90)
The kinematic boundary condition off the pressure
2 27r ./ ~—r = (3i 0 < x < 1, y = 0~ `~1y
The kinema.tie boundary condition on the suction side:
q + 1 1 judo = (~)* a < x < 10, y = 0+ (92)
E)u= or A (23)
with qu being the ordinate of the upper hydrofoil surface
as shown in Figure 6.
Equations 20 and 21 can lee reduced to the following
form:
2 27r ~ ~—~ = 2 ° < .~ < 1 t2~y
2 ~ 2~ 1 6 _ ~ = 02~') 0 < x < 1 (95)
with the use of the definitions:
where up+ is the horizontal perturbation velocity on the
wetted part on the suction side of the foil.
Equations 25 and 24 are in the same form as equa-
tions 4 and 5. Therefore, to invert these equations the
same methodology can lee followed as described in see-
tion 2.1. The perturbation velocity u+w, however, is still
an unknown.
To determine u+w for O < x < lo the l;inen~atie bound-
ary condition, equation 22, is applied. The solution for
U+W is described in Appendix B.
The analysis described in this section has been ap-
plied for a VLR section t16] and the results are shown
in Figure 7. The top of Figure 7 shows the predicted
cavity shape for a ~nidehord detachment at lo = 0.2. At
the lower part of Figure 7, the corresponding total source
distribution is shown together with the thickness source
distribution. Notice that the two source distributions are
identical for O < x < lo, as recluired by equation 56.
The described theory is applied for a VLR foil t16]
for a fixed cavity length I = 1.5 and for different values
of the detachment point lo. The predicted cavity shapes
and pressure distributions on the suction side are shown
in Figures ~ to 11. The cavities in Figures 8 and 9 are
unacceptable, because they intersect the foil surface. The
cavity in Figure 11 is also unaeeeptal~le because it pro-
duces pressures in front of the detael~nent point which
are smaller than the cavity pressure. rl he correct detaeh-
ment point seems to be the one corresponding to Figure
10. It appears to be the point for which the pressure
distribution in front of the detachment point has a zero
slope. No attempt has been made by the authors, how-
ever, to generalize this condition, since the detachment
point on the suction side should be determined by the
viscous flow in front of the cavity [10], rather than lay
any other potential flow criterion. Some more discussion
on ea.vityr detachment is given in section 6.
689

OCR for page 685

0.20*
0.10.
-~.100
-~.24)* '0* O.ZOO
tame I
..6~6
..Z..
_~.Z')a
_~.S,.0
~ end t .~. t ZOO
- Cavity Source Distribution
· Chicly ess Source Distribut ion
o. 2SO _
on. -
-cp
o. oso
I
-0.050
-O. 150
-O. 2
pOO 0.200 0.400 0.600 0. 800 1.000 1.200 1. 4001. X
X
Figure 10: Cavity shape and pressure distribution on the
suction side of a supercavitating foil with cavity detach-
ment at lo = 0.07. Sable foil as in Figure ~
Figure 7: Cavity shape, total and thickness source dis-
tributions for a VLR thickness profile with NACA a=.S
meanline sUpercavitating `` ith n~idchord detachment lo = Figure 11: Car ity shape and pressure distribution on the
.20. suction side of a. supercavitating foil with cavity detach-
ment at lo = 0.10. Same foil as in leisure ~
- . 258-t
too ' o.zoo o.'.oo ' 0.'600 o.aoo l.'coo ~ zoo 1.'4COl
x
0.2so
0.150
"Cp
0. OSC
-.OSC .
-O. 150 .
-O. 25n . . . . . . . . . . . . . .
0. 000 0. 200 0. 400 0. 600 0. 800 1. 000 1. 200 1. 400 1. 5C
Figure 8: Cavity shape and pressure distribution on
the suction side of a supercavitating foil with cavity de-
tachment at lo = 0. \'LR thickness fond and NACA
a=0.8 mea.nline, maximum thicl;ness/chord=0.04, max-
imum camber/chord=0.03, pi; = 0.001613, car = 2°,
1= 1.5
0.250
o.~=
up
o. 050
- . OSC
~ BY
-.258 f-
~ A ~
m 0.200 0.400 0.600 0.800 1.000 1.200 1. 4001. 5(
X
Figure 9: Cavity shape and pressure distribution on the
suction side of a supercavitating foil Title cavity detach-
ment. at lo = 0.01. Same foil as in Figure ~
690
2.4 Partial Cavities
In the case where the cavity is smaller than the chord
of the foil, as shown in Figure 19, the linearized cavity
problem can be formulated in a similar way as in the
case of the sllpercavitating foil, in terms of vorticity and
cavity source distributions t70], [1S], Aft.
For given cavity end, l, and cavity deta.chn~ent, lo, the
corresponding cavitation Mueller, the vo~ticity and cav-
ity source distributions can be given in terms of integrals
of u+, the horizontal perturbation velocity of the fully
wetted foil, between lo and l t93~.
2.5 The Leading Edge Correction
The linearized partial cavity theory is known to predict
that, for given flow conditions, increasing the foil thicl:-
ness results in an increase in the cavity extent and vol-
ume. This is contrary to experimental evidence, the non-
linear theory [3.5], and the short cavity theory t34~.
An alternative `va.y of including the non-linear thicli-
ness effects in the linear cavity theory can be achieved via
the leading edge correction t93], Em>. It essentially con-
sists of including Lighthill's correction t96] in the formu-
lation of the linearized ca.~'it~' problem. It can be proven
[2.3] that this can be ac]lieved bar modifying the linearized
dynamic boundary, condition on the ca.`ritv from
uc = 2 {ix; on the CCl~,ity (~)9)

