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OCR for page 654
An Experimental investigation of Cavitation inception and
Development of Partial Sheet Cavities on I we -
Dimensional Hydrofoils.
J. Astolfi, P. Dorange, JOB. Leroux, JAY Billard
(Ecole Navale, France)
ABSTRACT
The main results of m e perimental st dy
concerning the inception md the development of
partial sheet cavities on four Rio denenzoncl
hyd of oils are presented The conditions of
cavitation inception are measured together with She
minim m p~essme coefhcient in some cases The
effect of the Rey olds n mber is studied
Concerning cavitation development, depending on
the cavitation member or the male of incidence
various types of cavitation are observed es partial
sheet cavities, bubble, fingers, patches or
supercavitation patterns
For partial sheet cavities, the cavity lengths
w re measured on th foils for various conditions of
cavitation n mber md male of incidence A
attempt to correlate th cavity length date is studied
et She end of the pap r
INTRODUCTION
The physical process associated with She
inception md She development of ca itation is
complex md basic experiments on two-dimensiomal
hyd of oils remain m effective way to study She
fundamentals of cavitation md to mderstmd
propellers for mst mce Partial sheet ca ity is one of
the cavitation patterns that occurs on c two-
dimensional hydrofoil t picclly near th lecdmg
edge it corre ponds to th situation for which c
cavity of vapor extends own c Faction of She
hyd of oil's surface
For inception of partial cavities, the two
questions which arise are where md when do
cavitation occur ? it is generally acepted that
cavitation occurs on c f 11-scale lifting surface et
the position of the minim m pressure md when She
local minim m pressure falls to or below She vapor
pressure of She flowing liquid But m m my cases,
particularly on scale models, the incipie t cavitation
m mber ti is fo md to be different (often smeller)
from the opposite of th minimum pressme
coefhcient, Cpm~ generally obtained theoretically
by computation for m inviscid flow (A ndt 1981)
The main reason is gnat on scale models such es
hyd of oils or h~dd:3n i, long or short separation
bubbles, occurring et the leading edge, i fluence the
inception condtiorr, .. Aakeri (1975), ..;rtlreri et al
(1981) When et f 11 scale, he flow separation
bubbles are expected to disappear md h msition
will occur near the leading edge This phenomenon
is know to complicate the con canon of model Mnd
f 11-sccle cavitation scclmg, H mg md Peterson
1976, Billet et cl 1981) Aakmi (1975), Kilt
(1984) md Franc md Mich I (1985) indicated chat
attached sheet cavity development on hydrofoils
requi es the presence of c Seminar separation in
that case, according to Kilt (1984) She scenario is
chat "b md type cavitation occurred es bubbles w re
enbamed th ough the rernchrnent region, wh re
th y w re push d upsheam by She reverse flow" in
this quiescent legion, the bubbles increase
progressively es the cavitation n mber decreases
md form c vapor cavity attached near the leading
edge How ver, Cope m md Katz (2000) arg ed
that sheet cavitation c m occur also on attached
flow in that case, ocher parameters c m induce
favorable conditions, for m. tnce local pressure
di tobution, local surface imperfections, surface
m lens,
Concerning partial sh et cavity development, the
following pomts need to be st died:
the me m characteristics of th vapor cavity,
the inspection of the flow near the detachment
point md She surface of She cavity,
the examination of the closure ~ on of the
cavity together with She unsteadiness of She
cavity, ff my
M my authors have add essed these pomts in She
past Measurements or compntstiorr hue been
performed to determine the me m char cteristics of
sheet cavities, for e tmple le gth, height or vol me
J' et 91 1993, D shp mde et al 1994, Kimonos et al
1994, Farhat 1994, D mg md K iper 1998, Dor mge
et 911998) But it seems 60st little has been done to
compare experiment md computation results
i tensively
Arakeri (1975) md Tsssin Leger md Ceccio
(1998) have st died th cavity detachment region
The latter carefully e tmined She separated flow
over 9 t ries of bodies (mchding hyd of oils) near
th front of She midchord attached cavity
(occurrmg It 9 position of 37-42% of the foil chord
OCR for page 655
ienf h) They obser sd 6~t, m c6 it~ting flow, 6he
c6 ity sep6 6tion occurs up tresm of th sep6 6tion
of 6he bo md6 y kyer m the none6 it~tmg flow
Finclly, cert~in experiments h6 s focused on 6he
cloYure region of sh et c6 ities to tudy 6he
mechmisms responsible of cloud c6 it~tion 61cog
with sheet c6 ity dest~bilization: C611er~s~v4 et 61
(1998), K6~~arssmi et 61 (1998), Laberteaux md
C ccio (1998), 2hmg et 61 (1998), Kjeldx4n et 61
(1999), Gop61 m md Katz (2000) The presence of 6
re sotrmt jet is 6 reason ~ut probably not mique)
for sheet c6 ity de sbb i liza ti on md 6he re s 5t mg
cloud c6 itstioa
From 6 m meric61 pomt of view, impro sments
h6 s been prog essi dy inhoduced for m meric61
prediction of sheet ca ibtion tskmg into eco mt
the iscous 6 pects of the flow, md steady or
msteady c6 ibtion Kubob et 61 1992, Kimus md
Fire 1993, Deshpmde et 61 1994, Kimus 1998
smong odhers) Kimus et 61 (1994) show d 6~t 6he
predicted c6 ity extend md vohme incre6 x4d wi6h
the Rey olds n mber (for R = 2x106 to R =
2 x 107) md 6~t 6hey 6 e low r th~n th mvieid
predictioa R cently Brew r md Kim~s (1998)
show d 61sc that the tmmel w611s (confrement)
could modify c6 itation de dopment of p6 ti61
sheet ca ities considerably h p6 ticohr c6 ity
ienf hs 6 e fo md to be much 16 ger when tsking 6he
tumx41 w611s into ecomt Alhough mmeric61
simoktions pro ide sry ux4fu1 mformation, 6he
validation of the results obbmed though m merical
medhods reeds comp6 ison with e perimenbl
res 5ts
All th x4 works show that 6 carefu1
exper im enbl e sm inat io n of c6 ibt ion co ndit ions
togedher with th mspection of the flow re6 6he
suriee of foils p6 tic 56 Iy re6 the leading edge
for 6tbched sheet c6 itation) srv4 still recess6 y to
st dyc6 ibtion The6imofthispaperistop~esent
experimenbl results of 6 procedure of ir~sstigation
concermog the mception md the dew410pment of
p6 ti61 c6 ities on various hyd of oils Correktions
6 e tudied to provide mpi ic61 formulations for
the c6 ity lengths, which may pro s usef I for
f sther e smirurtion md comp6 ison with mmmeric61
res 5ts
EXPERIMENTAL APPARATUS
Test fadlity
The experiments w ~e conducted m 6he B ole
N6~1e C6 itation Tcmx41 fitted with 6 Im long md
h=0 192m wide square cross test section
Velocities of up to I 5 m/s md p~v4ssmv4s betw en 30
mb6 md 3 b6 m be 6chie sd The experiments
w re conducted on the hyd of oils show on Fig I
Th ee of 6hem 6 e of the NACA66 fsmily md h6 4
detemm med using the d6 t6 of V6 lentme (1974) Two
of th m h6 s the wme rekti s m6 im m 6hick ess
of 6%, but diffe~ent chord lengths of 100 mm 6md
150 mm re pecti dy The thi d has 6 ~elati s
m6 im m thick ess of 12% 6md 6 chord length of
100 mm Th fo sth h~s 6m Eppler section E817)
wi6hamasim mrekti sthick essofll%,6mdwas
6heoretic611y desig cd to impro s perform6 nce with
re pect to c6 itstion inception Eppler 1990) The
6heoretic61 c6 itation-fiee buckets of 6he foils 6 e
show on Fig 2 The materi61 for 6he foils was 6
polished Inox steel with 6 quasi v4ro-roughmess
0}=OOSxlO5m)
0 o:
01
005 :~
o:
04 06 08 1
04 06 08 1
0 o: 04 06 08 1
Fig. I Trsted foil sections. From top to bottom:
NACA66-6%, NACA66-12%, Eppler E817.
