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OCR for page 852
24th Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Experimental and Numerical Investigation of the
Cavitation Pattern on a Marine Propeller
Francisco Pereira~, Francesco Salvatore~,
Fabio Di Felice, Mauro Elefante2
Istituto NazionaTe per Studi ed Esperienze di Architettura NavaTe,
2 Centro Esperienze Idrodinamiche delIa Marina MiTitare
Rome, Italy
ABSTRACT
An experimental investigation on a cavitating pro-
peller in a uniform inflow is presented. Flow field
investigations by advanced imaging techniques are
used to extract quantitative information on both the
cavity extension and thickness. A refined map
of the propeller cavitating behavior is established.
Measurements are compared to numerical results
obtained using an inviscid flow boundary element
method for the analysis of blade sheet partial and
super-cavitation. The effect of the trailing wake vor-
ticity on the prediction of the cavitation pattern is
analyzed via a wake alignment technique.
INTRODUCTION
For two-dimensional and three-dimensional isolated
hydrofoils, the cavitation literature offers a limited
number of works where quantitative information has
been successfully obtained from the direct observa-
tion of the cavitation pattern (see e.g. Pereira et al.,
1998~. Even more complicated is the measure-
ment of the cavitation development on a marine pro-
peller. The problem is not new, but is very little
documented. Lehman (1966) seems to be the first
to report such measurements, using multiple ortho-
graphic views to build an estimate of the cavity vol-
ume on a model propeller. Yamaguchi (1985) reports
a stereo graphical technical to carry out the thickness
of the cavity. Using a technique that has been much
used in the case of isolated hydrofoils (Dupont and
Avellan, 1991), Ukon et al. (1991 ) report measure-
ments of the cavity thickness distribution. Full-scale
measurements on propellers are rare because of the
obvious difficulty: Tanibayashi et al. (1991) is, to the
authors knowledge, the only work of this kind.
In the present work, new developments are in-
troduced into the observation and quantification of
the cavitation pattern and are applied to a skewed
four-blade model propeller in a uniform inflow. Us-
in~ a novel cavitation analysis approach suitable for
field applications, the area and the maximum thick-
ness are estimated. A detailed chart of the cavitation
figures is also determined and analyzed.
A Wageningen modified type model propeller
(INSEAN E779A) was selected for the present re-
search project for two main reasons. First, this
model propeller has been widely studied with the
most advanced flow measurement and visualiza-
tion techniques, such as laser Doppler velocimetry
(Stella et al., 2000) and Particle Image Velocime-
try (Di Felice et al., 20001. A large amount of data
has been collected providing a thorough documen-
tation on the non cavitating flow characteristics: the
propeller geometry and LDV data are now available
at http://crm.insean.it/E779A. The present work is
intended to extend the existing database to cavitat-
ing flow conditions. Second, the E779A model ge-
ometric characteristics are very close to those used
to design propellers for twin-screw ships. Further-
more, the propeller performance and its cavitating
OCR for page 853
flow phenomenology are quite complex and provide
a complete and challenging benchmark for the vali-
dation of numerical codes.
The unique data collected in the present work,
covering a large number of flow conditions, provide
a deep insight into different flow features such as
sheet cavitation, tip and leading edge vortex cavi-
tation, bubble cavitation and supercavitation. A rich
cavitating flow database is being constructed, repre-
senting a powerful tool to investigate in detail the
range of applicability of new theoretical models and
to assess the accuracy of predictions. As a mat-
ter of fact, the validation of numerical predictions
against experimental propeller flow data is not well
addressed, even though the field is very active (Kim
and Lee, 1997; Mueller and Kinnas, 1997; Dang,
2000~. Unfortunately, and because existing analyzes
are usually based on experiments that cover a re-
stricted set of flow conditions, only a limited num-
ber of flow patterns are documented. Moreover, in-
adequate data formats do not allow for a convenient
exchange of information.
State-of-the-art theoretical analysis of cavitating
propellers is based on inviscid-flow models such as
lifting surface methods and boundary element meth-
ods. Viscous-how modeling by RANSE or LES ap-
proaches are still limited to simple geometries as
two- and three-dimensional hydrofoils in uniform
flow. Propeller flow applications seem to be yet
a long-term challenge. The theoretical formulation
used in the present analysis is based on a well estab-
lished boundary element approach that has been de-
veloped by several authors over the past two decades
(Lee, 1987; Kinnas and Fine, 1993~.
The paper describes, in a first part, the experi-
mental methodologies that are introduced to charac-
terize the cavitation development on a rotating pro-
peller. In a second part, a brief outline of the nu-
merical approach is given. Finally, the results of the
experimental investigation are used to assess the va-
lidity of the numerical method.
EXPERIMENTAL ANALYSIS
Experimental Setup
The experiments are carried out at the Italian Navy
cavitation facility (C.E.I.M.M.) The tunnel, a closed
type circuit, has a 0.6 m x 0.6 m x 2.6 m square
test section. Optical access to the section is pos-
sible through large Perspex windows. The nozzle
contraction ratio is 5.96: 1 and the maximum wa-
ter speed is 12 m sol. The maximum free stream
turbulence intensity in the test section is 2%, and is
reduced to 0.6% in the propeller blade inflow section
atr/R=0.7,whereR=D/2=0.117misthepro-
peller radius. The flow uniformity is within 1% for
the axial component and 3% for the vertical one.
