1
Cav03-GS-7-003 Fifth International Symposium on Cavitation (Cav2003)Osaka, Japan, November 1-4, 2003
Unsteady Tip Leakage Vortex Cavitation from the Tip Clearance
of an Oscillating Hydrofoil
Masahiro MURAYAMA, Yoshiki YOSHIDA*, Yoshinobu TSUJIMOTO
Osaka University, Engineering Science,
1-3, Machikaneyama, Toyonaka, Osaka, 560-8531, Japan
*Currently, National Aerospace Laboratory, Kakuda Space Propulsion Laboratory
1 Koganezawa, Kimigaya, Kakuda, Miyagi 981-1525, Japan
E-mail : [email protected]
AbstractTip leakage vortex cavitations from the tip clearance of
an oscillating hydrofoil with tip clearance were observed
experimentally. It was found that the fluctuation of the
cavity size delays behind the oscillation of the angle of
attack. The maximum value of the cavity size decreases
when the frequency of oscillating hydrofoil is increased. To
simulate the unsteady characteristics of the tip leakage
vortex cavitation, a simple calculation based on the slender
body approximation was conducted with taking into
account the effect of cavity growth. The calculation results
of the cavity volume fluctuation showed qualitative
agreement with the experimental results.
IntroductionIt is recognized that cavitation instabilities, such as
cavitation surge and rotating cavitation, are caused by the
unsteady characteristics of cavitation; i.e., mass flow gain
factor and cavitation compliance. Cavitations in unshrouded
pump impellers are classified into three types, cavitation on
blade surface, cavitation in tip leakage flow, and cavitation
in back-flow vortices. For the blade cavitation, unsteady
characteristics have been extensively studied, and it is now
possible to predict the mass flow gain factor and cavitation
compliance theoretically (Brennen [1], and Otsuka et al.
[2]). However, few studies have been performed for the
unsteady cavitation characteristics in the tip leakage flow,
although it seems that the cavitation in the tip leakage flow
plays an important role for cavitation instabilities.
In the last CAV2001, we presented the aspect of the tip
leakage vortex cavitation for a fixed hydrofoil as the first
step. In the present paper, the unsteady characteristics of the
tip leakage vortex cavitation; i.e., variation of the size and
location of the cavity, are presented with focusing on the
cavity response to the frequency of the oscillating hydrofoil.
Unsteady cavitations on the blade surface of oscillating
hydrofoils were extensively studied [3]-[5], but only a very
few number of results are related to the tip vortex cavitation
under unsteady conditions. To our knowledge, the only
results available on the subject are those of Mckenney et al.
[6], and Boulon et al. [7].
On the other hand, Rains [8] first proposed to apply the
slender body approximation to the tip leakage vortex. In this
method, a 3-D tip leakage flow is simulated by a 2-D
unsteady crossflow. Chen et al. [9] applied this method to a
compressor tip clearance flow by using the vortex method.
Watanabe et al. [10] recently extended this method to
include the effects of cavity growth. Higashi et al. [11] have
applied this calculation method to the tip leakage vortex
cavitation of a fixed hydrofoil, and examined the results by
comparing with the experimental results. In the present
study, Watanabe et al.’s method is applied to predict the
unsteady tip leakage vortex cavitation of an oscillating
hydrofoil with tip clearance. Discussions on the influences
of the frequency of the oscillating hydrofoil are made
through the comparison of the experimental results with the
calculations.
2
Experimental apparatus and procedureThe experiments were conducted using the cavitation
tunnel as shown in Fig. 1. The tunnel is a closed loop and
the base pressure and hence cavitation number, s, is
adjusted by using a vacuum pump connected to the top of
the pressure control tank. The cavitation number was
maintained constant s=1.0 throughout the present
experiments. The cross section of test section is square,
height and width are 70 mm and 100 mm, and length is 500
mm as shown in Fig. 2. A nozzle with area reduction ratio
4.65 is set upstream of the test section. The water was
deaerated by keeping at the lowest pressure (5kPa) for
more than 12 hours before the measurements. Although the
effects of Reynolds number were examined within free
stream velocity U=2.9 ~ 5 m/s (Re=UC/n = 2.6~4.5 ¥105),
the influence of Reynolds number on the test results could
not be identified in this range. Therefore, the free stream
velocity was maintained constant U=5 m/s throughout the
present experiments.
