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 : kryoshi@kakuda-splab.go.jp

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

References[1] Brennen, C. E., 1978, “Bubbly Flow Model for Dynamic

Characteristics of Cavitating Pumps,” Journal of Fluid

Mechanics, 89, pp. 234-240.

[2] Otsuka, S., Tsujimoto, Y., Kamijo, K., and Furuya, O., 1996,

“Frequency Dependence of Mass Flow Gain Factor and

Cavitation Compliance of Cavitating Inducers,” ASME Journal of

Fluids Engineering, 118, pp. 400-408.

[3] Tanibayashi, H., and Chiba, N., 1968, “Unsteady Cavitation of

Oscillating Hydrofoil,” Mitsubishi Heavy Industries Technical

Report (in Japanese), Vol. 5, No. 2, pp. 45-52, or Tanibayashi, H.,

and Chiba, N., 1977, “Unsteady Cavitation of Oscillating

Hydrofoil, “Mitsubishi Technical Bulletin, 117, pp. 1-10.

[4] Miyata, H., Tamiya, S., and Kato, H., 1972, “Pressure

Characteristics and Cavitation on an Oscillating Hydrofoil,”

Journal of the Society of Naval Architects of Japan (in Japanese),

132, pp.107-115.

[5] Franc, J. P., and Michel, J. M., 1988, “Unsteady Attached

Cavitation on an Oscillating Hydrofoil,” Journal of Fluid

Mechanics, 198, pp. 171-189.

[6] Mckenney, E. A., and Hart, D. P., 1993, “Experimental

Determination of Bound Circulation and Shed Vorticity Induced

by an Oscillating Hydrofoil,” ASME FED-Vol. 153, Cavitation

and Multiphase Flow, pp. 87-91.

[7] Boulon, O., Franc, J. P, and Michel, J. M., 1997, “Tip Vortex

Cavitation on an Oscillating Hydrofoil,” ASME Journal of Fluids

Engineering, 119, pp. 752-758.

[8] Rains, D. A., 1954, “Tip Clearance Flows in Axial Flow

Compressors and Pumps,” California Institute of Technology,

Hydrodynamics and Mechanical Engineering Laboratories, Labo.

Rep. No. 5.

[9] Chen, G. T., Greitzer, E. M., Tan, C. S., and Marble, F. E.,

1991, “Similarity Analysis of Compressor Tip Clearance Flow

Structure,” ASME Journal of Turbomachinery, Vol. 113, pp. 260-

271.

[10] Watanabe, S., Seki, H., Higashi, S., Yokota, K., and

Tsujimoto, Y., 2001, “Modeling of 2-D Leakage Jet Cavitation as

a Basic Study of Tip Leakage Vortex Cavitation,” ASME Journal

of Fluids Engineering, vol. 123, pp. 50-56.

[11] Higashi, S., Yoshida, Y., Tsujimoto, Y., 2001, “Tip Leakage

Vortex Cavitation from the Tip Clearance of a Single Hydrofoil,”

The Fourth International Symposium of Cavitation, CAV 2001,

Session A6-006, or Higashi, S., Yoshida, Y., Tsujimoto, Y., 2002,

“Tip Leakage Vortex Cavitation from the Tip Clearance of a

Single Hydrofoil,” JSME International Journal, Series B, Vol. 45,

No. 3, pp. 662-671.

[12] Gearhart, W. S., 1966, “Tip Clearance Cavitation in

Shrouded Underwater Propulsors,” AIAA Journal of Aircraft,

Vol. 3, No. 2, pp. 1852-192.

[13] Ido, A., and Uranishi, K., 1991, “Tip Clearance Cavitation

and Erosion in Mixed-Flow Pumps,” ASME FED-Vol.119, pp.

27-29.

[14] Laborde, R., Chantrel, P., and Mory, M., 1997, “Tip

Clearance and Tip Vortex Cavitation in Axial Flow Pump,”

ASME Journal of Fluids Engineering, Vol.119, pp. 680-685.

[15] Von Kármán, T. H., and Sears, W. R., 1938, “Airfoil Theory

for Non-uniform Motion,” Journal of the Aeronautical Sciences,

Vol. 5, No. 10, pp. 379-389.

[16] Fung, Y. C., 1969, “An Introduction to the Theory of

Aeroelasticity,” Dover Publication, Inc., p. 401.