The skewed four-blade model propeller has a
uniform pitch-to-diameter ratio of 1.1 and a forward
rake angle of 4°3". The blockage ratio in the test
section is about 10%. A sketch of the propeller ge-
ometry is given in Figure 1.
_\ ___~___~__= __
: W~ W
_ . . _i HI ~ ~ ~ ~ ~
Figure 1: E779A propeller geometry
In the following sections, the advance ratio J.
the cavitation number 60, the torque coefficient KQ
and the thrust coefficient KT are defined respectively
by vo/(nD), 60 = (Po—Pv)/(p vo/2), Q/(p n2 D5)
and T/(p n2 D4), where po is the reference pressure
measured at the propeller axis, Pv is the fluid vapor
pressure, p is the fluid density, vo is the upstream
velocity, and n is the propeller rotational speed (rps).
The measurements described here have been per-
formed across a large range of working conditions,
60 and J being the varying parameters, see Figure 2.
The measurement configuration is pictured in
Figure 3. The blade angular position is adjusted to
allow a full view of the blade face. A CCD camera
is dedicated to the measurement of the pattern area
and is oriented at an angle with respect to the test
section window. To minimize the aberrations intro-
duced by the thick window and the water/glass/air
interfaces, a glass tank in the form of a wedge and
filled with water is clamped to the window in such a
manner that the camera optical axis is normal to the
wedge front window. The propeller shaft is equipped
with a rotary encoder that supplies an electrical trig-
ger signal to the image acquisition system. This lat-
OCR for page 854
o.9
0.85
0.8
0.75
0.65
+ ~ 4h 4. ~ -
I,_ -
+*++4~+
+++ +~+ ++
+ +
+ ~
0.7 ++'+*+++ +
~11 1 1~ - 1- ~
+
+ +
+ +
+ + +
+ + 4.
+ + ~ +
~_, - + + ~ ~ ~ -4 ~ ~ + + + 4.
.
0.6
0 2 4 6
Oo [ - ]
8 10
Figure 2: Hydrodynamic conditions
ter controls the phase between the propeller and the
image capture, and pilots a pulse delay generator ac-
cordingly. This instrument sends the necessary trig-
ger signals, with the appropriate time delays, to the
camera and the illumination system. A single high
intensity flash lamp, with a 10 Us light pulse, is used
to observe the cavitation pattern. For every condition
shown in Figure 2, 128 images are acquired. Simul-
taneously, the flow parameters are recorded: So, J.
Go, the cavitation number (Sn based on the propeller
rotation speed, KT and KQ.
Cavitation Extension Measurement
We introduce a novel methodology to determine the
cavity area, designed to be implementable in field
situations where the experimental constraints do not
always allow the use of standard techniques of image
analysis. The common method in analyzing cavita-
tion images consists in enhancing the contrast be-
tween the cavitation pattern and the rest of the im-
age. This is usually done by thresholding the image
as used by Pham et al. (1998) for instance. How-
ever, it is well known that this approach is very sen-
sitive to variations of the illumination. The region
of interest needs also to be easily identifiable from
the background, either by running the experiment in
such conditions that only the cavity scatters light or
by removing a background image from the cavita-
tion image. In a complex environment such as a ro-
tating propeller, this approach is not robust enough:
tip vortex, scattering of the blade, fluctuations of the
illumination intensity, unwanted objects in the field
of view, are among many other causes that make the
analysis difficult or unreliable if done automatically.
The approach used here, and depicted in Fig-
ure 4, is based on the cross-correlation between a
template image (Figure 4a) and the image under con-
sideration (Figure 4b). The template image may be
the blade viewed in non-cavitating conditions. The
cross-correlation is a robust tool to make the compar-
ison between images: a high correlation peak would
indicate that the template and the image are simi-
lar, while a slight difference would drop the correla-
tion coefficient to low and distinct levels. However,
in order to localize the transition regions between
the cavitating and the non-cavitating situations, the
cross-correlation is performed on small image re-
gions, represented in red color in Figure 4a and Fig-
ure 4b, with the resulting correlation image repre-
sented in Figure 4c. The size of the correlation win-
dows is set to 7 x 7 pixels in our case. The convo-
lution operation is performed across the whole im-
age to produce a correlation image, see Figure 4d,
where the differences between the template and the
image being analyzed are clearly and uniquely iden-
tified as cavitation features: cavitating vortices from
the blade tips, contour of the attached cavity.