E cepted for th NACA66-6% foils, th foils
wsre mo mted such that th suction side of 6he foil
w6 s in fi ont of 6he horizontsl lowsr-w611 of the te t
section Note 6~t 6he suction side is referred to 6s
6he most cambered side of th foil They wsre
ckmped on ore of the wxtic61 w611s of 6he te t
section On the other w611, 6he foils w4'v4 sustsired
by ux4 of on 6 is (5 mm diameter) positiored 6t
xt = 0 25 A mechamic61 mo mting system em~bled
6he foil to rotste Y 6 gi sn 6mgle of inciderws on 6m
6 is of rotstion located 61sc 6t xt = 0 25 D e to
mw4rtainties in positiomog the foils, it was
eximated 6~t the 6mgle of mciderws was know
wi6h 6m 6ecm ey of + 0 14°
Fig. 2 TheoretieYI eurvr4 of - Cp,,.
The c6 itstion mception (or desinence)
conditions we~v4 obtsmed by visual impection of the
flow field ill mim~ted with 6 strobo eopic hfht The
6mfJe of incidence for cavibtion irwsption, *(~,
2
OCR for page 656
was determined by progressively increasing the
angle of incidence for a constant cavitation number
until cavitation was visible to the naked eye. Then
the desinent condition, A, was determined by
decreasing the angle of incidence until cavitation
disappeared. For partial sheet cavities, the cavity
length 1, was estimated from the cavity profiles
determined by image processing of visualizations
obtained by a laser sheet, formed by a 1W Argon
Laser source focused through a cylindrical lens. It
consisted of computing the mean value of the gray
level of a given pixel from about 20 acquired
images and of determining the frontier of the cavity.
It was then superimposed over the image of the foil
without cavitation to determine the relative lengths
of the cavity 1* = 11c (Fig. 3)
leading edge where the minimum of the pressure
coefficient is expected to occur. The flow outside
the thin boundary layer near the surface of the foil
may be considered as a potential flow. Thus the
Bernoulli equation can be used to determine the
coefficient of pressure from the local velocity.
Assuming that the normal pressure gradient across
the boundary layer is close to zero, the local
pressure coefficient on the surface of the foil can be
computed with Cp = 1- (Ue / Uoo) 2, where Ue is the
maximum velocity on the velocity profile along a
normal to the foil surface. The maximum value Ue
is assumed to be outside the boundary layer.
.
0.2
0.15
0.~
0.05
0
Ye
Fig. 3 Example of an image obtained from the -0.05
superposition of the surface foil trace image (non
cavitating flow) and the cavity image. Flow is
from the right.
In order to measure the near surface pressure
distribution, velocity measurements were performed
on both the NACA66 12% foil and the Eppler foil.
The longitudinal and vertical velocities were
measured on the suction side of the foil using a
DANTEC two component LDA system, providing a
0.5 mm long (Z direction) and 0.04 mm wide (X
and Y directions) measuring volume. It was coupled
with two DANTEC enhanced Burst Spectrum
Analyzers and a DANTEC Burstware software. The
remote mechanical positioning system had a
minimum translation step of 16 ~m. The origin
(X= 0, Y= 0) of the positioning system was at the
leading edge of the foil at zero angle of incidence.
Measurements were performed at Z= 45 mm
(spanwise direction) from the front wall of the test
section. The laser beams were aligned with the
spanwise direction allowing us to approach closely
the surface foil. Velocities were mapped along lines
normal to the foil surface. The mesh for the velocity
measurements was computed previously and rotated
with the foil in such a way that the measurements
were performed at the same location relatively to
the foil surface when changing the foil incidence.
The spacing between the measurement stations was
selected so that the definition of the velocity profile
was accurate in the vicinity of the leading edge and
of the foil surface to detect the minimum of the
pressure coefficient (see Fig.44. The normal lines
were selected to be located for x~ close to the
'4~411
l
0 0.05 0.1 0.15 0.2 0.25
Fig. 4 Example of the velocity field measured
near the leading edge. NACA66-6 % foil.
Re=0.4xlO6.
RESULTS
Cavitation inception
NACA 6612% 100 mm
Fig. 5 summarizes the conditions for inception
and desinence of cavitation for the N66-12% foil.
The abscissa denotes the cavitation number and the
ordinate denotes the angle of incidence for which
cavitation appears, ohm) Fig. 5.a, or vanishes,
~~) Fig. 5.b. The figure includes data obtained for
1
O 1
6 _
4 _
~ _
0t 0
-2
-4
. .
a) 1ncepnon
W41:__
pressure side ~=: ~
3 4
_ 1 1 ~
0 1 2
6
3
OCR for page 657
_ T
b) dest mce ~ ~
, ~
Fig. 5 ExperimentYI angle of cavitadon
ineepdon (deshent) versus the eavitadon
mumber for various Reynolds numbers.
fNACA66, 12%, Idd mm chord length).
various R y olds numbers togedher with 6he
themv4ticcl valu4s of th mmimum pressun4
coefficient, -Cp,,,,, As show, c dfffere 2e still
exi ts between the mception md desim4D2e
conditions 7be differffs2e of mgles betwoen
i 2eption cod desim4D2e is of ctout fi y = 0 5° 7he
diffeu4D2e betwen *(~ md c~(~ me ms 6~t et c
gi on olocity 6he sbtic presYcre u4quiu4d to indu e
cavitation must be sig ificmtly lowor 6 m that
requi cd to elimim~te it Howo or, the diffeu4D2e
c m be exphfin by the process of cc itation detection
based on visu~lizations which c m i fer bics on 6he
mgle of cavibtion mception it c m be cssumed that
the reel cc ibtion i 2eption (micro bu bles not
observm le) occu s before it is visuslly detected
56t , ~
Fig. 6 Angle of cavitadon ineption and
deshene as a function of the Reynolds mumber
for a = 1.8. Comparison with the theoretie
angle of heeption for Cp.,~,=1.8.