In the present experiment, a flat plate hydrofoil was
used. Figure 3 shows the configuration of the test
hydrofoil. The foil is made with stainless steel and its
surface roughness is about 1 mm. The chord C of the foil is
90mm, the span H is 67 mm for the design tip clearance 3
mm (the real measured value is 2.95 mm). The thickness of
the foil is constant 3 mm. Both the leading and trailing
edges were rounded with radius 1.5 mm. With a square tip,
“gap cavitation” (Gearhart [12]) in the clearance between
the tip and upper wall, and “sheet vortex cavitation” in the
shear layer of the leakage jet appeared clearly. To remove
these types of cavitation, the pressure side corner of the tip
was rounded to a radius 3 mm, as shown in Fig. 3 (Ido et
al.[13], and Labore et al. [14]).
The oscillating mechanism is shown schematically in
Figs. 4 and 5. A three-bar linkage oscillates the hydrofoil in
a nearly sinusoidal motion as shown in Fig. 5. It provides a
pitching motion of the hydrofoil around the mid-chord “O”
in Fig. 3 at a dimensional frequency, f, up to 16 Hz which
corresponds to a reduced frequency, k=2pfC/(2U)
=wC/(2U), up to 0.9. The unsteady angle of attack a(t) is
represented by a(t)=am+Da¥sin(2pft). All the results
presented here correspond to a mean angle of attack, am ,
3
equaled to 4 degrees, and an oscillating amplitude of the
angle of attack, Da, equaled to 2 degrees. Although a few
tests were conducted for other values (am, Da) to analyze
the effect of these parameters, the same characteristics of
the unsteady tip leakage vortex cavitation were confirmed.
Both side and top walls of the test section are made of
transparent acrylic resin, so that we can make visualization
systematically. Two types of observation were made. One
is by a movie with a high-speed video picture (250
frames/sec records), and the other is by a photo at a certain
instant angle of attack using a still camera with strobe light
(20msec).
Experimental ResultsObservation of the tip leakage vortex. Figure 6
shows the photos of the tip leakage vortex cavitation
observed from the top view at various angle of attack; i.e.,
a=2, 4(+), 6, 4(-), 2degrees sequentially, for the reduced
frequencies k=0 (steady), and k=0.45, 0.90 (unsteady), at
the cavitation number s=1.0. Here, a=4(+) degrees denotes
the angle of attack equals to a=4 degrees and the angle of
attack is increasing, and a=4(-) degrees denotes the angle
of attack is decreasing. Hereafter, we focus on the influence
of the oscillating frequency on the size and location of the
tip vortex cavitation.
4
(1) Steady case (k=0)
First, for the steady case shown in Fig. 6 (a), cavity size
develops at larger angle of attack, and reduces at smaller
angle. The angle made by the foil and the trajectory of the
cavity increases as we increase the angle of attack. Those
are reasonable results because the pressure difference of the
tip clearance increases, and the amount of the leakage flow
also increases at larger angle of attack.
At a=2 degrees, the tip leakage vortex cavitation is not
observed, although the blade cavitation at the leading edge
appeares slightly. At a=4 degrees, the cylindrical vortex
cavitation with constant radius develops very long. The end
of the cavity can not identified because the cavity grows
over the window. At a=6 degrees, the cavity in the shear
flow appeared near the tip clearance, and rolls up to the
vortex cavity with considerable large size. The tip vortex
cavitation twists and convects downstream over the
window. Other results for the steady case have been
presented in detail in the previous report [11].
(2) Unsteady case (k=0.45 and 0.90)
As we increase the frequency of the oscillating angle of
attack, the critical angle of attack when the cavity develops
to the maximum size varies from a=6 degrees (at k=0) to
a=4(-) degrees (at k=0.45), and a=2 degrees (at k=0.90). It
was found that the cavity in the shear layer of the leakage
jet does not appear for the unsteady cases, and the size of
the tip leakage vortex cavitation decreases remarkably, as
we increases the oscillating frequency. For the steady case,
we can not observe the tip leakage vortex cavitation at a=2
degrees. To the contrary, the tip leakage vortex cavitation
appears downstream of the foil at a=2 degrees for the
unsteady case, in particular, at k=0.90.
Variation of the cavity radius. To obtain a
better understanding of the influence of the oscillating
frequency on the cavitation behavior, the cavity size and
the trajectory of cavity were measured with the high-speed
video pictures from the top and side views as shown in
Figs. 7 (a) and (b). The size and location of the vortex
cavitation fluctuated considerably even at the same angle of
attack, so a single picture was not sufficient to measure the
size and location of the tip leakage vortex cavitation.