~ _ ~~ ~ ~ _^A
Figure 4: Image cross-correlation procedure: (a) tem-
plate image; (b) cavitation pattern image; (c) local cross-
correlation; (d) correlation image
The correct quantitative evaluation of the cavita-
tion area Ac is only possible if done in a known coor-
dinate system. The registered image is a perspective
representation of the three-dimensional blade sur-
face. To report this image information into a plan
view, where the area would be accurately measured,
OCR for page 855
Cavita~
patterned
Water
_
_
_
Control & acquisition
119~
Rotation
pulse
1 ----------I
- Camera
~ ~ Images
controller
Trigger signal
Figure 3: Experimental setup
one could perform a simple back projection using
basic geometric optics, and this would correct for
the variable magnification. However, this is only
valid if the system is free of optical aberrations and
requires an exact knowledge of the optical parame-
ters. In fact, the complex lens system composed of
the camera objective, the prism, the test section win-
dow and the water medium introduce focusing aber-
rations and optical distortions, which are the source
of non-linear magnification. If the focusing aberra-
tions can not be removed, image distortions can be
compensated through calibration procedures (Soloff
et al., 1997~. This operation, also very common in
image processing techniques where it is referred to
as warping transformation, is performed here. We
use a second-order perspective transformation to ac-
count for the non-linear effects. If x and y are the
original image coordinates of a point, the warped im-
age coordinates x' and y' are then defined by
, a8x+aOy+aO+a3x2+aOy2+a
X —
by x + bo Y + bo + ho x2 + bo y2 + bo xy
, _ at x + ai y + a2 + a3 x2 + at y2 + aS by
Y by x + by y + b2 + b3 x2 + b4 y2 + bS xy
. .
where at and by are the coefficients of the transform.
. .
al and by are unknowns that are determined by a
Nelder-Mead least-square minimization method, us-
ing reference points on the distorted image and the
known corresponding points in the plan view.
To perform this correspondence, a grid is printed
on a blade, with lines drawn spanwise at various
r/R, and radial lines regularly spaced chordwise, as
shown on the top image in Figure 5. Instead of de-
termining one unique transformation for the whole
image, as commonly done, Equation (1) is calcu-
lated for each polygon defined by the intersection
of the radial lines and of the constant r/R lines.
This approach refines the transformation and avoids
the side-effects inherent to the use of non-linear
polynomials in regions where calibration points are
not available. Three sample distorted polygons are
shown in the top image of Figure 5. Their correspon-
dent warped image, determined using the locally cal-
culated warping function expressed by Equation ( 1),
is shown on the bottom image, which is the final
undistorted image. For clarity, we show only four
corresponding points for each polygon.
<1~ The warping procedure is applied to the correla-
tion image (Figure 4d), on which standard threshold
techniques are utilized to track the cavitation pat-
tern. The use of a threshold is now possible with-
out the drawbacks outlined previously, for the corre-
lation operation described above unequivocally en-
OCR for page 856
Figure 5: Warping procedure: original (top) and warped
(bottom) images
__
trances the features of interest. Morphological oper-
ators may then be used to isolate the cavitation pat-
tern from small and undesired features, such as trav-
cling bubbles, tip vortices. This sequence of opera-
lions results in a pattern image (Figure 6), where the
cavitation region over the blade is clearly outlined.
The undistorted and thresholded image is thus
used to correctly estimate the area of the cavitation
pattern, with a common coordinate system with the
numerical grid used to perform the computations de-
scribed in the next sections. The cavitation area Ac
follows immediately and is expressed in percentage
of the blade total area Ao starting at r/R = 0.3.
We finally propose a rough estimate of the sheet
cavitation thickness by simply measuring the thick-
ness of the trailing section of the sheet or the size
of the vortex structure immediately downstream the
sheet. Figure 7 shows four cases. The thick-
ness is measured on the average image and at the
Figure 6: Cavitation extension
location of minimum cross-section of the cavita-
tion pattern. The measured dimension is then non-
dimensionalized by the propeller diameter, for the
purpose of comparison with the numerical solution
of the thickness.
1__
In__
_ -
_1
ma
_:
_ ~ 3
it_
-
Figure 7: Cavitation thickness: a rough estimate
THEORETICAL MODEL
Governing equations
Under the assumptions of inviscid and initially irro-
tational fluid, the perturbation velocity field is irro-
tational and hence it can be expressed in terms of a
scalar potential, ¢. In a frame of reference (Oxyz)
fixed to the propeller with the x-axis parallel to the
OCR for page 857
propeller axis, the unperturbed flow velocity reads
Vet = VA + Q x x. (2)
where VA is the incoming flow to the propeller, x =
(x,y,z), and Q = (Q,0, 0) is the angular velocity of
the propeller. Hence, the total velocity field is
q = v, + Vo. `3y
By incompressible flow assumptions, the poten-
tial o satisfies the Laplace equation V20 = 0 in
the unbounded fluid region Up surrounding the pro-
peller, its trailing wake and the cavity. In the frame-
work of potential flow modeling of bodies that are
capable to generate lift or thrust, the wake denotes
a zero thickness layer where the vorticity generated
on the body is shed, and represents a discontinu-
ity surface for the potential. The cavity denotes
the fluid region where vaporization occurs. In the
present approach the cavity is assumed to be a thin
layer attached to the blade suction side and, if super-
cavitation occurs, to the trailing wake surface.
The Bernoulli equation gives the pressure p and,
in the frame of reference fixed to the propeller, reads
aO+ {q2+P+gzo= ~V2~+Pp°, (4)
where t denotes time, q = I, Vet = v, g iS the
gravity acceleration and z0 denotes depth.
