Paper No.
65- FE-3
R. B. WADE
Graduate Research Assistant,
Division of Engineering and
Applied Science,
California Institute of Technolog y,
Pasad e na, Calif.
A. J. ACOSTA
Associate Professor of
M echanical Engineering,
Division of Enginee ring and
Applied Science,
California Institute of Technology,
Pasadena, Calif. Me m. ASME
Experimental Observations on the Flow
Past a Plano-Convex Hydrofoil
Some new measurements and observations on the nonwvitating and cavitating flow past
a plano-convex hydrofoil are p resen.tell. Under some conditions of partial cavitation,
strong, periodic oscillations both in the w vity length mul forces exerted on the hydrofoil
are observed. T he reduced frulu.ency of oscillation (lepends upon the cavitation number
and angle of attack; i t also depends somewhat on tunnel speed for the lower angles of
atlack but becomes snbstanlially independent of speed for the highest angle. The peakto-peak magnitud-e of the fo rce oscillation wn amount to about 20 percent of the average
force.
Introduction
W ITH t he adve n t of t he hydrofoil boal and t he demand for highe r s peed pumps and propellers, !l d e tai led knowledge
of t he pe rformance of hyd rofoi l ·ections under all conditions of
ca vi tation becomes necessary for t he proper unde rstanding of
events t hat may occur in e ngineering applic-ations. We have been
in terested for some t ime in t he cavitating flow t h mugh hy drofoils
arranged in eascade beca u~e of its obvious appli r-at iun t o pumps
a nd p erha ps prope lle r·s . In <·onside ring possible hydrofoil profi le
shapes for such experim e n t.~, we settled upon a pla no-convex
sha pe ( flat bottom a nd c-i rcula r >WC upper surfaee ) for it.s s implicity a nd econom.v of manufacture a nd a lso b ecause ex tensive use
is made o f slight va ri a nt ~ from th is form for propeller secti on ~.
An a dded b u t not e~~e n t i al .-onsideration is that a na lytic ealeulat ions a rc fa cilitated by a simple geometry.
As a preliminary step to t he undertaking of a full-seale C>1..~cade
ex pe rimen t, we deeided to st udy t he characteristic-s o f an individual foil of the ty pe chosen as t his would suggest po:;sible phe nomena to look fo r in c·asrade. A l~o, because of t he greater di ffic·ult_,.
of expe rimentation, it w a.~ not like ly th at t he pe rform:tnc·e in ca:;cade could be examined in as great detail as t hat of a n isolated
hy drofo il. Furthermore, such tests a re in themse lves of considerab le in teres t. Ex perimental st udies on plano-c-onvex profi les or
on cloRe ly relat ed ones are, however, not new. Bulhan ( 1], 1 for
example, repo rts cav itation experimen ts made on a series of
Karm an-Tre fH ~ p rofi les . Experimen ts on a s imila r ~er-tion were
made by W a lchne r [2 ] a nd mo re recently by K e n neen [3].
Generally , t he profiles o f t hese s t udies diffe r from t he presen t one
in cer tain respec ts- notably t he leading edge det.ai l- exce p t fo r
[1], in which the extent of c-avitation is mu~h l c~s t hnn t hat of t he
p resen t work.
' Numbers iu brackets des iguate Refereuce:; at e nd of paper.
Contributed by t he Fluids E n gineering Divisiou for preseutatiou a t
t h e Applied Mechanics / F lu ids Engineering C onference. Wa s hingtou,
D. C . , June 7- 9, 1965, of TnE AMER ICAN Soc rET>' o•· .\[ECHANICA r,
ENG INEE ns. :\[anuscript received at AS.\ IE H eadquarters, February 19, 1965. Paper No. 65--FE-3.
Plan o-eon vex hydrofo ils, like t he other members of the K a r manTrefftz series, h ave sharp leadi ng and trailing edges . At positive
angles of attack, a n infinitely negative pressure occ urs at t h e leading edge a nd norma lly a cavity would be expected to . prin g from
t he leading edge. T hil'<, however, doe;; not a lway s occur. At low
a ngles of attac k, below 4 deg for t he sect ion tes ted, the r·avi ty
a ppears downst ream of th e lead ing edge on t he low-pre:·sure
surface. In fact, fo r so me of these lower a ngles, t wo cavi t ie·· are
obser-ved simultaneously; one, very sho rt, springing from t he
leading edge a nd th e ot,her from a poin t near t he maximum t hic kness of the h ydrofoil. At a nd a bo ve 4 deg, t he cavity star ts at t he
leading edge for a ll <·ond it io n:; of cavitation ; a nd in t he present
work, t he emphasis is plaeed on a ngles of attack greater th an 4
deg. The ca vit,y-detachme n t poi n t is t hus fixed at t he leading
edge in t he range tested, resul t ing in a mo re readily anticipated
flow pattern and a more : t ra igh t forwa rd in terpretation of t h e
experime n ta l resul ts t ha n othe rwise wou ld have been t he c·ase.
One outcome of t his rest rict ion is that t h e profile b ecomes a fl at
p late for cavities longer than the chord, a configuration al read .v
test ed by P arkin (4].
Description of Experiments
:\tea.sure me n ts o f t he lifL force, drag, a nd moment on th e hy drofoil were made wit h no cav itation to de termine t his basic flow.