Therefore, the size and location of the cavity were averaged
over twenty frames of the high-speed video pictures at the
same condition; i.e., the same angle of attack.
Figure 7 (c) shows the variation of the cavity size; i.e.,
radius of the tip vortex cavity, R, at mid-chord Z/C=0.5 and
at the trailing edge Z/C=1.0 for various reduced
frequencies up to k=0.90. We evaluated the radius of the
cavity assuming that the cross section of the vortex
cavitation is circular. In addition, the cavity length, l, on the
blade surface was measured. The length of the blade
cavitation near the tip clearance was shorter than that at the
mid-span for all cases. The length of the blade surface
cavitation was measured at the mid-span as shown in the
Fig. 7 (b), and averaged over many frames with the same
condition. Figure 7 (d) shows the variation of the length, l,
of the blade surface cavitation for various frequencies.
From Figs. 7 (c) and (d), it might be of interest to note
that the variation of the cavity radius delays behind the
oscillation of the angle of attack as we increase the
oscillating frequency. The delay is larger when the angle of
attack is increasing than when it is decreasing; i.e., the
cavity size develops slowly and that shrinks rapidly. In
addition, the maximum size of the cavity decreases if the
oscillating frequency is increased. As a result, the
amplitude of the fluctuation of cavity radius for k=0.90 is
half of that for k=0. Such aspects; i.e., the phase delay
5
behind the oscillation of the angle of attack and the
reduction of the amplitude of the cavity fluctuation, can be
also observed in the blade cavitation as shown in Fig. 7 (d).
The amount of the phase delay of the tip vortex cavitation
was almost the same as that of the blade cavitation.
From these experimental results, it could be concluded
that: (a) significant delay and (b) amplitude decrease occurs
for the tip leakage vortex cavity fluctuation as we increase
the frequency of the blade oscillation.
Outline of the analytical methodIt is plausible that the tip leakage vortex cavitation
occurs in low pressure region in the vortex core formed by
rolling up of the shear layer between the tip leakage flow
and the main flow. To explain the calculation method, we
illustrate crossflow planes A, B, C, and D at different
chordwise locations a, b, c, and d, respectively, as shown
in Fig. 8. Location a is at the leading edge and d is at the
trailing edge. The tip leakage flow starts at location a. As
the crossflow plane moves through the foil, the vortices
representing the shear layer shed into the main flow from
the tip of the foil. The vortices roll up in the subsequent
cross sections, as illustrated in planes, B, C, and D. The tip
leakage vortex cavitation is expected to occur in the low
pressure region in the rolled up vortex core.
We assume that the crossflow plane moves downstream
with the free stream velocity U. Hence, the distance
between the planes analyzed is DS=U¥Dt, where Dt is a
6
time increment. The following assumptions were made in
the present calculation. The velocity of the tip leakage jet
on the crossflow plane at location S is simply assumed
Uj=(2Dp/r)1/2, where Dp is the instantaneous pressure
difference across the tip clearance. The pressure difference
Dp is assumed to consist of the steady component Dpmrelated to the mean angle of attack, am, and the unsteady
component Dpu related to the amplitude of the oscillating
angle of attack, Da. The steady component Dpm is
estimated from a non-cavitating 2-D incompressible
inviscid flow calculation around a thin foil. The unsteady
component Dpu is estimated from the unsteady airfoil
theory described below.
The analytical method to calculate the non-cavitating
flow around an oscillating foil in an ideal fluid has been
found first in Kármán, and Sears [15]. Here, we use the
results of 2-D non-cavitating flow analysis available from
Fung [16] to evaluate the unsteady pressure difference
across the blade. Figure 9 (a) shows the coordinate used in
the following theoretical calculation. The motion and the
foil are defined as follows. The foil surface is represented
by x=cos(q), so that the leading edge comes to x= –1 and
the trailing edge x= + 1. If we assume the pitching motion
around the mid-cord at x=0, the displacement h of the foil
surface can be represented as,
h = –D a ¥ x ¥ exp(iwt) , w=2pf (1)
The unsteady pressure distribution on the foil is expressed
as follows [16],
pu = rU2 ¥ Da ¥ - 1+
ik
2
Ê
ËÁ
ˆ
¯˜C (k) -
ik
2
ÏÌÓ
¸˝˛tan
q
2- 2iksinq
È
ÎÍ
+1
4k
2sin2q
˘
˚̇¥ eiwt (2)
where k (=2pfC/(2U)=wC/(2U)) is the reduced frequency,
and C(k) is Thedorsen function. Typical unsteady pressure
distribution on the blade is shown in Fig. 9 (b). Unsteady
component of the pressure difference Dpu is calculated
from pu using equation (2).