The Laplace equation for o is solved by impos-
ing boundary conditions on &{p. On the cavitation-
free portion of the propeller surface, namely the wet-
ted body surface ~Y'WB, the impermeability condition
yields q- n = 0, or, recalling Eq. (3)
aa0 =_v,~n on3°WB' (5)
where n is the outward unit normal to the surface.
Across the wake surface 3°w the pressure is con-
tinuous. By applying mass and momentum conser-
vation laws under non cavitating conditions, yields
( an ) on amp, (6)
where 3°ww is the cavitation-free wake portion, and
denotes discontinuity across the two sides of the
wake surface. Recalling Ap = 0 across the wake sur-
face and combining the Bernoulli equation (4) and
Eq. (6), one obtains that the potential discontinu-
ity /\o is convected along wake streamlines, and the
convection velocity is the averaged flow velocity on
both sides of Law
A further condition on o is required in order
to assure that no finite pressure jump may exist at
the body trailing edge (Kutta condition). Following
Morino et al. (1975), this is equivalent to impose that
the potential discontinuity at the wake trailing edge
equals the difference between potentials on suction
and pressure sides at the blade trailing edge
A0 (XTE) = ATE OTE - (7)
In order to take into proper account crossbow ef-
fects, Eq. (7) is coupled with a pressure-based itera-
tive Kutta condition as proposed by Kerwin (19871.
Boundary conditions on the cavity edge 3°c are
imposed by assuming that the cavity is a fluid region
where the pressure is constant and equal to the vapor
pressure Pv By imposing p = pv on Arc, and by
using the Bernoulli theorem (4), results
q = [(nD) on—
+gZO) +v,2~ ,
(8)
where CTn = (Po—Pv)/ 2 p~nD)2 denotes the cavita-
tion number referred to the propeller rotational speed
n (rps) and diameter D. Equation (8) is used to obtain
a relationship between ~ on 3°c and On. By consid-
ering on each blade surface a curvilinear coordinate
system (s, u) with s in chordwise and u in spanwise
directions, respectively, and s,u covariant base vec-
tors, one has
qS = qu cos ~ + ~ sin ~ ~ N/q2 _ q2 _ q2 (9)
where qS = q s, qu = q u and qn = q n. Decom-
posing qS by Eq. (3), and by integrating in chordwise
direction from the cavity leading edge abscissa SCIE,
yields
As
0(5,U) = 0(SCLE,U) + J (qS V! s)ds. (10)
S CLE
Equations (8) to (10) combined provide a non lin-
ear boundary condition for 0 on 3°c The present
derivation differs from Kinnas and Fine (1992) in
that quantity qn in Eq. (9) is not neglected in calcula-
tions. In the case of supercavitating flows, a similar
derivation leads to an expression for the velocity po-
tential on the cavitating portion of the wake surface
us
O (s, u) = 0 (STE'U) + J (qs —Vl S) ds, (1 1)
S TE
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Representative terms from entire chapter:
wake model
where STE iS the blade trailing edge abscissa.
The condition p = Pv and hence Eq. (8) are not
valid in the aft portion of the cavity where pressure
tends to wetted-flow conditions through complex
two-phase flow phenomena. In the present work,
a cavity-closure region is introduced in which pres-
sure is forced to vary smoothly from p = Pv to wet-
ted flow values behind the cavity trailing edge. The
fraction ~ of the cavity length occupied by the clo-
sure region is prescribed within a range that has no
influence on the flowfield solution (0.1 < ~ < 0.31.
An expression of the cavity thickness he is ob-
tained by imposing a non-penetration condition
on ~c. By combining the constant-pressure and
the non-penetration conditions, follows that 3°c is
a material surface. Denoting by 5~CB the cavi-
tating portion Of 5~B' and by ~ the normal dis-
tance to ~CB, the above condition corresponds to
(~/3t + q v) (ll—he)' or
aahC +Vshc q = at +v,
on 50CB~ (12)
where Vs denotes the surface gradient on JAMB- In
the case of supercavitating flow conditions, Kinnas
and Fine (1992) show that a similar condition applies
on the cavitating portion Ago of the wake surface.
Equation (12) represents a partial differential equa-
tion for he that may be solved once o, Do/On and
O¢/Ot are known.
Boundary integral formulation
The Laplace equation for the velocity potential is
solved by means of a boundary integral formulation.
By assuming that the perturbation vanishes at infin-
ity, the third Green identity yields at any point x ~ Dip
E(`x>)o(
Solution procedure
The proposed methodology for the analysis of cav-
itating propeller flows is based on the assumption
that the flowfield perturbation induced by the cav-
ity has a negligible impact on the location of the
wake surface. The validity of this assumption is
discussed elsewhere (Arndt et al., 1991; Kinnas and
Pyo, 1999~. Thus, the wake alignment procedure is
performed under non-cavitating flow conditions. A
wake geometry of prescribed type is used as the ini-
tial guess. Once the actual shape of the trailing wake
is determined, the cavitation model is turned on and
an initial guess ~OCB°) for the cavity shape is imposed.