CavitaLion experime nt:; were t hen carried ou t by fixing t he angle
o f attack a nd lowering t he tu nne l pres.~ure. The object of t hese
ex pe rime n ts was lo determ ine t he performa nce of t he hy drofo il
under cavitating condi t ions, i.e., lift, drag, a nd so on , a nd to
obse rve t he formation a nd developmen t of t he cavitation or t wophase regi on formed abou t t he hydrofoil. or special interest
h ere was t he measure ment o f cavity lengths. Under some condit ions, however, t he cavity on t he hy drofoil oscillated wit h a
re cognizable period ; it wa · a l o clear from t he v ibrations of the
t unnel appa ra t us a nd struct ure that there were considerable
fluctuations of t he forces o n t he h yd rofoil as well. A number of
expe rimen ts of a Jl reli m inary natu re were carried ou t to as certain
t he magnitude o f t he fo rces a nd associated frequen cies. For t hi
p urpose, a few moderately high- p eed 16-mm m otion-pi cture
----Nomenclature-----------------------------A
c
p la n fo rm a r·ea = (s X c)
= cho rd
dm~~:
J
/(
l e n~~:th
P - Pk
pV 2j"2
/)
eoefficien t
lift c:oefli cien t =
A p V' / 2
K ,.
f)
moment, cocffi c: ie n l - - - - about
A cp l' ' /2
t he m idchord point
drag fo rce on model
cavit.atiun numbe r ba.sed on va por
p -
. lpV '/2
p,
pressu re= p V'/ 2
.11
('.II
frequeney of oscillations
con·e cted ravitaLion number
/,
.11
p
cav ity le ugtlt
lift fo rce o n mod el
mome n t on model
<·orrecled t unn el static pressure
m easured cavity pressure
va por p ressure of wa te r
N
rad ius of eircula r surface of model
-~ =
span
t
t hickness uf lty drufo il
V
t.u nn el v e locity
a
augle of attack, degrees measured
front <·ho rd line
A
dist.nn ce of cen ter of pressu re from
leading edge
p
d ensity of waler
Jh
p,
Discussion on this paper w ill be accepted at ASME Headquarters until July 12, 1965
oequences were made. An Eastman camera was UHed and the
average frame speed waR about 1900 per se<·. The development
of cavities is, of coun;e, of great interel:'t: but of parlinalar imporlan<·e for the pre~ent problem is the relationship between the
geomelt")' of the growing and collapsing eavilation pattern and 1he
variation of the force on the cavity. Fot· this purpo~e, the highspeed motion pictut·e~ recorded the output of a strain gage
mounted on the hydrofoil simultaneously with the eavil\' mot ion.
Several film ~ trips were assembled into a short sound mot ion
t>i!"lure' whieh illustrates this nonsteady <·avitation proees~.
This unsLeady euvitation phenomenon ha;, been not.ed before in
referetwe (4) and also by i\[eijer (5), who earried out "imilt~r experiments on very thin hydrofoils. Observat,ions on the frequetw.'·
and amplitude of t hese osc-illations do not seem to have been reported HS yet although a somewhat simihu· phenomenon ha,~ been
desnibed by Knapp (6). In his experiments, parli:d <·avitation
waR observed on hemisphere-cylinder bodies of revolution and
thick, I wo-dimensional, nonlifting bodies <·onsisting of a flat plate
with a half-round circular nose.
Fig , 1 (a)
Model and base plate
Experimental Procedure
The tests were conducted in the high-speed water tunnel at the
Hydrodynamic:; Laboratory [7). The test model used, as previously mentioned, was a plano-convex hydrofoil, the dimensions of
which are shown in Figs. 1 (a) and I (b). The leading and tmiling
edge:; of the hydrofoil were left perfedl.v sharp. The 14-in-dia
cylindrical section of the t unnel was eonverted to :m :tpproximately 14-in. by a-in. two-dimcn~ional red angu lar SC<'tion by the
u se of inserll;, as explained in
The model was integral!~· maehined with the base p late shown
in Fig. l (a) which was bolted to the spindle of the for<'e b!dance .
.. hims were used so that the plate was made flush with the tunnel
wall. A circular gap of approximately 0.020 in. was left between
the model attachment piet·e and the surrounding tunnel wall.
The fon·e balanc·e and readout equipment whi<'h were u~ed to
measure the steady values of lift, drag, und moment on the hydrofoil were the l'ame as in (3] . A d etailed de.~cription o[ th e forcemea~uring equipment i~ given in [H).
Normally, two kinds of experimenll; were m:tde for the stead,·
forces. Fully welled data were obtained at <·on~lant lunn~l
speed and pressure, and the angle of allR<·k was varied over the
region of intere~t, in the present ease - 4 to 15 deg angle of attack
with itwremenls of 1 .') min of arc. This was repealed for ~eve­
rat velocities to give a range of Heynolds numbers. Cavitation
experimenh, however, were made with u <'Onstant, angle of
attack and the ambient pressure within the tunnel varied to obtain the full range of cavitation. These lest_.; covered a range of
angles of attack of 4 to 10 deg and velocilic>< from 15 to 40 fps.
The expet·imental ~elup is shown in Fig. 2. Two of the forC'emeasuring balancef' are shown in this illu;;lration, a..~ well a.~ the
recording camera:;. For both cavitation and fully wetted measurements, 3.'}-mm ca mera,~ were u sed to record the readings of the
force console. Additional camera.~ were also used to record,
simultaneously with the foree measurement8, elevation and plan
form views of the cavitation on the hydrofoil. The lengths of
c·avilies were measured from these photographs.
The end gap between the model and the faeing wall of the twodimensional test section was adjusted to O.OOfi in. and kept approximately at this value t hroughout Lhe experiment. Although
the end gap d id vary slighil y throughout the exper·imenl, it was
found ll11tt I he variation of the fon·es with end ga p for full y welled
flows over a range of 0.00.5 to 0.01 Of) in. was less than 5 pe;·eent for
the lift and drag and negligible for the c·asc of moment. During
a lest ruu , the variation of the end gap was never greater than
0.004 in. ; henc·e this eiTeet was sufiieienlly small to be <·onsidered negligible. Similar result;; wcre obtained in (3), a lthough a
r:n
.