OCR for page 685

- c,
Figure 12: Partially cavitating hydrofoil .,.~0~,
to
2. 000 . . · · · ~ ~ · .
- - P=d Method
L~ =~~
without I`E corn
0 ' 0.200 ' o..oo U peon z/c I
Figure 13: Pressure distributions on an NACA 16-009
section with a 50% cavity at or = 3° frown panel method
L
/x + pr/' and linear theory without leading edge corrections. Cp =
uc = ( ~ \/ —~ ~ COO; on the cavity (30) p—p=, /pU2 /2
where x is Else distance from the foil leading edge and pa
is the leading edge radius.
The modified boundary value problem with the intro- -C,
auction of equation 30 has been solved and the solution
has been expressed in terms of integrals of known quan-
titles t234.
A direct comparison of the linear cavity theory, with
or without the leading edge correction, and the non-linear
theory is shown in Figures 13 and 14. The cavity shapes O ax,
as predicted by the linear theory, with or without th
leading edge correction, are added normal to the foil and
the produced foil geometry is analyzed with a poten-
tia.l based panel method t17i, where the exact l;inema.tic
boundary condition is applied on the exact foil or c< ity
surface. The pressure clistril~ut.ions produced from the
panel method are shown in Figures 1:3 arid 14, together
with the linearized pressure dist~il~.~tio~.~.s from linear the-
ory with or without tl-~e leading edge correction. In these
Figures the pressure distribution from the linear theory
with or without the leading edge correction is constant on
the cavity, since this has l~een required via the dynamic
boundary condition. The pressure distribution from the
panel method, however, is not exactly constant on the
cavity and this is a measure of the accuracy of the lin-
ear cavity theory. Comparing Figures 13 and 14, the
substantial improvement of the linear theory, when the
leading edge correction is included, becomes apparent.
691
.
/
Palm Method
Shear Theory
with At: corn
Figure 14: Pressure distributions on an NACA 16-009
section with a So-so cat ity at ~ = 3° from panel method
and linear theory with leading edge corrections. Cp =
P p~/pu2 /2