Fig 5 shows also 6~t 6he irwoption (or desment)
cu os for th su tion side tend to be closer to 6he
themv4ticcl cu o es 6he R y olds number mcrecY4s
7 he s sme tende 2y is obser od on the pu4ssun4 side
(lowor part of 6he cu os), but the discrepsD2y
betwoen 6he e periment md th th ory u4mcim
large e on for 6he h~rge t R y olds number 7he
i duo 2e of the R y olds number is depicted
cleYIy on Fig 6 As show, * (t I 8) md
c~(t 1 8) demv4cse when the R y olds number
i xv4cses, md are cbY4 to the 6heoretical prediction
( y= 4 65° for -Cpm~= 1 8) when 6he R y olds
°n~
number is equ~l
Fig 7)
0 4 (a, a~y a~
02 ~ \4
^\
o~
^~4
~4 06 08 lO6Pe I
or larg r 6 m 0 8xlO (see also
~ di2 ~ 4ml<~ib ~
Fig. 7 Relative difference hetwen the
e~rpenmentYI and theoretieYI angle of ineption
as a funedon of the Reynolds mumber for two
hydrofods.
The pu4ssue coefficient dishibution on the
su2tion side resuting from 6he olocity
mecsu ements is show on Figs 8 for
R = 0 4 x 106 md Re = 0 8 x l O~ The presYcre
dixribution e h~bits c shYp peak r~oar 6he lecding
edge Th peak mcg itude (Cp,,~n)e 2 iDorecY s md
its locYion (de 2ted x*~) mo os towards the
lecding edge when mcrecsing the R y olds number
The valu4s are repo ted on TY le I to be compared
to th cavibtion conditions For R = 0 8 x 106, the
mcg itude of (-CP,,~n) 2 is foumd to be close to the
desment cavitstion number md to 6he theoreticcl
valu4 of - Cp,,,~ for m mviscid umboumded flow
For Re = 0 4 x 106, (-CP..~D) 2 is foumd to be larger
6 m t, both valu4s bemg lowor th~n 6he theoreticcl
valu4 of Cp,,.~ The diffeu4D2e is of en ctinbuted to
6he pu4serw4 of c lecding edge xparction bubble
which is not bken mto a.2coumt for m i 2id flow
O
Cp
~ oc=6'
~r
00 02 04
· Re=04x 106
^Re=08x 106
06 08 1 0
x/c
Fig. 8 Pressure eoeffieieat on the Yuetion side of
the NACA66 12% foil.
The photog cphs on Fig 9 showing the
cavitmion pattem et 6he lecdmg edg tend to
4
OCR for page 658
confirm the presence of a separation bubble.
Moreover, at inception for Re= 0.4x106 and
Re = 0.8x106, Fig.9.a shows that a difference exists
between the cavitation patterns. As shown, for
Re= 0.4x106, glossy longitudinal cavities
characterize cavitation inception whereas, for
Re = 0.8x106, the inception appears as a narrow
spanwise cavitation band. Moreover, the location of
cavitation inception moves towards the leading
edge for Re = 0.8x106 in accordance with the
location of the experimental minimum pressure
coefficient.
For the same cavitation number below cavitation
inception, (see Fig. 9.b), it is shown that for
Re=0.8xlO6, the cavitation pattern is a thin glossy
cavity followed by a perturbed cavitating region
downstream contrary to Re=0.4xlO6. This can be
attributed to a laminar flow extending over a larger
distance on the foil for the lowest Reynolds
number. The photographs on Fig. 10, showing the
development of the cavitation for the two Reynolds
numbers at the same cavitation number, confirm
this. It is shown (see dashed line) that the interface
of the cavity with a glossy aspect (which can be
representative of the laminar flow) is much larger
for Re = 0.4x106 than for Re = 0.8x106. In the latter
case, it is very small and is very close to the leading
edge. This is accompanied also by a net increase of
the cavitating region downstream on the foil
surface.
Re (Cpminjexp X~kmin <; (1 Cpmin
(inviscid
0,4 106 -2,69 0.004 2.32 -3
0,8 106 -3,1 0.002 2.99 -3
Table 1 Values and location of CPmi7, If= 6 ).
Comparison to the desinent cavitation number
and the theoretical value of CPmi7~.
NACA66-6% 100 mm and 150 mm.
Fig. 11 shows the inception curves for the two
NACA66-6% foils. As it has been observed
previously, the angle of cavitation inception
decreases with an increase of the Reynolds number
(see also on Fig.7 for the NACA66-6%- 1 50mm
foil). Moreover, a difference is recorded between
the two foils. For a given cavitation number and for
nearly the same Reynolds number, the cavitation
appears first on the largest foil when increasing
progressively the angle of incidence. This means
that for a given angle of incidence the cavitation
number at inception is larger on the largest foil.
This can be due to the confinement effect, which
tends to increase the peak of the minimum pressure
coefficient at the leading edge. This observation is
confirmed qualitatively by the theoretical minimum
pressure coefficient that is computed taking into
account of the tunnel walls (see Fig.124. However,
the increase of the theoretical minimum pressure
coefficient is found to be smaller than the increase
of the experimental inception cavitation number.
(a)
(b)
Fig. 9 Photographs of cavitation. a) inception for
Re = 0.4x106 (left c; = 2.16) and O.Xx106 (rigth c; =
2.854. b) developped cavitation c;=1.98+0.02,
Re=0.4xlO6 (left), Re=0.8xlO6 (rigth) or = 6°.
NACA66 12% foil. Flow is from the left.
Fig. 10 Photographs of cavitation development
for Re = 0.4 106 (left) and 0.8 106 (rigth), same
cavitation number c; = 1.31 + 0.01, or = 6°.
NACA66 12% foil. Flow is from the left.