' The film, entitled "Some Non-Steady EfTe<·t~ in C'avity Flows"
Report No . 1-;-79.6, may be borrowed from the llydr~dynami~s
Laborato ry, Karman Laboratory of Fluid :\fe chnni~s and J et Propul~ion, C'uliforn ia Institute of Technology, Pasadena, C'alif.
2
Fig . 1 (b) Definition of positive sense of forces and moments. Dime nsi ons of mod el u sed are C = 2 .77 in., t = 0.19 in., R = 5 in., s = 2 .85
in.
somewhat larger drag variation wa:< obtained there probably
because of the greater thickness of the model.
The readings of t h e force gages were corrected for tunnel ~tatic­
pressure intera('(ions and fort he tare forces on the mounting disk.
These tare fon·es, although ~mall in the case of lifl and negligible
in the <"fb~e of moment, c-omprise, under certain cir<'umslan<'es, n~
much a.~ ao pei·cent of the total dmg force. The details of the
tare-fon·e determinations and the method of obtaining these
res ult.~, by mounting the model from t h e opposite wall of the tunnel, nrc described in [:J].
The dynamic head p V 2 /'2, and hence the tunnel speed V, wn.~
determined by measu ring the pres:;ure drop bet ween the piezometer ring at the .'i-ft.-dia circular ~eclion of the tunnel upstream of
the tunnel nozzle and the two-dimensional section itself. This
pressure diiTercn<"e wa.~ rc<·orded on the force read-out console.
The static· pre;;.-;ure in the tunnel was measured at the t wo-dimensional sec·tion by means of a mercury manometer.
The diiTeren<·c bPI ween the working-section static pressure a nd
the pressure within the ('>tvity was measured with n mercury
manometer. The c·avit.v-pressure orifice was lo<·>tted 0.2 in.
behind the leading edge on the suetion face of the hvdrofoil at
midspan. Ow in~ tot he frothy nature of the c·avily, w:~ter tended
to e nLer the tubing (·onnc<·ting the c·avity orifice with the manometer, thus causing fa lse readings. To insure a ('OtTeC't reading of
the cavity pressure ftl all times, this line was kept dear of water by
eonstant purging with a s mall amount of bleed air. ' The resulting e rror in pressure was alwa~· s less than O.R in. of wate r, whid1
• This portion of the work was carried out before the work of G .
Gadd, "An Air-Blowing Technique for :\!ensuring Pressures in
\Vater," National l'hysienl Laboratory Report Sll.R24/ 61 , 1961 , wru<
available. The present method , essentially the same as Gacld's. "''"'
devised b~' T. Ki~eniuk.
Transactions of the AS ME
fig . 2
Tu nn el w orking section together with manom eters and recording cam era s
corresponds to an error in the determination of t·avitation number
of about J percent or less in the worst <·asc.
The present working sct·tion presents severe limitations in t he
measurement of the relevant pressures that are needed to present
the results in useful form. It is frequently impractical to have a
two-dimensional section of suflicicn t length to measure t he static
pressure that would prevail an infinite distance upstream, even
though this is what is needed in the reduction of force coeflicienis,
and so forth. In tho present case, the static pressure in t he working section was measured at an orifi<-e 2'/ • chords upstream of the
midpoint of the h ydrofoil and 1.4 chords above the level uf t he
hydrofoil. However, a di!Terent orifice only 1.8 chords from the
model <'enter line and on the center line of the flow was used as the
working-section, static-pressure reference for the measurement of
cavity pressure. Both orifi<-es are sulli<'iently close to the hydrofoil so thai it can be anticipated that the pressures there will be
noti('eably a!Teeted by the lift for<'c and possibly by the cavity
size. 4 As all fol'(·e coeflicients a nd <'avitation indexes arc based
on the working-section pressure, it is ne<'cssary that the foregoing
effects be taken into account. Therefore, an extensive tunnel
calibration was carried out to provide this information. The
appropriate details can be found in [lO]. Briefly, the pro<'edure
was to use, as a calibrating pressure orifice, a point nearly midway
between the test seetion and the upstream inlet to t he nozz le.
Then, presuming the influence of forces on the hydrofoil not to
affect this pressure, the ratio of the pressure diiTcrences between
the tunnel total pressure and the various points in question was
determined for vnrious values of lift, drag, and eaviiaiion number.
This procedure is nearly the same as that used by Kermeen in
making his measurements.
As previously mentioned, experiments were made to measure
the forC'e variations on the hydrofoil during the nonsteady c~tvity
oscillations. The force balance used lo measure the steady forces
was entirely too massive to respond to these mpid fluctuations.
' ' isual observation of the flow with a "Hirobota<'" indi<'aled that
the frequen<'ies <'ould vary from about I 0 to 30 cps. To study
t his region more thorough ly, therefore, it was ne<'cssary to employ
transducers of moderately fast response. At the same lime, it
was realized that it would be expensive lo manufacture a threecomponent, fast-responding force balance for these preliminary
tests. To get a measure of the £1u<·luaiing fon·es and to measure
the frequencies more accurately than could be done with a
Htrobotac, we flush-mounted a semiconductor, half-bridge strain
gage at the root of t he hydrofoil. The strain gage was located at
• The recent calculation of Fahula [9] shows that the effect of the
cavity can he very important in the measurement of tunnel static
pressure by the dire<·t means chosen here.