In the present analysis, leading edge cavity detach-
ment is prescribed by experimental observations.
By solving Eq. (13), O¢/an on ~OCB°) is obtained
and an estimate in(°) of the cavity thickness is eval-
uated by Eq. (12~. A free cavity length approach is
used here. The cavity trailing edge is determined by
imposing he = 0 at each surface strip in chordwise
direction. If this condition is not verified, it is as-
sumed that the guessed planform is too small and the
cavity shape is extrapolated. Thus, an updated cavity
planform CAB) is obtained and the procedure is re-
peated until the difference between cavity volumes
computed at two successive iterations is less than
a prescribed value Marc = 1 x 10-5 in the present
analysis). Convergence after more than 12 iterations
is infrequent and usually occurs only for those flow
conditions that are outside the proper range of appli-
cability of the present cavitating flow methodology.
As a fundamental step to assess the accuracy of
the present methodology to predict non-cavitating
and cavitating flow features, the effects of surface
discretization on the numerical results have been
investigated and are briefly addressed here. A
family of grids characterized by a constant ratio
between the number of blade elements in chord-
wise direction MB' in spanwise direction NB' and
the number of wake elements in streamwise direc-
tion per turn, MW is used. In the present anal-
ysis, MB/NB = 4, MB/MOO = 4/5, with MB =
{24, 36, 48, 60, 72, 84, 96) is considered. Hub sur-
face is discretized by a number of elements ranging
from 528 (MB = 24) to 1248 (MB = 96~. Figure 8
shows the effect of grid refinement on the calculated
propeller thrust and torque in the case of advance
coefficient J = VA /nD = 0.71 in non-cavitatina and
cavitating flow conditions at On = 1 .5 1 5.
1
0.300
0.290
0.280
0.270
0.260
0.250
0.240
0.230
non-cavitating flow model 0
cavitating flow model 0
~3
n Awn - - -
,
0.03 0.04 0.05 0.06
1/MB
0.00 0.01 0.02
0.050
0.048
0.046
by 0.044
0.042
0.040
non-cavitating flow model 0
cavitating flow model o
jV ~
EGO A a A ' A
0.038
0.00 0.01 0.02
0.03 0.04 0.05 0.06
1/MB
Figure 8: Effect of discretization on calculated propeller
thrust (top) and torque (bottom) coefficients. Non cavitat-
ing flow at J = 0.71, and cavitating flow at J = 0.71 and
On = 1.515
Smooth convergence 1S observed under non-
cavitating flow conditions, whereas oscillations are
present if the cavitating flow model is included.
Figure 9 illustrate the effects of grid refinement
on the calculated cavity area AC and on the cavity
volume Vc (AT denotes the total blade area). Results
are reported in the following table:
MB 36 48 60 72 84 96
C .305 .265 .225 .183 .176 .172
AT
.
V 3 .103 .383 .318 .276 .242 .230 .221
Using MB > 72, discretization errors are less
than 10% of corresponding extrapolated values for
MB ~ °° All the numerical investigations described
in the next section have been obtained by using the
MB = 72 surface grid. In such case, the computa-
tional time required by a non-optimized code to per-
form one step of the cavitating flow solution scheme
is 53 sees. on a 800MHz PC.
°8r
0.7 ~
\
/M~ _ 36 ~
/ . / , .. ~ ~ ~
/ ME; = GO
/ P~S— {2
/ MB=~4 —
/ MB = 96
Figure 9: Effect of discretization on the calculated cavity
planform (J = 0.71,c,,' = 1.515)
0.6 ~
0.2 t
0.1
O _
\
0 0.2 0.4 0.6
J
KT (Exp.) [a
lO*KQ (Exp.)
KT (Num.)
10AKQ (Num.)
to
Hi\
~E' C
EN
0.8 1 1.2
Figure 10: Open water characteristics of model propeller
E779A.
0.50
0.48
FLOW FIELD INVESTIGATIONS 0.46
First, the non cavitating flow features of the model
propeller E779A are considered. The experimen-
tal data have been taken from previous works per-
formed at the CEIMM facility. Figure 10 shows
the open water characteristics. In the range of J
considered in the present investigations (0.65 < J <
0.88), the numerical predictions of the thrust coeffi-
cient are in good agreement with the measurements,
whereas discrepancies between computed and mea-
sured torque coefficient are observed for J < 0.8.
This may be motivated by the approximate approach
used to compute the viscous flow contributions to the
hydrodynamic loads that, in the considered range of
J. are relevant for torque and almost negligible for
thrust.
Figure 11 shows the location of the tip vortex
for two values of the advance coefficient. The vor-
tex spatial location has been determined by means
of the vorticity field measured using the PIV tech-
nique. The good agreement between the measure-
ments and the numerical results confirms the valid-
ity of the present wake alignment technique in de-
termining both the trailing wake contraction rate and
the pitch.
~ Due to the low blockage ratio of the facility ( 10%), no tunnel
correction has been considered in the numerical calculations.
or\ J=0.748 (Exp.) O
I \N J=0.748 (Num.)