5
OCR for page 659
r r T I
4
4
· N66- lOOmm, Re = 0.5 10 6
~ N66- lOOmm, Re = 0.8 10 6
+ N66-150mm, Re = 1.2 10 6
Theoretical - Cumin
6j
Fig. ~ ~ Experimental angle of cavitation
inception versus the cavitation number for
the NACA66 - 100 mm and NACA66 -
150 mm foils.
I.5
2.0 ~ ~ ~ ~ ~. O
1.0
0.5
OC 0.0
-0.5
-1.0
-1.5
-2.0
o
' "ax"
,1 O NACA 66-6% - 150 mm,
:_ bounded flow, h/c=1.28
In_
Cp,lim
\
\.
1
2
Fig. 12 Theoretical values of - Cp,,,,n with and
without the tunnel walls.
Eppler E817- 100 mm.
For the Eppler foil, Fig. 13.a shows also that the
experimental cavitation-free bucket is wider than as
the one that is predicted in theory. However, when
it is compared to the experimental value of - Cpmin,
the agreement is good, excepted as the angles of
incidence is larger than 6°. In that case the
experimental value of - Cpmin is found to be smaller
than <;i. This is clearly shown on Fig. 13.b showing
the difference between the experimental cavitation
number at inception and the measured minimum
pressure coefficient as a function of the angle of
. .
inch Pence
10
8
6
4
2
O
-2
4
-6
-8
-10
1 ~ 1 1 1 _
· ~ _
- ~ I /\ cavitation inception +
· experimental value of ~CE\nn - _
theoretical value value of -C~~nin~ -
_ A ~~ ~ ~ ~ ~ - .
- ~ ~ /\ ~ ~ _
o
1 1 1 1 1
2 ~ or ~Cpmin 6
Fig. 13.a Experimental angle of cavitation
inception versus the cavitation number for the
EX17 foil. Re = 0.5106.
The discrepancy originates from the development of
a separated flow region at the leading edge when
the angle of incidence increases. This is shown on
Fig. 14 depicting the velocity vectors for 10°. In
that case, a large reverse flow region can be
observed near the leading edge.
1.8 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
1 4 ~ Hi—(-Cpmin) ~:
1.2 - /
1.0 ~ _
0.8
0.6 /
0.4 ~
0.2 _ _~ _
0.04
-0.2 1 1 1 1 1 · 1 1 1 1 1 1 1 -
0 1 2 3 4 5 6 7 8 9 10
oc
Fig. 13.b Difference between the experimental
inception cavitation number and the
experimental minimum pressure coefficient as a
function of the angle of incidence oc. E817 foil,
Re = 0.5106.
Fig. 14 Experimental velocity field on the suction
side of the E817 foil showing the separated flow
region at the leading edge, oc = 10°, Re = 0.5106.
6
OCR for page 660
7 - l
6 -
5
4 -
OC 3 ~
2-
1 -
O-
-1
Fig. 15 various types of cavitation on the NACA66 -12%-100 mm foil.
4
or n
-6
-
= = = , ~ A_ ~
~ ~ ~ · ~ ~ ._ ._-
_ Cavitation inception
_ ~ partial cavity
_ Inception of pulsating partial cavities
_ · pulsating partial cavity
O bubbles
_ Xfingers
_ ~ pulsating fingers
_ · patches
_ ·supercavitation
~ ASIA ~ .
2 2,5
inception
()4 ().6 ()S 1 () 1.2 1.4 1.6 1.S 2.0 2.2 2.4 2.6 2.8 3.0
Fig. 16.a Various types of cavitation on the NACA66 -150mm foil.: 1 development of partial cavity, 2
fluctuating partial cavity, 3 pulsating sheet cavity, 4 attached shear cavitation, 5 bubbles, 6
supercavitation.
Fig. 16.b type 1, c; = 3, oc = 4°. Fig. 16.c type 2, c; = 1.1, oc = 4°. Fig. 16.d type 4, c; = 2.6, oc = 6°
Fig. 16.e type 3, c; = 0.81, oc = 4°
7
OCR for page 661
Cavitation types
Concerning cavitation development, several
types of ca anon on have 1 een observed with various
combinations of cavitation n mber md male of
incidence (see on FiglS or Figl6c md The
associated photon cphs on Fig 1 6b to 16 e) For all
the foils, it was observed that partial cavities of
leng h low r thm coo t half The chord length have c
relatively stable behavior, vifh c shedding of U
shaped vapor tructmes Fig 16 c) The U-shaped
sire tmes e habit c ce tam deg ee of orgmization m
the p tnwise di ection md were inclined et m male
of cutout 45° with th me m flow
Long sheet cavities (/* larger thm cutout O 5)
e habit c pulsating behavior while sh dding large
vapor-filled tructu es h f is case, The position of
the closure pout of the cavity was shongly
flu tufting md th length cavity was not
detemmmed As the cavity closet e region reached
the foil hailing edge (/* ~ 1) c t piccl behavior was
observed vifh c periodic appearance md
disappearance of the cavity et low flequ ncy of
cutout few He to Figl 6 e) This indu ed c shong
fl id- tru tu e interaction phenomenon For
relatively large male of incidence (larger 6 m
cutout 4°) md large cc itation n mber (larger thm
cutout 2), c pe ulisr ntsitttion pattem, rumed
ttttched shear cavitation, was observed (see zone 4
on Figl6 al The ntsitttion inception appeared
away from The foil surface, probably in spmwise
vortices exi tmg in the sh ar layer developmg on
the interface of the lecdmg edg bubble separation
By decrecsmg the cavitation n mber, the cavitation
pattern developed es c ptnitl sheet cavity but vifh
c cloudy Interface in that case, org ml zed ca vibting
sire tmes rolled up m th wake of The cavity (see
Fig 16 d)
It was observed that bubble cavitation was
limited to c lezmn corre pending to low cavitation
m mbers md mode He males of incidence Some
peculiar cavitation pattems appears also wish
bubbles es - finger'' or patches (see for m. an e on
Fig I I for cr of cutout 2° to 2 5° md a low r thm
05)
Partial sheet cavity characteristics
The dab of cc itylenghsare reps ted onFig 17
es c f motion of t for various valu s of a m linear
scale Fig 17 c ) md in log-log scale Fig 17 b)
Conehtions have been Investigated concernmg
partial cavity development Su h m attempt to
correlate The date could be very useful to provide
crudytical tools which could be used to predict The
development of ptnitl sheet ca itation or to
compare the e perimental resuln with m mericcl
resu ts
10
100 F
01
15
~ · ~
o ~ 2 ~ A · 8 ~ ~ ~ ~
30 35 30
.
b)
10C
Fig. 17 Relative cavity length as a fuoctiort of the
cavitation mmmber for various angles of
iocidrocr and foils: a) liocar scale hi log-log
scale.