Journal of Engineering for Power
12
I I
'0
09
0
08
,; of
~~
~"
8~
~"
"
07
06
REYNOLDS N U M80t
+
Re: • 07~ • •o"
Ra • 06 2 • 10"
Ra • 046 • to"
10
ANGLE ,
'
a,
DE GRE ES
r
12
I]
,.
±
fig. 3 Force coeffi cients a s function s of angle of aHack for noncavitating
flow at se v eral Re ynolds numbers for plano- convex hydrofoil
the cenler of the hydrofoil and was waterproofed with the bonding
agent used, a form of epoxy cemen t.
The output of the strain gage was recorded with a direet wriiingrccordjng oscillograph. No attempt was made to analyze separately t he effects of moment, drag, and lift force on the output of
t he gage. lt was assumed that the o utpu t would be proportional
to the lift force and that this proportionality would be the same
for static as well as dynamic conditions. For purposes of the present experiment, these assumptions seem reasonable. The strain
3
50
1.0
45
0 .9
40
0 .8
\
-- - ·-·-
\
I
...!..-
0 .7
35
f\
0
"'
<l
30
~
<l
0
I
......
::::;
!
20
10
5
-2
ll.
... 0 .5
\
a:
w
... 0 .4
z
w
""""
0
2
4
6
~
c
-
~
0
""---,
u
I
v
~~
\
w
a:
\
I
I
15
0.6
(/)
~
I
25
~
\
a:
"'a:
w
a:
,~ ,
~ ..__
0.3
~
8
0 .2
~
10
12
0.I
0
- 4
I4
-2
0
Fig. 4 lift-to-drag ratio a s a function of angle of aHack for non cavitating
flow at same Reynolds numbers indicated in Fig. 3
4
2
10
8
6
12
14
ANGLE , a , DE GRE E S
AN GLE , a , DE GRE ES
Fig. 5 Varia ti on of center preuure location X/ c w ith angle of aHack for
noncavitating flow at same Reynolds numbers indicated in Fig, 3
gage was t he n ca librated by co mparing ill; ou Lp u t wiUt Ute ou tp u t of the exterua l force bala nce when t ho flo w was swady.
Experimental Results
l~or each datu point, lift, drag, a nd momen t coe lliciou Ls were
calculated .
' o corrections we re made fur t unnel in Lorfcrence
elfecLs such as wa ll b lockage, wa ke b lockage, a nd luugit udi na l
p ressure gradien t . All t hese effecLs are s ma ll for t he p resen t
experiment excep Lpossibly for t he fu Lly cavi tating flo w.
Le t us cons ider t he full y wetted characl-eris tics of t he hydrofoil.
I n F ig. 3, t he lif t, drag, and mo ment coeffi cienLs a re plot ted vers us
a ngle of attack . T he po in Ls a re s hown for 1\ Heyno lds number
ra nge of from 0.46 X 10• to 0.75 X 10 6 based on chord. Over
t his range, t he re is very lit tle s ignifican t cha nge in any of t he
force coeffi cients.
I t is seen t hat at abou t 1 deg angle of attack, t here is a s light
stalling eff ect in t he lift curve wi t h a con espo ndi ng increase in t he
d rag. T his effect is cha racteristic of certa in s harp-nose aero foils
a n d is due to t he type of bound a ry-layer sepa rat ion occu rring on
t he foil (11 , 12, 13]. This " wave" in the lift cu rve comes abou t
because of the ty pe of la mina r separation of t.he bou nda ry layer a t
t he leadi ng edge and iLs s ubsequen t turb ulen t reattach men t.
T h is hump in t he lift curve can be removed by increasing t he
R eyno lds number to approxi ma tely 6 X JO• or by increas ing t.he
nose s urface ro ughness. These effecLs a re discussed in d etail in
t he foregoing references. The lift slope is somewhat less t ha n
211' belo w the h u m p and d ecreases further a bove it .
F igs. 4 and 5 s how, respect ively, t he variation of t he lift-t.o-drag
rat.io a nd t he cen ter-of-pressure location wi t h a ngle of a ttack, t he
kin ks in t hese curves being d ue, once a gain, t.u t.he boundary-layer
separation .
T o d eterm ine how co ns istently t he cav it,y JH'essu re, or cavitat.ion n umbe r, could be recorded a nd how t.his reading com pared
wit h t hat based on vapor pressure, a plot of K agai ns t 1\. , was
mad e for vary ing velocit ies a nd a n gles of at.t.ack. As seen in F ig.
6, t his reading is q ui te repea tab le. The d isc•·epancy be tween t he
two read ings increases w it,h increasing cav itation n umber. It
w\\\ be noted t ha t t he cavi ty p ressu re is a lways hi ghe r t ha n t he
vapor p ressu re. This result i:s t.o be expected as t he gases in solu-
4
.E
0
!
( AVIfAfiO N
ilf U III I(II
l ASlO O N
V APOIII
P II[S S UII[,
It•
Fig. 6 Comparison of m easured cavita tion number to tha t based on
vapor pressure
tion con tribute to t.he p ressu re wit hi n t he ca vity. T hese results
also check wit,h t,hnse obt.a ined prev iously (3 1.
F or cavita ting flow, t he va lues of t he for ce coell'icien Ls as a funct.ion of t he merumred cavi tat.ion num ber a re shown in Figs. 7- 10,
each graph being fo r a differen t a ngle of attack. T he s ubseq uen t
p hotograp hs indicate t he degree of cav itation occurring on t he
hy d rofoil a t a few d ifferen t cav itation numbers whi ch a re ma1·ked
on t he graphs.