A vortex method was used for the calculation on the
crossflow plane. A source qj =2¥Uj is distributed at the tip
clearance, which represents the tip leakage jet. The discrete
free vortices G representing the shear layer between the tip
leakage jet and the main flow are released from the corner
of the tip. The strength of the vortices is determined as
G=Uj2¥Dt/2 from the leakage jet velocity Uj.
It is assumed that a single cylindrical cavity starts to
develop at the location of the minimum pressure, where the
pressure becomes lower than the vapor pressure pv first.
The growth of this cylindrical cavity, radius R, on the
crossflow plane is determined by the unsteady version of
Bernoulli equation. The effect of the cavity is represented
by a source qb=2pR¥dR/dt at the center of the rolled up
vortex. Watanabe et al. [10] have presented the complete
details for the calculation method. The effects of the
sidewall of the test section are ignored for simplicity.
Boundary conditions on the upper wall and the foil are
satisfied by introducing the mirror image of singularities
within 0£S/C£1.0. However, the mirror image with respect
to the foil is not considered at S/C>1.0 where there is no
foil surface. The initial radius of the cavity is set to be
R/C=0.00011, and the time increment is Dt=(C/U)/2000 in
the present calculations.
Although there are several assumptions in this simple
calculation, the qualitative agreement could be found for
the location and size of the tip vortex cavitation for a fixed
hydrofoil. In the previous report [11], the influences of the
cavitation number, angle of attack, blade loading, and
amount of tip clearance on the size and location of the
cavity have been verified by comparing with the steady
experimental results.
7
Calculation resultsCavitation behavior. Figure 10 shows the cavity
shapes calculated for steady (k=0) and unsteady cases
(k=0.45, 0.90). The conditions (s =1.0, am=4 degrees,
Da=2degrees, k=0, 0.45, and 0.9), and the arrangements of
the figures are the same as those of experiments results as
shown in Fig. 6.
First, for the steady case (k=0), the cavity size is larger
at larger angle of attack. This general behavior agrees well
with experimental result except that the tip leakage vortex
cavitation grows rapidly near the leading edge at a=2
degrees. It seems that the estimation of the steady pressure
difference, Dpm, across the tip clearance estimated from the
2-D inviscid flow analysis is responsible for the
overestimation. However, the present 2-D unsteady flow
model based on a slender body approximation can
qualitatively predict the location and the size of the cavity
for the steady case.
For the unsteady cases (k=0.45 and 0.9), the tip vortex
cavitation grows to the maximum at a=6 ~ 4(-) degrees.
From these results, it is clear that the cavity development
delays behind the foil oscillation. Here we focus on the
instant with a=2 degrees, it can be observed that the tip
leakage vortex cavitation grows again downstream for the
unsteady case (k=0.45, 0.90), although the tip leakage
vortex cavitation disappear near the trailing edge for the
steady case (k=0). These characteristics agree with
unsteady experimental results qualitatively.
Figure 11 compares the trajectory of the cavity for the
steady and unsteady cases. For the steady case (k=0), the
angle made by the foil and the trajectory of the cavity
increases with increasing of the angle of attack. For the
unsteady cases, the cavity trajectory becomes meandering
with increasing of the oscillating frequency. For the
unsteady case k=0.90, the trajectory at a=2 degrees is far
from the foil, and trajectory at a=6 degrees is close to the
foil at Z/C=2.0. It is contrary to that for the steady case.
Similar behavior of the cavity trajectory could be observed
in the experiments.
8
Cavity volume. Figure 12 shows the variation of
the estimated cavity volume compared with experimental
results. Both cavity volumes were estimated by integrating
of the partial cylindrical cavity from the leading edge
(Z/C=0) to twice of the chord length (Z/C=2.0). In this
estimation, it was assumed that the cavity shape is
cylindrical. Although the agreement between the
experiment and calculation is not sufficient quantitatively,
the same tendency can be found with respect to the
influence of the oscillating frequency. When we increase
the oscillating frequency, (a) the fluctuation of the cavity
volume delays behind the oscillation of the angle of attack,
(b) the amplitude of the fluctuation of the cavity volume
decreases.
In this chapter, we have compared the experimental
results with the calculation results. Although the simulation
uses various simplifying assumptions, it was found that the
present simulation can predict the influence of oscillating
frequency on the unsteady behavior of the tip leakage
vortex cavitation qualitatively.