A fi st correlation is show on Fig 18 are
plotted on which the cavity lenzlhs me plotted es c
f motion of the ratio a/cr (or in den ees) As shown,
The date path r togeth r very w 11 for c given foil
In that case, the cavity length evolves es (a/cr)
wish the valu s of the e ponent m given on Fig 18
How we, The dab are separated when w compare
The data for two different foils Moreover, as shown,
The exponent m varies sigmficmtly hetwen about
3 md 5) md m "u iqu " pow r how, describi g the
partial cavity development whatever the studied
foil, is not determined
Wish a md n as scaling parameters, the
con hymn does not take i to account of a
characteri tic of th foils A fir t characteristic of a
given foil is the male of zero aft, .-, md
particularly the difference (co .-.,) which cm be
considered also to be proportional to The lifi
coefficient Fig 19 shows the chatty lenzlh databy
mtrodu ing th difference (co m) As show, the
dab are less t attered particularly the dab of the
NACA66 6% md The 31' foils are w 11
regrouped) but There is a not signfficmt
improvement compared to Fig 18
8
OCR for page 662
Al
old
loo
of
lo Do
Fig. 18 Rrladve cavity lengths as a function of
a/cr. m is the exponent of a power law feting the
data for a g en foil. Or h degrees.
10
o/(~ ho)
100
Fig. 19 Rrladve cavity lengths as a function of
a/(cr - .-.) where .-0 is the theoretical angle of
zero lift coefficient.
It c m be as mmed that for z given foil md for z
given cavitation m mber, the amount of She cavity
development depends on the difference betw en She
mgle Or It which the foil operates md n,(a) She
mgle of cavitation Inception for the given
cavitation mmmber This is taken i to accou t on
Fig 20 on which She cavity le gth dab are plotted as
z faction of a/ [(cn n,(a)] his rep~esenbtion
takes mto account on the inception condition (ie l/c
= 0 for Or = of) in At case, it is fouled that She
exponent m is much smaller 6 m in She previous
representation, md is close to 2 How ver, the valet
of the exponent varies sig ific mtly once zgam
when comparing th foils
By replacing th experime tal mgle of
inception It z given cavitation number by She
theoretical valet n, ~ obtsired from Fig 2 for
in lance), it c m be observed on Fig 21 that dab
scattering is red ~ ed md that only She data of She
NACA66 vifh z chord length of 150 mm discards
apart of She overall date How ver, as f is foil has
the largest chord length, it c m be assumed to be She
mo t i flu need by the tunnel walls (co finement
effect) A compuation of the theoretical valu s of
-Up,,, his been carried out by taking into account
of the turn I walls (see Fig 12) As show on
Fig 12, for z given valet of -Cp,,,., th theoretical
valet of th inception mgle is low r for the
bounded flow he difference (cq it, no ) is fouled
to be of about 0 4 deg ee m th r mge of the
cavitation mmmber studied Fig 22 shows She cavity
length data as z f motion of a/ [(cn .-= ~ (a)] with
n,~ = An, - O 4 for She NACA66-6%- I 5~: mm
It is observed that the correction betw en the
me Ill data is improved Although data scattering
still exists, it zppe s At all the dab evolve as:
llcocL t 1
Ra ~~ - (' )]
Al
(1)
with m exponent which is close to -2 his
conception apple s to be w 11 representative of the
cavity development on the tested foils for:
I < a < 3 5, 3 5° < Or < 6° md as long as She cavity
lengths do not exceed half She foil chord length
100
old
10
100
GI(cc e a)
Fig. 20 Relrdive cavity lengths as a function of
a/(cr - e (a)) where n,(a) is the angle of cavitation
heeption for a given value of a. m is the
exponent of a power law feting the data for a
-
ool ' ' '
Al 10 100
a/(u ~p(a)
Fig. 21 Relrdive cavity lengths as a function of
a/(cr - nail (a)) where cr,~,(a) is the theoretical
angle of cavitation inception for a given value of
By.
9
OCR for page 663
ol
ol (p p ~(~))
T~=
loo
Fig. 22 Rrladve cavity lepgths as a fupctiop of
(~/(P ~ P`~m.((~)) where P ~,~, (~) is the theoretical
apgle of cavitadop ipeptiop corrected of
eopfipemept effect.
CONCLUSIONS
7he mcm results of m experiment~l study
cop ermog the mception ad the development of
particl shet cavities on fou two-dimensiopal
hyd of oils are presented Cop rning fhe mception
conditions, it is show fP~t, for c giv n cavit~tion
m mber, the mgle of ip idep e for cavit~tion
ip eption d precses es fhe R y olds number
ip ~ecses ~etw en aou 0 4xl0 to 1 2x105) it is
foumd to be larger th m fhe themeticcl valu
dedu cd from comput~tion for m m iscid flow
How v r, the su fa pressu e coefficient, dedu cd
of v locity mesurements on two foils, indicate th~t
the minimum p~essme coefficient is close to fhe
ip eption cavit~tion number e pepted when c large
separated region is observ d at fhe foil lecdmg
edge
Cop erning cavitation dev lopment, acordmg
to fhe cavitation number or fhe mgle of ip idence,
various types of cavitation are observ d es particl
sheet cavities, bubble, fingers, patch s or
supe pavit~tion patterns
For p pti~l shet cavities, the cavity length
w re mesu cd on th foils for various conditions of
cavitation number md mgle of ippidep e A
cttempt to correlate th cavity length date is studied
et the end of the pcper 7his study indicates th~t fhe
cavity lengths tend to evolv es ((fi(o P ((~))
with m expop nt m close to - 2 where P (~) is fhe
ip eption mgle et c giv n valu of th cavit~tion
m mber ~ for c giv n foil A close examipation of
the p pti~l sheet cavity on op foil NACA66-12%)
shows that fhe cspect of fhe cavity c m be v ry
diffe~e t when the R y olds m mber ipprecses
pcssing from 0 4105 to 0 8 105) it is observ d that
the t msition point on the cavity su fae for which
the mte fae passes fi om c glossy cspect to c cloudy
cspect mov s towards fhe foil lecding edge when
the R y olds number ip ~ecses
Aeknowledgemepts
7he cubhors wish to express thei deep
cppreciction for fhe mpport of the techmiccl taff of
fhe IRENAV md of the mppo t of the E ole pavale,
Mmi try of D fense, France A D Long iclle
who condu ted c part of fhe experime t du mg his
engip er project is g cteful aknowledged
REFERENCES
Arakeri, V. H 1975 "Vi pous Effects on the
Position of Ccvitation Separction from Smooth
Bodies", Jou pal of Fluid Mech mics, Vol. 68, part
4, pp 779-799
Arakeri, V. H., Carrol, J. A., apd Holl, J. W.,
1981 "A Note on 6he Effect of Sho t md Long
Lcmipar Separction Bubbles on D sment
Ccvit~tion", Joumcl of Fluid E gmeering, Vol. 103,
March, pp 28-32
Arpdt, R. E. A., 1981, "Ccvitation in Fluid
Mahip ry md Hyd mlic Struptu es", A m R v
FluidMech, 13:273-328
Bdlet, M. L., apd Holl, J. W. 1981 "Sp~le
Effects on Various Types of Limited Ccvitation",
umcl of Fluid E gipering, Vol. 103, pp 405-
Brewer, W. H., apd KGppas, S. A. 1995
"E perime tcl md Comput~tiopal Ip stigation of
Sheet Ccvit~tion on c Hyd of oil" Th 2nd Jomt
ASM JSME Fluids E gip ering Co fe~eppe On
Lcser A emomeLv, Augut 12-13, Hilton Heed
Isl md, Soubh Carolipc
Callepaere, M., Frape, J.P., Michel, J.M.