Figs. 11- 14 s how graphs of t he cavi tat ion number divid ed by
a ng le of a t ta ck as a functio n of cnvi ty length. T hese poin Ls we re
obtained fmm t he 35-mm p ho tograp hs taken o f t he cavity for
each data poin t . T he solid poin ts a re t hose occun·ing in t.he unstead y flow regime. As ca n be seen, t he unsteady region occu rs
over a reg ion of a pp roxima w ly 0.6 l /c to J .2 1/c, regard less of
angle of attack. T h is region of unsteadiness is ind ica ted on t he
grap hs o f t he force coeffi cien Ls as well. H ere the forces are
fluctun ting v iolen t ly, and t he poin Ls s hown plo tted a re "avemge"
for ces recorded b y the bala nce. A lt hough we believe t hese
a verage forces a re rep resen tat ive o f t he t rue t ime avem ge, no
systema t ic investigat.ion of t his point has yet been ma de.
Transactions of the AS ME
18
..
! ,,,
0
0
O>
J
..t U-i_
0>
~ e~
--------~··~-- -----------------
-fU1..l't
[
•
10
c•
-------:~--·-;-·--,
0
u
s 8z
~~
~
~£lf(O
l'M 15
•
.
08
ru1tY
llf'L 1 rro
l MI TS
06
0 4
O>
-,
/
. . ..
'
A;/oTAT
fig . 7(a) Force coefficients as functions of cavitation number at an angle
of attack of 4 deg for a plano-convex hydrofoil of 7 percent thickness.
Note that all drag points are flagged .
Fig . 7(b)
Cavitation occurring on a plano-con-
vex hydrofoil at an angle of attack of 4 deg at
various cavitation numbers, K. Letters are those
referre d to In Fig. 7(o) .
4
00
16
ZO
ll
Z4
26
:AVITAT• 'II Nu'-181 A, K
fig. S(o) Force coefficients as functions of cavitation numbe r at an
angle of attack of 6 deg for a plano -convex hydrofoil of 7 p ercent thickn e ss. Note that all drag points a re flagged .
fig. B(b) Cavitation occurring on a plano-conv e x hydrofoil at an angle of attack of 6 d e g at
various cavitation numbers, K. lette rs are those referre d to in Fig. S(a).
..,
'8
,.
•,
..:
.s r
oz."
~~~
10
...
08
•• 0
•• u
~ ~
u u z
' 0. ••
...
~~
([_
'~·
o•
;~·
-FULlY
LIM
W[ Tf£0
rs
)4
Fig. 9(o) Force coefficients as functions of cavitation number at an
angle of attack of 8 deg for a plano-convex hydrofoil of 7 percent thickness. Note that all drag points are flagged .
Theoret ic·ul (•urves obtained from linearized free-streamline>
theory in the regions of fu ll eavitation [ 14] and par tial cavitation
[ 15] on ajlat-plale hydrofoi l are a lso shown in Figs. 11 - 14. We
see that fo r fully eavitating flow the agreement is better than for
the partial c·avitating ease. This, however, is to hP PXpPc'ted
Journal of Engineering for Power
Fig. 9(b) Cavitation occurring on a plano-convex hydrofoil at an angle
of aHack of 8 deg at various cavitation numbers, K. Letters are those
referred to in Fig. 9(a).
si rwe, in t he former easE>, the hydrofoil aets ex:wlly like a fiul.
plate whereas, in partial eavilalion, c·amher and thic-kness efTeets
p lay a role.
E:xper imenlal results on a similar serlion are reporlNI in [2] for
nonrav ita t in11: and cavitatinv; rondilions for angl('s of altaek up
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Fig. 1O(o) Force coefficients as functions of cavitation number at on
angle of attock of 10 d e g for a plano-convex hydrofoil o f 7 percent thickn ess. Note that all drag points a re flogged .
Fig. 1 O(b)
Cavitation occurring o n a plano-convex hydrofoil at on
angle of attack o f 10 d eg at various cavitation numbers, K. Letters a re
those referred to in Fig. 1 O(o).
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cavity length-to-c hord ratio for a plano-co nvex hydrofoil a t an angle of
attack of 4 deg
Fig. 12 Cavitation number d ivided b y angle of attack as a function o f
cavity length-to-chord ratio for a plano-convex hydrofoil at an angle of
attock of 6 d e g
to 5 d eg. The cavitation number in these experiments is based
on vapor pressure a nd, hence, a direct eomparison with t he
present data cannot be made exaelly, and the thickness of the
profile is slightly difTerent. Nevertheless, quite a favorable agreement is found in the common region c·over!'d by both investigations.
angles of altac·k equal lo or greater lhan 4 deg is similar, and Lhe
general dcvelopml'nt of the nnnsteady process is the same for all
angles.
From I he fully welted rondilion to a cavity length of 60 perrcnt
chord, the c·avities arc steady in the mean ; t he <"avily is not glassy
rlear, however, but is filled with a fmthy mixture of air and water
and has no definite structure, sueh as a reentra nt j et, for example.