- 1 ~ ~ J=0.880 (Exp.)
I ~ rum)
0.44
0.42
0.40 _
.
-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
x/D
Figure 11: Tip vortex location in non cavitating flow con-
ditions for two advance conditions.
Using the measurement chart shown in Figure 2
and analyzing the individual cavitation images, the
E779A model propeller can be described in terms
of distinct cavitation patterns. Figure 12 represents
a map of the cavitating behavior of the propeller,
also known as a bucket diagram, where the main
figures of cavitation have been identified. The pro-
peller presents four main patterns: bubble cavita-
tion/traveling cavitation, supercavitation, sheet cav-
itation, leading edge (LE) and tip vortex cavita-
tion. The bubble cavitation is present at low val-
ues of the cavitation number (fin and is coexisting in
most of the cases with supercavitation. This situa-
tion can be observed for J = 0.65,on = 0.528 and
J = 0.88,on = 0.387: bubble cavitation occurs in
· Bubble and supercavitation
~ Partial and tip vortex cavitation
· Supercavitation ~ Leading edge vortex cavitation
0.9
0.S
1
3.('
2.5
2.0
1.5
c)" 1.0
0.5
-0.5
. .
~ 1~,
tic
0.2 0.4 0.6 0.8 1.0
Yl~
1
~ ~ ~ ~ cavitation flow model
i. I_ convergence limit
, ti—-—~—- - · ~
0.7
0.6
\ \
~ ~1
I; ·
!~.
r`~e ·~. · ~ o
.; · · \
_ ~ ·) · ~ ~ \
. · ·\
, / , 1 \ , 1 \ 1 1
0 / 1\ 2 \ 3
J = 0.65, an = 0.528
I r/R=0.4 —
I ..R ~.~j
I r/R=0.8
~ Bubb~ec<.~v a, . ~..
^\
J=0.88,(5n=0.387
r/R=0.4
!','~...~~
r/R=0.8
Bu 1: bte GHAN.'. Or) - ...~.......
6n [ ]
4.0
3.0
2.0
1.0
0.0 . — ~
Cav. riR=O.9
Wet. r/R=O.9
-1 .0
0.0 0.2 0.4 0.6 0.8 1.0
x/C
J = 0~77,
not take into account bubble cavitation). At higher
on, the Cp presents a steep gradient at the leading
edge, responsible for a sharp start of the leading
edge cavitation. This attached sheet cavitation mixes
with the vortex structure downstream and, for the
highest values of (fin, transforms into the so-called
leading edge cavitating vortex, for it starts far from
the tip and along the leading edge, as visible for
J = 0~77~on = 2,082 and (Sn = 3.268. The pro-
peller also presents a form of streak cavitation, see
the case J = 0.88,6n = 0.387. This pattern is char-
acterized by two or three vapor streaks detaching at
precise locations along the leading edge. This sit-
uation is thought to be caused by irregularities of
the model blade geometry (i.e. local roughness) that,
combined with the flat Cp distribution, become the
place of early and very localized cavitation incep-
tion. A curve dentin = Thin fJ) is also plotted in Fig-
ure 12 to identify the locus of (5n values such that, if
on < anon, the predictions by the present numerical
cavitation flow model are not reliable. In such cases,
the iterative cavitating flow procedure usually fails
to converge.
Images of the cavitation pattern for 9 selected
flow conditions are shown in Figures l 3 to l S. These
images have been obtained using the warp procedure
described earlier. The calculated cavity areas by us-
ing both a free wake model and a prescribed wake
model are also shown for comparison.
It is apparent that the predicted extension of the
cavitating area is affected by the shape of the trailing
wake used in the calculations. In particular, if a cor-
rect location of the trailing vorticity is obtained by
means of the wake alignment technique (free wake
model), numerical results (red plots) are in closer
agreement with experiments as compared to those
obtained by using a prescribed wake model (blue
curves).
A quantitative comparison between the observed
and the calculated cavity extensions is made possi-
ble using the area measurement technique described
in this work. Results are shown in Figure l 6, where
the data referred to four values of the advance coef-
ficient are shown. The area is expressed as the ra-
tio between the measured cavitation area Ac and the
blade planform area Ao for r/R > 0.3. The experi-
mental data are represented with the corresponding
standard deviation. The cavitation area fluctuations
are in general very small, in the range of l to 3%,
except for the cases at the lowest values of the cav-
6n = 1.00
:7
On = 1.51
On = 2.02
Figure 13: Planform view of the cavitating blade and pre-
dicted cavitation area at J=0.71: prescribed wake model
(_), free wake model (_)
I. ~
- ~ - ~ ~ - - ~
a
-
- ~ ~ ~ - ~O ~
- ~ ~ ~~ ~ -
_ ~ it_ ~1
- · - - ~ ~ ~ - ~ -
_ ~ ~~ _I 1 ~
1 ~~
On = 1.19 On = 1.38
_
e _
__ _
_, 1 .
a_ - _
·_n—_ mu__
rem_ I~ _
l ~!
d ~'
~~a _~'
___ ~ ;_
___ _
; ~~ _
[~ ~~
On = 1.78 On = 2.06
1 1 1
~ _!