1998 "h flu p e of Ccvity Thickness md Pressure
G cdient on the Unstecdy Behavior of P. pti~l
Ccvities" Thi d Interpatiopal Symposium on
Ccvit~tion, April 7-10, G'enoble, Fpu e
Dapg, J., apd Kuiper, G. 1998 'Re~nh mt Jet
Modellmg of Particl Ccvity Flow on Two
Dimensiorul Hydrofoils", Third Interpatiopal
Symposium on Ccvit~tion, April 7-10, G'enoble,
Frappe
Deshpapde, M., Fepg, J., apd Merlde, C. L.
1994 "Ccvity Flow Predictions Bcsed on 6he Euler
Equ~tions", Jou pcl of Fluid E gip ering, Vol. 116,
pp 36-44
Dorapge, P., Astolfi, J. A., BiBard, J.-Y.,
Frumap D. H., apd Cid Tomas, 1 1998
" Ccvitati on Ip epti on md D v l opment on Two
Dimensiorul Hydrofoils", T ird Interpatiopal
Symposium on Ccvit~tion, April 7-10, G'enoble,
Frappe
Eppler, R., 1990, "Al foil Desig md Date",
Sprmger-Verlag
Farhat, M. 1994 "Contribution ir 1'6tude de
1'6rosion de p~vitstion: mbp~msmes
hyd odynamiqu s et pr6diction", These N°1273,
E ole Polytechmiqu F6dGrcle de Lcusarme,
Switzer md
Frape, J.-P., apd Michel, J.-M. 1985
"Attahed Cc itation md the Boumdary Lcyer:
10
OCR for page 664
E periment~l Ip tigation md Numerical
T'eatment", Jou pal of Fluid M ch mics, Vol. 154,
pp 63-90
Gopalap S., Katz J., 2000, 'Flow st petme md
modeling issu s m the closu e ~egion of zttahed
cavitation", Phy ics of Fluids, Vol. 12, N°4, pp
895-91 1
Huapg, T. T., apd Petersop, F. B. 1976
"h flu p e of Viscous Effects on Mode Full-Scale
Czvit~tion Scalmg", Jou pal of Ship Research, Vol.
20, N°4, pp 215-223
Katz, J., 1984, "Czvit~tion phenomenz withm
regions of flow separation", Jou pal of Fluid
Mech mics, Vol. 140, pp 397-436
Kawapamd, Y., Kato, H., apd Yamaguchd, H.
1998 "Three-Dimensiorul Characteristics of 6he
Czvities Formed on z Two-Dimensional
Hyd of oil", Third htep~tional Symposium on
Czvit~tion, April 7-10, Gr poble, Frappe
KGppas, S. A. 1998 'The P'ediction of Unsteady
Sheet Czvit~tion', Third Interpational Symposium
on Czvit~tion, April 7-10, Gr poble, Frappe
KGppas, S. A., Mishdma, S., apd Brewer, W.H.
1994 'Non-lmear Apaly is of Viscous Flow A oumd
Czvit~ti g Hyd of oils", 20th Symposium on Naval
Hvd odypamics, August 21-26, Univ rsity of
Czliforni~, S mtz Barbma, C A
KGppas, S.A., apd Fipe, N.E. 1993 "A
Numerical Nonlip ar Apalysis of the Flow aroumd
Two md Th ee-Dimensional P. pti~lly Czvit~tmg
Hyd of oils" Jou pal of Fluid Mech mics, Vol. 254,
pp 151-181
Kjeldsep, M., Aradt, RE.A apd Ellertz, M.,
1999, "mv stigation of umsteady cavit~tion
phenomenz", Proceedmgs of the 3'3 ASME/JSME
Joints Fluids E gmeermg Co ferep e, July 18-23,
1999, S m-Fpsppi po
Kubota, A., Kato, H., apd Yamaguchi, H
1992 "A New Modeling of Czvitating Flows: z
Numerical Study of Unsteady Czvit~tion on z
Hyd of oil Section", Jou pal of Fluid M chmics,
Vol. 240,pp 59-96
Le. Q., Frape, J.-P., apd Michel, J-M. 1993
"Partial Czvities: Global Behavior md M m
P'essu e Di tobution", Joumal of Fluid
E gip ering,Vol IlS,pp 243-248
Laherteaux K R., apd Ceeeio, S. L.1998
"Flow in the closure region of closed partial
zttsched cavitstion", Third Interpational
Symposium on Czvitation' April 7-10, G enoble,
France
Tassip Leger, A., apd Cep lo, S. L. 1998
"Emmination of 6he Flow p ar the L zding Edge of
Attsche d Czvitat i on Part I D tschment of Two -
Dimensional md A isymmetric Czvities", Jou pal
of FluidMechmics, Vol. 376, pp 61-90
Valeadpe, D.T. 1974 ' he effect of Nose
Rzdius on 6he Czvit~tion Ip eption Characteri tics
of Two-Dimemional Hyd of oils", Report 3813 of
6he Na~l Ship Research md Dev lopment Center,
Bethesdz, Maryl md 20034
Zhapg, Y., Gopalap, S., apd Katz, J. 1998 "On
6he Flow Shuptue md Tubulep e m 6he Closure
Region of Att~ched Czvit~tion" 22th Symposium
on Nzval Hvd odypamics, August, Wzshmgton
DC
NOMENCLATURE
c
h
h,
X = hori ontal coordinate (st~eamwise)
Y = v rtip~l coordinate
Z = sp mwise coordipate
x = coordinate ziong 6he foil chord
foil chord leng h
= te t section width
= me m su face rougimess
cr = mgle of mcidep e
c6 = th oretical :mro Ifft mgle
* = mgle of mcidep e at cavitation ippeption
c~ = mgle of mcidep e at cavitation desip p e
U= = f~ee sheam v locity
PO = te t section static p~essme
Pv = vapor pressme
p = fluid density
v = kmematic viscosity
R = F~y olds number, U= c /v
t = cavitation number, PO Pv)/q
t, = cavitation number zt ip eption
t s = cavitation number zt desipence
u = me m st~eamwise v locity
v = me m v rtical v locity
U = modous of local v locity, (u +v2)
U~ = local potential v locity zssumed to be the
maximum of U on z Ime normal to the
foil su face
~ cavity length
m _.+ ~. ~
Cp = foil suface pressu e coeffici pt. I ~/
U-) ~
cpm~ = minimum of Cp
x*, ~ = locationof Cp-~
6h md exp zs z subscript me ms 6heoretical md
experimental valu
sp~lmg by 6he foil chord lengh or the fiee tream
v locity are d poted with ~ zs z super pript. xt =
~c.U*=U/U=.