Incidentally, at the very first stages of cavitation when the
<·avities are not longer than 2- 3 percent of the c·hord, a relatively
high-pitc·hed OSC"i llation and no ise develop with a frequenc·y of
about 270 cps. This is though t lo be assoeialed with one of t he
fundamental vihmtory modes of t he hydrofoil itself. Although
audible noise is generated, the resulti ng forc·e osc·illations are small
as m<'asnred by t he I'm bedded strain gage. The eavity rel.ttins its
Nonsteady Cavitation
As men tioned previously, oscillations in cavity length and
hyd rody11amic force developed when the c·avity was about 60
percent or so of t he chord and persisted until the c·avity was at
least 1.2 times the chord. A description of the development of
this process now seems in order. The geneml behavior for all
6
Transactions of the AS ME
12
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cavity length-to-chord ratio for a plano-convex hydrofoil at an angle of
attack of 8 de g
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Fig. 1 S This sequence of photographs shows in plan view behavior of
cavity during one cycle of oscillation beginning with top left-hand view
and proceeding down each column . flow is from leh to right, at a tunnel
speed of 27 fps , with leading edge of plano-convex hydrofoil to le ft In
each case. Angle of attack is 6 deg and cavitation number 0 .90. Time
lapse between each photograph is 0.0042 sec.
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cavity length-to-chord ratio for a plano-convex hydrofoil at an angle of
attack af 1 0 d e g
Journal of Engineering for Power
frothy chamc·te r unti l just before oscillation eommenres, at whi c·h
point the portion of t he cavity near the leading edge hec·omes clea r
and glassy. Hhorlly thereafter , the cavity begins to oscillate.
These initial oscillations are of s ma ll magnitude, both in ex tent
a nd foree, a nd arc relati vely high in freque ncy. For the present
tunnel rondilions, these frcque nc·ies may range fro m 50- 60 <'ps.
This stage of oseillalion seems lo b e rather tra nsitory; and with
a slight decrease in tunnel pressure, Lhe oscillation c-hanges over
into a more rhamC'leris liC' luw-frequenC'y, large-amplitude dis-
1
Ol 0
tu l'lxmre. T he oseill ntions t hen typ if'ally ha ve a double a mplitude of about one half chord. Typical frequencies in t his sta~e
·r-under the ("Onditions of our tests were about 12 to 2.') ep d(''
pcnding upon velocity and angle of attac k. T he oscillations
0 .20
I-fpersisted wit h furt he r loweri ng of tun nel pressu re until t he cavity
tl
was about one fo urth longe r than t he chord. Gene ra lly, t he
v
1a mplit ude of t he cavi ty and force osci llation dec reased. T he
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flow t he n became quite steady wit h pmpe r fu ll cavitaLion
-::::::.
0 . 82
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developed . During t his e n t ire process, the forces, average :.t nd
-o;:::::
0 , 70
1---t-nonsteady, f·i1-st increased a nd t hen dec•·eased. The maximum
average for("e an d nonsteady fo ree occu rred at or near Lhe condi!-'
tion of max imum oscillation in Lhe f'a v ity.
One eycle of the cav ity oscillation is shown in F ig. 1.'). This
0
,. ,.
<2
30
"
"
fig ure shows t he p lan form of the de velopi ng cavi tation at a n
'II- fi /UC
angle of attack of 6 deg. T he flow is left Lo r igh t wi t h t he lead in)!:
edge being at t he left, in both !"Olumns. The sequence starLs at
Fig. 16 Re duced fre que ncy during phase of maximum forc e o s cillations
as a function of tunn el sp eed for varying angle s of aHack.
Reynolds
t he uppe r left a nd t ime inc reases downw a rd. T hese photogmphs
number range is from 0 .62 X 106 to 1.05 X 106•
are selecLed from a test fi lmstri p taken at 1200 frames per sec.
SLnrting at Lhe m inimu m caviLy le ngt h, the cavity grows s mooLhl.v
and, as it approaches t he e nd of the hydrofoi l, a reentrant jet. is
seen to for m a nd g radu a lly fi ll Lhe rearward portion of Lhe cavity.
On reaching the end of t he foi l, t he cavity surface beco mes uneven
and irregular and s ma ll vort iee may b e s hed from Lhe end of Lhe
r·>wity, caus ing small fl uctu a tio ns in the fo rce o n t he hydrofo il.
I
.
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'
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-20
The flow wi t,hin t he cav ity t he n appears to become uns tab le a nd
a la rge volu me of cavity is a b rup tly shed into t he stream ; t he
(·y de is t,hen repeated. T h is sequence of e ve n ts is similar to t h:1 t
nf nons teady (•f\vitation reported by K nap p IGI. H owever, t hc l"('
n.re im po rtan t d itfere n res: The cavity is not co mp letely f-i lled
hy t he reen tran t jet, alt hough oscilla t ions from inrnmpletely
filled cavities a re re por ted by hi m. In t he present case, a nd as
r·an be seen in F ig. 15, there is a lways a cavity at the leading edge
of t he hy drofoil. The geneml stages of Lhe osei llittion a re rerLain ly very mu ch as desc ribed by Kna pp: "(a) Forma Lion an d
growt h, ( b ) fi lling, and (c) b reakoff." The othe r di fferenl'e in our
view is t hat, when breakoff OC!"urs, a large ('h a nge in force e ns ues
a nd t hem is a la rge con esponding eha nge in eirculnLion . T his
Fi g . 17 Pe rce ntage fo rce ftuctuation s and cavity le ngth o s cillation a s a
wou ld lend one to sus pect t hnt t he d y n amif"s of t he p resen t p hefunction of time in region of ma x imum o scillatio n s for an angle of aHack
no menon a re related t o t he Lime-vary ing circulation.