1 em_ 1 1 ~
1 __ 1 1; · I,.
1 ~ ' 1 i ~ ~'
1 __ 1 1 ~_'
I row I I .~'
_ 1 _
1 l~ 1 1 l~
1 1_~— 1 1 1_D"
1 · ~ ~ 1 1 . _ ~~
1 _ ~ 1 1 _1 _
1 ._''' ~3—~ 1 1 _' .. *'_
1 1 1
On = 2.38 1 1 On = 2.75
Figure 14: Planform view of the cavitating blade and pre-
dicted cavitation area at J=0.77: prescribed wake model
(_), free wake model (_)
Figure 15: Planform view of the cavitating blade and pre-
dicted cavitation area at J=0.83: prescribed wake model
(_), free wake model (_)
tation number (fin, where the occurrence of bubble
cavitation (see, for instance, the plot at J = 0.88) or
of streak cavitation (J = 0.77) is the source of im-
portant changes of the pattern aspect. The numerical
results are in quite good agreement with the exper-
imental data, in particular at high (Sn The results
tend to diverge for lower values of (fin. A higher
discrepancy is observed at J = 0.71, yet the differ-
ence is within only 15% from the experimental data,
and can be explained by the limited resolution of the
computational grid (see the table above).
50%
40%
ct
a'
c'' 30%
o
co
-
ct
Cal
a)
-
20%
10%
0%
.... . ~
i,
......
... .. .... ....
_ ._
· J=0.710 + 4 E-4
* J=0.769 + 6 E-4
· J=0.830 + 2 E-4
· J=0.879 + 3 E-4
—J=0.71, free wake model
- ~ `` —J=0.77, free wake model
~.~ \ —J=0.83, free wake model
- J=0.88, free wake model
:...~. ~
...:~...
0 1 2
On [~]
3 4
Figure 16: Effect of parameters J and On on the cavity
area Ac. Comparison between measurements and numeri-
cal results (free wake model)
A general feature of the present numerical results
is that the predicted cavities are overestimated com-
pared to the measurements. Large discrepancies oc-
cur at low (fin where attached cavities are thick and
strong cavitating vortices are present at the blade tip,
and at low J values, where the propeller blades are
heavily loaded.
The characteristic of the present numerical invis-
cid flow model to overpredict the extension of the
cavitating flow region may be further explained by
existing results that highlight the effect of viscous
how modeling. Salvatore and Esposito (2001) show
that the inclusion of viscous flow effects determines
a reduction of both cavity extension and thickness as
compared to inviscid-flow calculations. Specifically,
for the case of a three-dimensional hydrofoil in uni-
form inflow (60 = 1.148, Re = v0c/v = 9 x lO5),
viscous flow cavity area is up to 15% smaller than
the inviscid flow one.
The calculated cavity volume Vc and maximum
thickness he are displayed in Figure 17. Results ob-
tained using both the prescribed and free-wake mod-
eling are compared. As observed in the case of the
cavity area predictions, the extension of the cavitat-
ing flow region is generally overestimated when us-
ing a trailing wake that is not correctly aligned with
the flow. This trend is confirmed by both the cavity
volume and the maximum thickness predictions.
The procedure described above to provide an es-
timate of the cavity maximum thickness on the ba-
sis of the measurement of the trailing vortex thick-
ness has been used to compare against the numerical
predictions. Though only approximate, the results
shown in Figure 18 are found to be fairly represen-
tative of the true situation. In particular, the levels
and trends are compatible with the numerical results.
Based on the good agreement found on the cavity
area between the numerical and the experimental re-
sults, see Figure 16, we can confidently consider the
trailing vortex (or the downstream cavity) thickness
as a fair estimate of the true cavity thickness, in a
first order approximation.
The cavitation literature reports a number of
experiments where the cavity length and the cav-
ity thickness have been measured, though es-
sentially on two-dimensional hydrofoils and in
quasi three-dimensional foils: Dupont and Avel-
lan (l991),Pereira (1997), Laberteaux and Ceccio
(1998~. These works show that the cavity maximum
thickness is found to vary in a closely linear trend
with the cavity length, at least in the steady situ-
ations (i.e., free of cloud cavitation), which is our
case here as per the low fluctuations observed and
reported in Figure 16. Figure 19 displays the maxi-
mum thickness, which we recall is only a rough es-
timate of the cavity true maximum thickness, versus
a length Ic taken as the square root of the attached
cavitation area Ac; hence, the values of IC are only
indicative. The numerical results fall on a straight
line, whereas the experimental data follow the same
linear trend, with the additional dispersion due to the
coarse thickness evaluation. This result seems to in-
dicate that steady leading edge cavitation on a com-
plex geometry presents the same macroscopic fea-
tures as on simple geometries. Yet, such a result
needs to be confirmed by an accurate measurement
Be-04
6e-04
2e-04
Oe+OO
awns
_ _
2e-02
1~ n
. — — —
Oe+OO
l
Free Wake, J=0.71
Free Wake, J=0.77
Free Wake, J=0.83
Prescr. Wake, J=0.71 -
Prescr. Wake, J=0.77 - O
Prescr. Wake. J=0.83 ~
O _
0.5
1 1.5 2 2.5 3 3.5 4
Free Wake, J=0.71 ~
Free Wake, J=0.77 0
Free Wake, J=0.83
Prescr. Wake, J=0.71 -
Prescr. Wake, J=0.77 o
'< Prescr. Wake,J=0.83 ~
0.5 1 1.5 2 2.5 3 3.5 4
On
Figure 17: Effect of parameters J and on on the computed
cavity volume Vc (top) and cavity maximum thickness he
(bottom). Comparison between free wake model and pre-
scribed wake model.
of the three-dimensional shape of the cavity.