ll
OCR for page 665
DISCUSSION
R A ndt
University of Mim sota, USA
Could you tell us what the design lift coefhcients
are for your foils Your experiments for The 66
series foil appear to be carried out for lift
coefficient greater f m The design Ifft
coefhcients For example, ff you refer to Maines
Ed A ndt, FE 1997, you will see that for the
66~-415 foil, the position of C m moves forward
to the leading edge for C, >0 6 (d ag bucket is
0 2-0 6) his will make quite a deference since
Chat design lif coefficient is about x/c = 0 6!
AUK HOR'S REPLY
The main geometric characteristics of The foil are
ah 1311 wmg:
the chord length is c = 100 mm or 150 mm
mdthe sp His 0 191 mm,
the maximum thick ess is e= 12 mm or 6
mm (~12% or 6%) at 45% from the
leading edge,
the maximum relative camber is 2% for all
the foils, the leading edge radms divided by
the chord length is given bypbt = 0 674 /,
the ft~eoreti 91 lift coefficient at a given
mgle of Incidence n is given by
Cl = 0 1092 (I - O 63 a) (n +2 35) for m
in iscid unbounded flow, Valentine (1974)
Clearly, the coordinates for the NACA 66 with
12% relative Thick ess are given in table I
It is right that our experiments for the 66 series
foil w re carried out for lift coefficient greater
thm the design Ifft coefhcient How ver, this
corresponded to the situation for which partial
sheet cavities occurred
Table 1 Coordinates of the NACA66 foil se tion
OCR for page 666
DISCUSSION
M Billet mdW Strakc
Applied Research Laboratory
he Pffmsyl mid State University, USA
Fir t, we would like to con rat late She mthors
on m out t mdmg experimental effort in
characterizing the inception Ed development of
partial cavities on four different hyd of oils he
mthors have provided c very complete set of
date for cavitation inception Ed desinence
conditions on two-dimensiomcl hyd of oils
In the introduction, the mthors mention the value
of this type of basic experiment as m effective
way to understand She cavitation m more
complex sit ctions such es marme propellers Ed
w agree with f is However, m issue till
for th ee-dimensiomtl sin ttions Would the
mthors please exp Ed on the applicability of
these date to propeller blades, especially m the
presence of the sp mwise loading gradients Ed
the f ee-dimffnsiomrlity associated with
propeller designs How will these effects
i fluence the characteristics of the cavity such es
its length Ed unsteadiness?
he paper also illustrates the importance of
"reel" flow physics on cavitation by comparing
the experimental results with She theoretical
inviscid predictions Please comment on the
flow phenomena that recall in the Urger
discrepancy betw en the experiment Ed
predictions with negative Ogles of attack
conditions competed to positive Ogles of attack
Pigures 5 Ed 11)
AUTHOR'S REPLY
Concerning the recall in She larger discrepancy
betw en She experiment Ed predictions with
negative Ogles of attack conditions compared to
positive Ogles of attack, Pigures 5 Ed 11), w
have to say that no pecu liar attention was paid on
this point How ver, the effect of c separation
bubble that could be m me impo t mt for negative
Ogles bee mse of the shape of She foil c m be
assumed How ver, m of her effect es the
co finement effect (t m I walls) c m be also
suspected
OCR for page 667
DISCUSSION
S. C io
University of Michig m, USA
The mthors present m interesting study of
partial cavitation on r..-o-dimensionci hyd of oils
The mthors compare the inception cavitation
mmmber with She minimum p~essme coefficient
computed for the non~vitating flow md
conclude that the C, ~ to for flows without
leading edge separation Visual cavitati m calls
w re made, md She hors suggest that the
actual value of inception may be at higher
cavitation mmmbers since it may be difficult to
detect small microbubbles at inception Could
the mthors provide more details on how
cavitation Inception was called?
Also, the mthors report Nat inception occurs at
high r cavitation mmmbers as She attack mgle is
increased md leading edge separation occurs
Did the mthors detect ~ separation bubble during
the LDA measurements near the leading edge of
the NACA66 hyd of oils (as show in Figure 14
fm She f 5 I ~ he d of oil)?
The mfhors Investigate the i fluence of
Rey olds mmmber variation on the inception md
appearance of the ca vity F igure I O suggests that
the boundary layer near the region of cavity
detachment has been sufficiently modified to
chmge the appearance of the cavity as the
Rey olds mmmber is chmged by ~ factor of two
Can the mthors deduce what He state of the
boundary layer is upsheam of the cavity
detachment? Specifically, is the boundary layer
inge ted by She cavity ItmirLtr, t nsiriorLtl, or
turbulent? This is m admittedly difficult
observation to make e. 2 rimentally for cavities
detaching close to the leading edge) Also, are
the mthors co fident that the fieesheam
turbulence level will remain unchmged as the
Imettretm velocity is mcreaiedS It is interesting
to note how the extent md appearance of the
cavity is i fluenced by He chmge in the
Rey olds mmmber Could the mthors comment
on what physical processes they suspect are at
work?