OCR for page 685

3 The Cavitating Hydrofoil- The
Numerical Method
The nun~erical method consists of discretizing the chord
end the cavity into a. finite Hunter of segments on which
the vorticity and source distributions are a.pproxi~:na.ted
with point Vortices a.nc! sources respecting ely.
The spacing of the panels is l~a.lf cosine on the foil
and constant in the wa.l;e. The a.rrangeme~t of the vortex
and source panels is shown in Figure 1.5. The following
notation is used:
VPsi = boundaries of source panels
nisi = positions of point so~`rce.s
-\'Pi = boundaries of vortex panels
Ads = position of pout vortices
Xki = position of ki'~.ematic boundary condition
collocat'.o~. point
Aid = position of dy~.amic bou~dc~ry condition
collocation po'.~2.ts
The arrangement of the vortex and source panels is
such that the expected source and vorticity singularities
at the leading edge of tl~e foil, as well as the square root
singularity of the source distribution at the trailing edge
of the cavity, are modeled accurately. The collocation
points for the application of the l;inematic and dynamic
boundary conditions are chosen such that Else Caucl~y
principal value of the involved singular integrals is com-
puted accurately. The detailed analysis for the selection
of Else panels and control points is given in (1S], t6] and
t94.
The presented nu~nerica.l Netted ``as developed orig-
ina.lly for pa.rtia,ll~r and supercaxita.ti~g l~>drofoils with
the cavities starting at the leveling edge ore the suction
side and at the tailing edge on the pressure side of the
foil id], t18], t9~.
To extend the numerical method to also predict face
and/or midc.hord supercavities, we assume that the de-
tachment points are Xs on the suction side and ,X'p on
the pressure side. The points Vs and X'p coincide with
any of the source panel boundaries ,X'psi on the foil.
By separating the total source q into the thickness
source qC and the calcite source I, the corresponding
boundary integral equations become:
The kinematic boundary conditions
2 + 27r j; ~ - ~ ~ ~ do
~ = 0+ 0 < x < His
c = vortex pOSitions
I. .
= source positions
0 = dynamic c. pi
~ = kinematic c. p.
X = 0 X.
~ to tot I
_ . ~ ''1~ ~ 1~ ~ 1 ~ ~ i~ ~ j
I 1 1 1
X, 1 1
I 1
31 Figure 15: Discrete singularities method for superca.`i-
( ) tasting foil `` ith arbitrary detachment points
qc 1 f1 7(~)d: = t: dl1 ?J = 0- 0 < X ~ -~v
where 11(~) is the foil mean ca,ml~er surface.
The dynamic boundary conditions
—UOO 2 + ~~—27r j; , —x
(3:~)
— Uth ~ = 0 AS < X < I
(3.1)
2 2 27; 1; ~ — T lath ~ = 0 ~ < X < 1
(3,5)
where uth is the horizontal perturl~a,tion velocity due to
foil thickness, given as:
lath = 1 jIqw(~)df (36)
27r 0 ~—~
To discretize the above integral equations, we mal;e
the following definitions:
· N = number of discrete vortices
· M = number of discrete cavity sources Qi
NS = number of fully wetted panels upstream of
Xs
(39) · NP
Alp
692
= number of fully wetted panels upstream Or

OCR for page 685

· M discrete cavity sources Qi
The strengths of the discrete vortices and cavity sources
are related to the corresponding vortieity and source dis- . .
.1 . ~ ll · 1 cavitation number a
trll~utlons as IOllOWS:
pi = )(-\V' ~ ('~Pi+~ - 'HEY') (3
Qi = 4: ('~Pi+~ - 'APE
Allele If is the mean goalie of the cavity source at the
corresponding source panel.
At this point, we will assuage, without loss of gener-
a.lity, that UOO = 1.
The discretizecl boundary conditions become:
The kinematic boundary conditions
a) On the suction sidle:
Qi 1 N
_
IS _,~S are 9~ ~ V~ -
i = 1, , NS (39)
- ·Yv' ( d~ ) i
b) On the pressure side:
_ Qi _ 1 it, Fj _ {dr1~1
2(`Xpi+~—Vpi ~c' Or I= -Yki—Xvj \y dX J i
i = 1, ..., lVP (~40)
where of is defined as: of = qc/qc(\si) and is approx
i~na.ted Fitly its value for a flat plate cavitating at the
same cavity length [6] and t94.
Tl~e dynamic boundary conditions
a) On the suction side:
al
_ IS + rz + ~ ~ Qi ~
i = NS+1, , IlI (41)
b) On the pressure side:
a Pi
2 2( ~pi+:—gyp. ~
The cavity closure condition:
~1
~Qi=0
i=1
There are N+~+1 unI;nov~ns:
· N discrete vortices lTi
27r ~£ i, X ttth
i = I\rP + 1, ? N (~4~>)
There are also N+LI+1 equations:
· NP + NS kinematic boundary conditions
.
M - NP + N - NS - 1 dynamic boundary conditions
· 1 cavity closure condition
· 1 equation relating F~ to Qua
The last equation, which relates the discrete singu-
la.rities Qua and Hi, replaces the Fist d~nan~ic boundary
condition [1Si. However, in the case of midchord detach-
ment, there is no dynamic l~ounda.ry condition to be sat-
isfied on the first source panel and this relation is not
applied.
The convergence of the described numerical method
is shown in Table ~ for different numbers of elements on
the foil. The ana.lytica.l results shown in Table 1 have
been found by using the analysis described in Section 2.
Finally, the predicted cavity shapes front the analyt-
ical and the numerical method are shown in Figure 16.
..
~ of ElemntS
5
0
20
do
80
100
.-
N~rerical
Sit
.3057
.2082
~ .
.2111
.2131
.2095
_ .2095
Analytical
S.: -
.2357
.2070
.2110
.2131
.2100
.2099
Volure
.0988
.0883
.0852
.0843
.0858
.0859
Analytical
Volure
.0771
.0872
.0853
.0844
. 0858
.0858
Table 9: Convergence of the numerical method. Su-
(43) percavitating Joul;owsl;i thiclir~ess form with parabolic
meanline, maximum tl~icl;ness/chord=0.04, maximum
camber/cl~ord=0.09, cat = 3°,1 = 1.5
693