The propeller thrust and torque coefficients for
three values of the advance coefficient are shown in
Figure 20. A common feature is that both KT and
KQ increase as the cavitation number is increased.
This trend is more evident at relatively low J and is
associated with extensive cavitation on the blades.
Numerical calculations are also given for compari-
son. For flow conditions characterized by moderate
to light cavitation, i.e. Gn > 1.5 at J = 0.71, 0.77, and
2%~
1%
0% _
o
· J=0.710+4 E-4
~ J=0.769 + 6 E-4
· J=0.830 + 2 E-4
· J=0.879 + 3 E-4
J=0.71, free wake model
J=0.77, free wake model
J=0.83, free wake model -
J=0.88, free wake model
3 4
Figure 18: Effect of parameters J and 6,' on the cavity
maximum thickness ho. Comparison between measure-
ments and numerical results (free wake model)
4%—
q0/^ ~
_ .
=~2%
1 oak
O
· .
· Experiment
· Free wake model
Linear regression
· e:
ON
.~.
_ :.~.
.~e ~
0% 2%
1
4% 6% 8%
Ic
Figure 19: Behavior of the cavity maximum thickness he
with the cavity length Ic. Comparison between measure-
ments and numerical results (free wake model)
On > 1.0 at J = 0.83, a similar degree of accuracy as
the one observed in Figure 10 for non cavitating flow
conditions is obtained. At lower (fin values, both KT
and KQ are underpredicted. This may be related to
the overestimated cavitation patterns at low (fin, as
discussed above.
0.5
° 0.3
y
0.2
0.1
o
0.6
0.5
0.4
0.3
0.2
0.1
o
0.5
0.4
0.3
0.2
0.1
o
KT (Num )
KT (EXP )
K`: (Num.)
VOODOO O O O O O KQ (EXP.)
Oo~
O 0.5 1 1.5 2 2.5
o
o
<,n
3 3.5
4 4.5
K (Num )
AT (EXP ) O
- Kit (Num.)
Q (EXP) O
c~aooo(3o 0 0 0 0 0
c' ~ ' - _
000~
O 0.5 1 1.5 2 2.5 3 3.5 4 4.5
On
KT (Num )
KT (EXP.)
K`, (Num.)
O C O O O ~ O ~ G O G KO (EXP ) 3
~ -
, , ,
O 0.5 1 1.5 2 2.5 3 3.5 4 4.5
On
Figure 20: Effect of cavitation number Cyn on propeller
thrust and torque coefficients at J = 0.71 (top), J = 0.77
(center) and J = 0.83 (bottom)
CONCLUDING REMARKS
An experimental investigation on a cavitating pro-
peller in a uniform inflow has been presented. The
cavitation figures have been analyzed through new
and robust analysis methods, which have been found
to provide information otherwise difficult to assess.
The propeller cavitating characteristics have been
quantified in terms of cavitation extension. A mea-
sure of the cavity thickness was established, in a
first order approximation, from the measure of the
dimensions of the downstream section of the sheet
cavitation. A complete mapping of the propeller hy-
drodynamic characteristics has been established.
Measurements are used to assess an inviscid flow
boundary element method for the analysis of blade
partial sheet and super-cavitation. The methodology
includes an accurate evaluation of the trailing wake
vorticity path by means of a wake alignment tech-
n~que.
Comparison with the experimental measure-
ments shows that the present theoretical methodol-
ogy is able to accurately predict the sheet cavitation
area across a wide range of advance ratios and of
cavitation number values. In addition, the predicted
cavity thickness is found to be in satisfactory agree-
ment with the experimental data. The importance of
an accurate evaluation of the trailing wake vorticity
is highlighted and the range of applicability of the
present prediction model is also clearly identified.
The present work constitutes a first step of a joint
experimental and theoretical research project, with
the following objectives: i. the improvement of the
cavity thickness measurement; ii. the definition of
a volume measurement technique; iii. the extension
of the cavitating flow model to unsteady propeller
cavitation; iv. the inclusion in the theoretical model
of viscous effects; v. the modeling of the tip vortex.
A database accessible to the public domain will
be regularly updated with the latest results and de-
velopments: http://crm.insean.it/E779A
ACKNOWLEDGMENTS
The authors are grateful to the CEIMM personnel.
The work was supported by the Italian Ministero dei
Trasporti e delta Navigazione in the frame of the IN-
SEAN 2000-2002 Research Program.
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