AUTHOR'S REPLY
Concerning ~ separation bubble during the LDA
measmements near the leading edge of the
NACA66 hyd of oils, we have to say that
velocity measurement w re carried out only on
the NACA66 with 12% rektive thickmess No
observable negative velocities w re detected
which should be the condition for ~ separation
bubble to exi t So the presence of ~ separation
bubble was not clear How ver, the cavitation
inception formed of ~ thin p mwise bubble b md
seemed to Indicate the presence of ~ separation
bubble
Concemmg the extent md appearance of the
cavity which is i fluenced by She chmge in the
Rey olds mmmber Fhotogmphs of cavitation
patterns showed that such ~ h msition is rented
to ~ tr msition pomt on the ca vity so face, which
moved fo ward to She leading edge when the
Rey olds mmmber Increased it c m be assumed
that this c tnt:e related to ~ h msition from ~ long
separation bubble to ~ short separation bubble
when She Rey olds mmmber so e. How ver,
the effect of the turbulence intensity (as rightly
suggested by the discusser) which slightly
increases with the velocity Fig 1) cm be also
responsible of such ~ phenomenon This point
willbe studiedclosely
0, 4
umms'
0,02 ~
O ~
_ ~
+ Smls
+ 8ml5
0 0,2 0,4 z. 0,6 0,8 1
Ftgmre I Turbulem e intensity as ~ Sum tion of
the 5paovise dttreftort measured at om chord
length upstream of the leading edge of the fed
for two veloeides. NACA66-6%-lSt mm foil.
OCR for page 668
DISCUSSION
J P Franc
University of Grenoble, France
The mthors have to be con r.uulc~ed for thei
cared I piece of experimental work on partial
cavitation The comparison of She results of
various tests conducted on four different
hyd of oils is particularly interesting for She
crurlysis of the inception Ed development of
sheet ca vities on r..-o-dimensiornl hydrofoils
The velocity measurements they performed on
the two thicker hyd of oils (NACA 66 12% Ed
Eppler E317) by LDV, use c very line mesh Ed
led to c d tailed description of the velocity field
near She lecdmg edge in particular, the mfhors
succor de d m determ inmg , from LD V, the
minimum of the pressure coefficient, which is
k own es m essential parameter for the
interpretation of incipient conditions
On figure 13 b, the mthors show that, for the
E317 hyd of oil et c Rey olds mmmber of 5 106,
the experimental minimum p~essme coefhcient, -
Cpmin, detemmmed fiom LDV measur me ts, is
in good agreement with She measured cavitation
inception parameter, s remat rely, regarding the
prediction based on LDV measurements The
mfhors give m interesting interpretation of this
difference From Their measured velocity fields,
they conelate this discrepancy to the
development of c separation bubble et the
lecdmg edge, which is k own to appear only et
high enough Ogles of attack
We c m conjecture chat c shear Icyer exists
betw en the free stream Ed the separation
bubble, Ed consequently that vorticcl structures
develop in this shear Icyer These coherent
stmct res have c given vorticity, so that the
pressure in then core should be low r f m the
ambient pressure As the mfhors have carried out
detailed measurements of the velocity field near
the leading edge, would it be possible to estimate
c characteri tic vorticity Ed c characteri tic
length of these structures? If so, it should be
possible to give m estimate of the pressure d op
in She core of these vorticcl structures Ed
compare it to the difference betw en the
minimum pressme coefficient Ed She cavitation
inception parameter Did the mthors attempt
such mapprocch?
At THOR 3 REPLY
As commented by the discusser, it is
right that She spatial resolution of the velocity
field was good enough particularly et the leading
edge A characteristic spatial step between two
consecutive velocity measurement pomts in the
vicinity of the foil surface Ed near He leading
edge (x*=0 01) is Xt = 0 01 Ed
Ye = 0 004 Thus, en estimate of She non-
dimensiom~l sp unwise vo ticity, ~ clU,
given by,
avow auto
axe a
(1)
cm be computed This was done fiom m
interpolation of She velocity data together with c
spatial numerical derivation in Xe Ed Ye
directions (note that ~ es c superscript
corresponds to non-dimensiom~l values scaled by
the foil chord length Ed She i flow velocity) it
must be pointed out that, in some cases, bee mse
one of She laser beams intersected to the foil
surface, She vertical component of the velocity v
was not measly at lea Thus, in that cases,
estimated only Through the second term of Equ
I A example of the calculation is given in
Fig I showing She lines of con d mt sp mwise
vorticity for m Ogle of attack 10° cone pondmg
to She case for which c flow separation was
clearly detected it c m be observed c streamwise
stret bed vorticcl shuctme exlendmg from the
leading edge up to about xt = 02 along the
boundary of the separated legion (depicted by
the dashed line on Fig 1) Becmse of the
vorticcl flow, it m be conject red, es mentioned
rightly by the discusor, Nat the pressure should
be low r ohm the one deduced only from
velocity measurements md She Bemoulli
equation (assumption of potential flow) This c m
be written formally by adding c conective temm
to the potential pressure coefficient taking mto
account of the vorticcl flow:
Cp = Cp~, + Cp~
(2)
The first term on the right hmd is the pressure
coefhcient deduced fi om She tr mslation velocity
measmements The second one on nobles from
the vorticcl flow it cm be assumed that the
contribution of the vorticcl component cm be
expressed as:
Cp~ = (Q: a )
(3)
OCR for page 669
which can be considered as the pressure
coefficient at the core center of a vortex of radius
an and a vorticity of At . A characteristic scale
of the vorticity As and a characteristic length
scale as can be inferred of the the vorticity
profile (see Fig.23. Here, an corresponds to the
characteristic length scale in which the vorticity
is concentrated and At is the mean value of the
vorticity in this region.
These values are reported on Table 1 together
with an estimate of CPQ . As shown, in that case,
there is a good agreement between the inception
number and the opposite of the minimum
pressure coefficient taking into account of the
vertical flow. Although, this result can appear as
a rough estimation, it indicates that the vertical
flow involved in the case of a flow separation
can play a very important role in predicting
cavitation inception.
-0.10
-0.15
_
'2"~
oc=10°
X~
0.00 0.05 0.1 0 0.1 5 0.20
Figure 1 Non dimensional spanwise vorticity
near the leading edge of E817 foil, = 10°
Re=0.5 106.
0.08S
0,08 '
0,075
0,07 '
Y*
0.06S
0,06 '
0,05S
0.05 '
a*=0,01 4
Qz*
(J)7
0 20 40 60 80 100 120 140 160 180
Figure 2 Vorticity profile, estimation of a
characteristic length scale and a characteristic
vorticity. x*= 0.04, = 10°, Re = 0.5106.
*
a z Ha
Zip CPQ (cppot +CPQ i
0.014 96.88 -1.84 -1.60 -3.44 3.3
Table 1 Estimation of the rotational
component to the minimum pressure
coefficient and comparison to the inception
cavitation number. x~ = 0.04, = 10°,
Re=0.5106.
Representative terms from entire chapter:
separation bubble