of 6 d e g, tunne l s pe e d of 2 7 f ps , and cav itation number of 0 .90
T he t races of the strain gage (to b e discussed presently) were
used to meaRure t he frequency o f the "strong" cavi t~ttio n oscillatio ns. These a re show n for various angles of attac k ~tnd t unnel sequenf'e of prints fro m the 16-mm fil m. :\ l casurements ('Ould be
velocities in F ig . 16. T he freq uen cies are reported in terms of ft
made readi ly from t he o riginal film, however , by projeeLing the
dimensionless Strouha l nu mber, c hord times frequen cy over t u nnel
fil m fmme b y fmme on a Sf" reen. B y t h is means, measuremenLs
speed. T he ra nge of t his p a rameter is fro m abou t 0.07 fo r a 4of for<·e a nd <"ttvity le ngth were mad e, a nd a typical ex:\mp le of
deg a ng le of attack to a bout 0. 14 at an 8-deg angle of atta<"k.
suf" h a measure me nt is s hown in l•'ig. 17. The re it is seen Lhat
Alt hough the re is some variation with speed at t he lowest angle,
t he double a mplitude of t he fo rr·e os<"i llation is about 20 pe rcen t
t he redueed frequ ency is relatively co nstant at t he h ighest a ngle.
of t,lw mea n. T he maximum fo l"(·e oceurs at t he max imu m
T his wou ld suggest that t he freque ncy of osr:i llaLion is not st ro ngly
etw ity le ngth a nd t he mi ni mum fol"(·e at the m inimu m ("avity
depend en t on t he rigid ity of t h e surrounding t u n nel str ucture.
leng t h. Gene rally speaki ng, t here is no substantia l phase c ha nge
There is Lbe basic q ue t ion, however, of t he effect of t he tunnel
between the oscillations of cavity length a nd Lhe oscillaLions of
a nd flow "complia n ce" on su ch transien t cavity flows as described
force. T he lnek of an appreciable p hase cha nge be Lween t he
herei n. F or exam p le, if the t u nnel were p erfectly rigid a nd if
force a nd cavity oscillations su ggests Lhat possib le ine rt ia l effects
there were no free surface othe r tha n t hat of t he cav ity itself,
of the fluid in the t un nel circui t a rc no t large, as has bee n indi cated
Lhe n a n infin ite pressure differe nce ( in an incom p ressible medium )
a lready, a nd Lh at t he tunnel bou nd a ries are in effect not rigid .
wou ld be required to create t he cha nging cavity volume. T he
Os<"illogra ph recordings of t he fo H·e a re s hown in Fig. 18 for one
t u n nel is comp lia n t ho wever; nume ro us pockets of vapor co llect
angle of attar· k ( 6 d eg) and vary ing tu nnel p ressure at consta n t
in the di ff user; a nd fro m the p hotographs in Figs. 7(b) to LO(/>),
speed. T he sequence of even ts pre vious ly described is borne ou t
it can b e see n t hat Lhe re a re e n t rained vapor-a ir b ub bles in l he
by this figure. For example, the hi gh-freq ue n cy oscillations can
flow. All o f t hese effects evidently provide !L eush ion fo r I he
j ust, b e disce rned in Lhe top traee ; an d as Lhe p ressure is lowered
fl uc tuating cavit y volume.
t he s mall-amplitude fast oseillations develop . These lead in to t he
It is in terest ing to n ote t h at, t he freque ncies observed by K napp
ch anwteristic la rge osci llation s hown in t he fo u r t h Lrace. This
are mu ch h ighe r t ha n those of t he prese nt wor k ; in his wo rk, Lhey
tra!"e is near the p oin t of max imum oscillation nnd maximum lift
ranged from .51 to 2 00 cps. Calc ulaLions o f Lhe redu ced frefon·e, as ean be ve rified from F ig . 8. Finally, the oscillations die
q uen cy, based on t he length of t he caviLy of t h ese oscillations, are
away wit,h fur ther reducLion in pressure.
abou t twice t he presen t valu es. Again , t he reduced frequ en cy o f
We have a lready men t ioned t he relative in depende nce of reh is observations is s ubstan tially inde penden t of tunnel s peed .
duced frequency up on t unnel s peed fo r Lhe la rge os(·i llations.
It was me n t ioned Lhat the ouLput of t he strain gage was pho toThe higher-frequency oscillations (the ser·ond and third Lraces in
graphed on t he motion-pic-ture fil m su eh as s hown in F ig. I r..
F ig. 18) a re, ho we ver, mo re or less indepe nd ent of tunnel s peed.
This m ises the possibility t.ha t t h ey are related to t he dy nami cs of
U nfortu nate ! ~', the tmr·e was too dim to he reprodur·('d in th is
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Transactions of the AS ME
Fig. 18
Trac es of oscillating forc e a s recorded from strain-gage output
as a function of time. Each tim e divis ion re pre s e nts 0 . 1 sec . Zero fo rce
datum is als o illustrate d at boHom of e ach trace . Tunne l s p eed is 31 .4
fps at angle of attack of 6 d e g . Se que nce b e g in s with top trace taken a t
a cav itation number of 1.69 and proceeds downw a rd wit h co rresponding
ca v itation numbers of 1.18, 1.03, 0 .9 3 , and 0 .48.
force oscillation is give n by fourth trace.
Point of maximum
l ite fo rte lmla ru·e or of the tun nel. T o in vestigate t his poinl, lite
fon·e bala n<·e with model altlwhed was shock-ex<·iled but no
evidenee of the 50 60 <"PS os<"illa t ions seen in Fig. IX were obser ved . I n fal't, the lowest well-defi ned structu ral frequem·y
obse rved was abou t I 00 cps.
T he unstE-ady cavitation behavior is by no means restricted to
single hyd rofoils. A s imilar p he nomenon O<'l'Urs in t he case of
<·aviLaLing <·as<·ad es, us horn e ou t by exper iments rel'en lly carried
o u L in Lit<' l fydmdynamits Laboratory. The series of events
desl'ribed here octUI'l:! again with little change qualitatively.