OCR for page 685

0.1SO~
O. OSO
- . OSO,
-.~
- HULUlC
~ ~~ W" Em-, DISCARD] "Stall
~7//~/m7~ ; ~
V.l..~ UNLESS P - EMU Ace. - ~-2
DEtACH~J 061 raft IS,Y£R SIDE ~ X-. ~
DESACRNW 011 rue UPPER SIDE ~ Xe. 1
0.2S8 300 0.200 0.;00 0.600 0.~ - I.~ t.~ t.4~1. ,
Figure 16: Cavity shapes from numerical and analytical
method
4 The Unsteacly Cavitating Pro-
peller Numerical Method.
A numerical method has been developed for the unsteady
sheet cavitation of marine propellers in spatially non-
uniform wakes t95i, t4i, t1Si. The corresponding com-
puter code is called PUF-3. The complete three di~nen-
sional linearized unsteady cavity problem is solved for
given propeller geometry, inflow wake and cavitation num-
ber. The propeller cavitation nuder is defined as:
where:
In = e f2D2 (~44)
· Pshaj~ = pressure at the axis of the propeller shaft
· Pv = Vapor pressure
· n = propeller revolutions
· D = propeller diameter
The flow around the blades and the cavities is rnod-
eled by a lattice of vortices and line sources located on
the mean camber surface of the blades and their trail-
ing vortex wal;es, as shown for one blade in Figure 17.
The chordwise arrangement of the vortices and sources
is the same as in the hydrofoil case, described in Section
3. The spanwise spacing is constant with quarter inset
at the tip. Details of the numerical grid can be found in
t254 and t18~.
The time.history of the cavity shapes is determined
for each blade strip by applying the three-dimensional
linearized unsteady cavity boundary conditions t25~. The
extent of the cavity on each strip is detern~ined iteratively
until the pressure on the cavity becomes equal to the
vapor pressure Pa The effect of the other strips on tl~e
same blade as shell as on the other I'lades is accounted
for in an iterative sense.
\
Figure 17: Numerical grid on one propeller blade and its
wake.
The leading edge correction, described in section 2.5,
hats also been implemented in the numerical method for
the propeller, in order to account for the non-linear blade
thickness effects [1Si.
In the present work, the numerical method for the
unsteady propeller cavitation is extended to predict cav-
ities with prescribed midchord and/or face detachment.
This has been accomplished by a direct application in
the propeller problem of the numerical method for mid-
chord and face hydrofoil cavitation, which eras described
in Section 3.
The modified PUF-3 has been applied for the DTRC
N4497 propeller t194. The advance coefficient is J =
l7SHIP/n/D = .S and the cavitation number 0.n = 1.5.
The time history of the cavity volume is shown in Figure
18 for different detachment points on the suction side of
the blades. The three-dimensional Respective plots for
some blade sections and their cavities are also shown in
Figures 19 to 21 for diFererlt deta.cl~ment points. Tl~ose
figures show the effect of the detachment point on the
cavity extent and shape to be substantial.
694