9
DiscussionsIn the previous chapters, we found that: (a) the phase
delay of the cavity response occurs, (b) the amplitude of
the tip vortex cavitation gets smaller, when the oscillating
frequency increases. In this chapter, we will discuss about
the reason through the examination of the calculation
results.
It has been assumed that the tip leakage vortex
cavitation occurs in lower pressure region in the vortex
core formed by rolling up of the shear layer between the tip
leakage flow and the main flow. Therefore, we will
examine the vortices representing the shear layer that
eventually rolls up to the tip leakage vortex core. In the
present calculation, the discrete free vortices G are released
from the corner of the tip on the cross plane. The total
amount of the shed vortices was calculated and compared
between the steady (k=0) and the unsteady (k=0.9)
condition.
Figure 13 shows the total amount of the shed vortices.
These figure shows that the total amount of the shed
vortices, S G, when the crossflow plane reaches to the
trailing edge at Z=C, at the instant angle of attack a=6,
4(+), 4(-), 2 degrees respectively. We focus on the amount
of S G for the trailing edge at Z/C=1.0. For the steady case
k=0, this value equals to SG/UC=CL/2=p¥sina at Z/C=1.0
theoretically. Comparing the unsteady result with the
steady one, it was found that (a) At a=6 degrees, S G for
the unsteady case is less than that for the steady case. (b) At
a=4(-) degrees, S G for the unsteady case is larger than
that for the steady case. To the contrary, at a=4(+) degrees,
S G for the unsteady case is less than that for the steady
case. (c) At a=2degrees, S G for the unsteady case is larger
than that for the steady case. These tendencies of S G at
Z=C agrees with those of the unsteady cavity radius, R,
observed at Z=C as shown in Fig. 7 (c).
From these results, it could be concluded that the
amount of the circulation of the tip leakage vortex core
consisting of the free vortices becomes smaller for the
unsteady case than the steady case. Also, the fluctuation of
the total amount of vortices delays behind the oscillation of
the angle of attack. It was inferred that the unsteady
behavior of pressure difference across the tip clearance is
responsible for the characteristics of the circulation of the
tip leakage vortex, and affects the growth and decay of the
unsteady tip vortex cavitation.
ConclusionsResults obtained in the present study can be
summarized as follows:
1. When the frequency of oscillating hydrofoil increases,
(a) significant phase delay and (b) decrease of the
amplitude occurs for the fluctuation of the size of tip
leakage vortex cavitation. These characteristics were
observed both in experiment and calculation.
2. The present 2-D unsteady flow model based on a
slender body approximation can qualitatively predict
the unsteady behavior of the tip leakage vortex
cavitation of the pitching hydrofoil.
3. The unsteady behavior of the pressure difference
across the tip clearance influences the amount of the
circulation of the tip vortex core, and affects the
growth and decay of the unsteady tip leakage vortex
cavitation.
10
AcknowledgementThe authors would like to express their sincerer
gratitude for Mr. Seiji HIGASHI who made many valuable
discussions for the calculations, and Dr. Tatsuro KUDO of
National Maritime Research Institute who showed them
many reports of unsteady cavitation in propellers.
NomenclatureC = chord
C(k) = Thedorsen function
CL = lift coefficient = Lift/(rU2C/2)
Cp = pressure coefficient = (p - p1)/(rU2/2)
f = frequency
H = span
i = imaginary unit
k = reduced frequency =2pfC/(2U)=wC/(2U)
l = cavity length of blade cavitation
p = pressure
p1 = pressure at inlet
pv = vapor pressure
Dp = pressure difference across the tip clearance
Dpm = mean component of pressure difference across
the tip clearance
Dpu = unsteady component of pressure difference
across the tip clearance
qj = strength of source representing the leakage jet
qb = strength of source representing the cavity
growth
R = radius of cavity
Re = Reynolds number =UC/n
S = distance along the chord
DS = increment of S
T = period
t = time
Dt = time increment
e, h = coordinates, defined in Fig. 9
U = free stream velocity
Uj = velocity of leakage jet flow
V = cavity volume
X = distance from the foil
Z = distance from the leading edge
a = angle of attack
Da = amplitude of the angle of attack
am = mean angle of attack
G = strength of vortices
n = kinematic viscosity
r = density
s = cavitation number = (p1 - pv)/(rU2/2)
t = tip clearance
w = angular velocity of oscillating foil
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