Conclusions
T he chantl'lerisli<·s of a plano-convex hydrofoil have heen
dcsnihed fo r· non('avilaling a nd l'avilaling fiows. lL is found
that the l'>Lvilalion behavior tan he divided into t hree regimes:
A pa r t ia lly l'aviLating region , a fu lly ('av itaLing region, and a
region sepamling t hese two in wh idr the How is a lways unsteady.
ln t he partial and fully eavilaling regions, t he forces are sleady
a nd are well defi ned in terms of the cavitat ion n umber and angle
of aLlack. In Lhe unsteady zo ne, however , t he fon·es fi u1·tuale
and t he <·<wily oscillates vio le n tly. T he flu l'lual ing no rmal for('e
on t he hydrofoi l meusur·ed in the present experiments has an
a mp litude variation of ± 10 perl'ent of its mean value. The rcdu <"ed frefi uen<"y of Lhe fo rce osl'i llaLions appears lobe !t function
p r ineipally o f t he angle of atlac·k. Redu<"ed frequencies based on
l'hord length and tun ne l velo!'ily are in the range of().!() to O.:W
for angles of atla<"k of 10 deg or less a nd fo r t he tunnel veloeilies
used . FOI' the present les ls, lhe <·aviLy fl u('tuaLions a rc in phase
wit h l he fo r<"e oseillalions and t he variation in cavity length is of
t he order of 60 pen·enl of the dwrd.
The p resent. invest igations on t he unsteady region of lire
l'avilalion on a hydrofoil a re o f a p reli minary nature, t he aim
be ing lo a('(tuire some information on the p ro('esses in volved and
to obtain a general q u alitative and f!Uant itat ive pi<"turc of the
unsteady phe nomenon. F u ture work on t he efTel'IS of tunnel
boun daries a nd possib le free s urfa<"es are dearly ne<"essary.
Acknowledgments
T he par lil'ipation of the m(•mbers of t he H ydrodynami1·s
Laboratory starT, and in par lil'ular Llrat of T . Ki!'eniuk, during
the l'ourse of t his experimen tal in vestigation is gratefully appre(·iated. This work was suppo rted by the Department of the
Navy u nder Co n lra<·l ~o n r 220(24).
References
1 J. Balh an, " ~ l ctingen aan l~nige hij ~dleCp:;hc.:hrocven (:crhruikelij ke Profielen in \ 'lakke Htrollling met. en zoncler Ca.vilat..ie,"
Nede rl andHC flche<•p,houwku udig l'roe fHLation Tc \.Yagcuingen , l'uhlieation No. 09, 195 1.
2 0. \\'akhner. " Profilme,.,ungen hei Kavitation," ll.vdrornedtaniHt·he Problerue deH SchifTsantriebs, edited by G. Kempf and
E. Foerster, Hamburg, 1932, pp. 256-2fj7; Engli ~h ab•tract, pp.
420-4~1.
3 R. W. Ken neen, " Water Tu nnel Tests of NAC'A 44 12 "'"'
\\'a ld111er Profile 7 ll ydrofoit. in :-<on-C'•witating and Cavitating
FlowH," ll ydrodynami<·s Lahorutory, ('alifomia Institute of Te<"hnology, Pu.,adena, Calif., Report No. 47-5, 1956.
Journal of Engineering for Power
9
4 B. R. Parkin. " Experime n ts on C ircular An: a nd Flat Plate
H ydrofo ils in Non-C avitating a nd Full Cavity Flo ws," H y drodynamics Laborat ory, California Institute o f T echnology, Pasadena,
C a lif .. Report 47-6, 1956.
5 l\1. C. Meijer, "Some Experime n ts o n Partly Cavitating H ydrofoils," I nternational Shipbuildino Prooress, vol. 6, n o. 60, Hl59,
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6 R . T . K napp, " Recent In vestigations o f t h e Mech an ics of
Cavitation and Cavitatio n D amage," TnANS. ASME, vol. 77, 1955,
pp. 1045- 1054.
7 R. T. Kna p p, J. L evy, J. 1'. 0 ' e ill. and F . B. Brown , "The
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8 G. M . Hotz and J . T. McGraw, " The High Speed W ater
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47- 2, 1955.
9 A . G. Fabula. "C hoked F low About Vented or Cavitating
H yd ro foils." J ournal of Basic En oineerino. TnANS. ASME , Series
D, vol. 8G, 1964, pp. 5G I- 568.
10 R. B . W ade, "Wat er Tunne l Observations on t h e Flow Past "
P la no-Convex H y drofoil ," H yd ro dynamics Laboratory, C aliforni>L
Instit u te of Tech nology , P asadena , Calif., Report No. E-79.6, February, 1964.
II D. H . Williams, A. F. Brown, an d C . J. W. Miles. "Tests o n
Four Circul a r-Back Aero fo ils in t h e Compressed Air Tunnel ," Aeronau t ic>Ll Resear ch Co uncil , Technic>LI R eport R and M , No. 230 1,
1948.
12 G. B. M cCull ough and D. E. Gault , " B oundary Layer a nd
Stalling C h aracteris t ics o f t h e N ACA 64A006 Airfoil Sect ion,"
NACA TN 1923, 1949.
13 D. D. Carro w, " A Note on t he Bounda ry L ayer a nd Stalling
C haracteristics o f Aerofoils," Aeronautical Research Council, C P
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14 T. Y. Wu, " A Note on t h e Linea r and Non-Linear Theories for
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15 A. J. A costa, "A N o t e on Part ia l Cavitation o f Flat P late
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Printed in U. S. A.
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