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~ -
nln
Is 8
.035
.030
Suction side detachment
in ~ of local chord length
0. 7~ _ -
1 it, a\
Blade Angle in degrees
:120
Figure 18: Cavity volume per blade fol different de-
tacl~ment points on the suction side of the N4497 pro-
peller as predicted by the modified PUF-3, An = 1.5,
J= VSHIP/n/D = 0.~.
W=7
pressure side
suction side
Figure 19: Cavity shapes for propeller N4497 at blade
sections No. 3, 5 and 7 as predicted by the modified
PUF-3. Detachment on the suction side, at 0.7% of the
local chord; Blade angle = 12° from the top, On = 1.5,
~ = VSHIP/n/D = 0.8
Figure 20: Cavity shapes for propeller N4497 at blade
sections No. 3, 5 and 7 as predicted by the modified
PUF-3. Detachment on the suction side, at 3.2% of the
local chord; Blade angle = 12° from the top, An = 1.5,
J = VSHIP/n/D = 0.8
f;7
Figure 21: Cavity shapes for propeller N4497 at blade
sections No. 3, 5 and 7 as predicted by the modified
PUF-3. Detachment old the suction side, at 8.4% of the
local chord; Blade angle = 12° fiom tile top, an = 1.5,
J = VSHIP/n/D = 0.8
695

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5 Conclusions
The following has been accon~plisl~ed in the presented
work:
The cavitating genera,! shape h drofoil problems with
arbitrary detachments, has been formulated in terms
of singular integral equations of unknown source
and vorticity distributions. Those equations are in-
verted analytically and the cavitation number, cav-
ity shapes and pressure distributions, are expressed
in terms of integrals of known quantities.
The effect of the detachment point on the cavity
solution has been investigated. In the case where
a supercavity detaches for~ravrd of the trailing on
the pressure side of a. hydrofoil, an equation for
the location of the cavity detachment point has
been found.
· A numerical discrete vortex and source method hats
been developed to predict the cavitation on h~dro-
foils with a.rbit~a.ry~ cavity detachment points.
The numerical method has been extended to pre-
dict unsteady propeller sheet cavitation with ar-
bitrary midchord and/or face cavity detachment.
The effect of the location of the cavity detachment
on the cavity volume and the cavity shapes has
been investigated.
6 Future Research
.
Perform experiments on car ita,ting propellers which
show midcl~ord and Ol' face cavity detachment. De-
termine the detachment lilies from the experiment
and run the modified PUF-3 `` ith those detachment
lines as input. Compare the predicted car ity shapes
and propeller forces from pITE-3 with else expe~i-
ment.
· Employ the cavity detachment criteria. in PUF-3.
7 Acknowledgements
Support of this research was provided by the AB Volvo-
Penta Coorporation. At this point the authors wish to
thanl; Professor Justin E. I(erwin of WIT, Mr. Lennart
Brandt and LIr. Ted R.osendal of Volvo-Penta for Blair
valuable comments and discussions during the course of
this work.
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The following items need to be addressed in order to
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Perform experiments on cavitating hydrofoils with
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in order to validate the presented theoretical re-
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Improve on the prediction of the cavity detachment
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[39] T.Y. Wu. A demote on tile Linear andl1Vonlinear T1le-
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Appendix A
Details of Face Detachment
The Villat-Brillouin condition t36i, [53 at the cavity Appendix B
detachment point requires the cavity to have the same
slope and curvature with the foil. This condition satis-
fies, locally, the requirements that the cavity does not
intersect the foil and that the pressures on the foil are
larger than the cavity pressure. If tic and Of are the or-
dinates of the cavity and the foil at the vicinity of the
separation point s, as shown in Figure 2, then we should
have:
=
dyc( +) dYl( -) o
d2yc( +) 42yc( ~) O (46)
Equations 45 and 46, via the lcinen~atic boundary con-
dition on either the cavity or the foil, become:
Finally, by using equations 50, 52 and 53, we get:
[do] 2 ~ (54)
Thus, in order for the condition 4S to be valid we
should have:
A(s, 1) = 0 (55)
Details of Midchord Detachment
The solution By and q to the system of equations 24
and 25, is given by the expressions 10, 11 and 12 where
43` has to be replaced by 02, as defined in equation 27.
To determine the unknown UC+w for O ~ ~ ~ lo, the
kinematic boundary condition 22 must be applied on the
upper wetted part of the hydrofoil.
Combining equations 22 with 21 it can be proven that
`45' the condition 22 is equivalent to:
qfx)=qu,( ~ for O<~

where
to
~1(~) d—f If
and
(qw—~o[J~4O)dy
(1+712)(71—Z)
--- (63)
N(Z) d f | \/~/7 · R(r1)dI1 (64)
Equation 62 is the solution of the integral equation
59 which satisfies the condition that up+, - 2 = 0 at lo.
The cavitation number, a, is obtained by applying
the cavity closure condition 7. It can be shown that t9~:
31
= a0—
with the following definitions:
(65)
1`l def 84r. 1~/~ ~+17~/7i~ 3]F(~)d71
and
N def ~ Mar
where:
and
(66)
r3(r2 + 1) ,/ i. V; Hi+ 9~-NF(~)d~
(67)
]/IF (z) d f /. ~/2 ~2
o
HI (id (6S)
l
N ( ) def I )/(~ + t)(to—no) N( )d (69)
o
The integrations in equations 66, 67, 68 and 69, are
performed numerically with special care tal;en at the sin-
gularities of the involved integrands t9;.
699

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DISCUSSION
by H. Kato
I appreciate the authors' effort in
calculating sheet type cavitation. The authors
did not compare their results with
experiments. Therefore I am afraid that some
of the assumptions and conditions are
different from experimental observations.
Firstly the sheet cavitation is closely
related with boundary layer separation. The
leading edge of sheet cavitation coincides
with the separation point of boundary layer
according to the observation by Franc and
Michel[10] and Yamaguchi and Kato [A1,A2]. We
can not choose the location of the cavity
leading edge arbitrarily as the authors did in
the paper.
The authors also mention that the pressure
distribution shown as Fig.11 is not realistic.
However, we usually observe a negative
pressure peak in the front of the cavity where
the pressure is lower than the cavity
pressure.
The third point I would like to point out
is the cavity closure condition; Eq.(7). A
sheet cavitation is followed by wake flow
which can not be neglected in many cases. The
calculation under the assumption of Eq.(7)
does not agree with the experiment especially
when the sheet cavity length approaches to the
foil length.
[Al] H. Yamaguchi and H. Kato: A Study on a
Supercavitating Hydrofoil with Rounded
Nose, Naval Architecture and Ocean
Engineering, Soc. Naval Arch. Japan,
Vol.20 (1982).
[A2] H. Yamaguchi and H. Kato: On Application
of Nonlinear Cavity Flow Theory to Thick
Foil Sections, 2nd Int. Conf. Cavitation,
I Mech E, Edinburgh, (1983) pp.l67-174.
Author's Reply
First, we want to thank Prof.Kato taking
the time to read our paper and for making
comments on it.
It is correct that we did not compare the
results of our method with experimental
results, and we are actually planning, as
stated in Section 6 of our paper, a systematic
series of experiments in the future. The
objective of this paper was to produce a
consistent and convergent numerical method for
the midchord and face unsteady propeller
cavitation.
Concerning Prof. Kato's comment on the
pressure distribution of Fig.ll, we do not
state that the pressure distribution "is not
realistic". We rather say that it is
"unacceptable" according to the conditions
imposed in the beginning of Section 2.2. In
addition, at the end of Section 2.3 as well in
Section 6, we also state, as does he, that the
midchord detachment point in front of the
cavity[10].
Finally, in eq.(7), we assumed cavity
closure at the trailing edge of the cavity.
this assumption is in accordance with a
linearized Riaboushinsky or reentrant jet an
cavity model. We agree, however, that more
physical model, is an open cavity model with
the "openness" supplied from further knowledge
of the cavity viscous wake. This "openness"
does not affect much the predicted cavity
shape and the cavitation number, in the case
of supercavitating flows. Thus, in the
presented analysis of the supercavitating
hydrofoils, we decided for simplicity, to take
the cavity " openness" equal to zero. For
partially cavitating hydrofoils, however, the
cavity wake is important and should be
included. An experimental analysis of the
cavity wake in the case of partially
cavitating hydrofoils is included in [9].
700