Modeling a cavitating vortex

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MODELING A CAVITATING

VORTEX

Prof.dr.ir. G. Kuiper, TUDeift I MARIN

Report 1119-P

June 1997

ASME Fluids Engineering Division Meeting,

FEDSM'97. June 22

- 2,, 1997

TU Deift

Faculty of Mechanical Engineering and Marine Technology

Ship Hydrornechanics Laboratory

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NOMENCLATURE

Martijn X. van Ri Jsbergen Propulsor Group

Maritime Research institute Netherlands 2, Haagsteeg, P.O. Box 28 6700 AA Wageningen, The Netherlands Phone: +31 31,7 493441, Fax: +31 3i7 493245

eMail: mvrijsberg@marin.nl

ABSTRACT

The purpose of this paper is to develOp a description of a cavitating vortex at various pressures. No consistent model for

such a situation is available in literature. To develop such a

description, systematic tests in

the MARIN high speed

cavitation tunnel have been carried ot with a cavitating hub

vortex.

The thickness of the cavitating core and the tangential

and axial veloctity distribution are measured at several tunnel

pressures at a Reynolds number of 6.5 . iO5.

The radial momentum equation is used to integrate the

tangential velocity distribution to give a relation between the

cavitation number and the thickness of the cavitating core.

Taking conservation of kinetic energy and angular

momuentuni into account the relation between the tangential velocity distribution and the cavitating core diameter has been

investigated.

a,, cavitating core radius Cg gas concentration in water Cp,,, minimum pressure coefflcient D diameter of the test section E kinetic energy

L angular momentum

Po pressureatr = R

vapor pressure

r radius

R radius of the test sect ion

U axial velocity in the test section tangential velocity in the testsection

[m] [ppm] [-i [m] [J] [kgm2s I [Pa] [Pa] [m] [ml [mis] [mis]

1997 ASME Fluids Engineering Division Summer Meeting FEDSM'97 June 22- 26, 1997

FEDSM97-3266

MODELING A CAVITATING VORTEX

Gert Kuiper ITUDeift

R&D Department

Maritime Research Institute Netherlands

2, Haagsteeg, P: Box 28

6700 AA Wageningen, The Netherlands Phone: ±31 317 493273, Fax: ±31 317 493245 eMail: g.kuiper@marin.nl V, et 13 V p a a1

tangential velocity in the test section angle of attack

Henry's constant

wavelength

dynamic viscosity of water density of water

cavitation nümber

PoPu

cavitation number at inception

pU2

INTRODUCTION

Cavitation On a ship propeller begins nearly always with cavitation in the tip vortex. The ship speed at which cavitation

begins 'is called the inception speed. For Navy ships this

inception speed is very important because cavitation increases

the radiated noise significantly It is therefore important to

contrOl cavitation inceptionof the tip vortex.

Tip vortexcavitation isa most complex form of cavitation on a propeller A drastic simplification is the two dimensional cylindrical vortex. Even after this simplification, inception of cavitation remains rather 'intractable because it depends on the Reynolds number and on the nuclei content of'the flow. Since neither parameter can be properly scaled, strong scale effects occur when model tests are carried out, see ITFC (1993); In

practice the nuclei content of a tunnel is kept as constant as

possible and a simple emperical relation between the incepient cavitation number and Reynolds number is used (McCormick,

1962).

The condition at which the cavitating vortex disappears

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Figure 1. Adaption of test section for LDV

(desinence) can be different from the inception condition

(hysteresis). However, there are indications that the radius of a cavitating vortex is uniquely related to the cavitation number and that this relation is independent of the Reynolds number and of the nuclei content (Kuiper,198l). Measurements of the

cavitating core are also given by Gowing et al. (1995). They

also added polymers and this did change the mentioned

relation, which may indicate a Reynolds dependence. Both

Kuiper and Gowing observed that the radius of the cavitating

core was rather independent of the position behind the tip,

although Gowing points out that at high air contents, diffusion occurs and the pressure in the cavitating core becomes higher

than the vapor pressure.

The relation between the radius of the cavitating vortex

and the cavitation index may be used to define an inception

condition from the cavitating tests. Very often the radius of the

cavitating core is taken at the location where the vapor

pressure occurs, e.g. Arndt and Keller (1992). In that case,

using a Rankine vortex, angular momentum is conserved but kinetic energy is not conserved, however.

Therefore this relation between cavitating core radius and

pressure (cavitation number) has been investigated in more

detail.

EXPERIMENTS

In the present research the behaviour of a cavitating hub

vortex is studied, instead of the (usual) tip vortex of an

elliptical foil, studied e.g. by Gowing (1995). The high speed cavitation tunnel of MARIN is particularly suitable to generate this kind of vortex. Because of it's small test section, both the tangential velocity distribution and the cavitating core radius are relatively easy to measure.

The tunnel, build of 5" stainless steel pipes, is driven by a centrifugal pump which induces velocities in the test section up to 60 ni/s. A heat exchanger can be used at high speeds to keep the tunnel water temperature constant. The test section is

a ventury with a contraction ratio 1:10 and has a constant circular cylindrical diameter of 40 mm over a length of 60

mm. The walls of the test section are adapted to make laser

Figure 2. LDV setup with: 1=Iaser, 2=mirror box, 3=beam spacer, 4=rotatlng device, 5= focussing lens

velocimetry possible, see Fig. 1.

The laser beams enter the test section through a flat

perspex plate, followed by a box, filled with water. Next, they pass through a film with a thickness of 0.2 mm to finally enter the flow. Thus the refraction of the beams in the test section is minimal.

Generation of a hub vortex

A vortex is created in the test section by a stator

positioned in the stilling section, upstream of the test section. The stator consists of four stator blades mounted on a central

hub of radius 0.2 R. The stator blades have a NACA 0020

section at a fixed angle of attack of 15°.

During the first tests of the experiments it was observed that the cavitating core wandered too much. This wandering was reduced by giving the trailing edge of the hub of the stator

a flat back (instead of a sharp point), thus creating a sharp

edge where separation is well defined. A further reduction of the wandering was attained by placing a receiving stator in the difusor with a sharply pointed hub in the upstream direction. In this way the vortex core was guided to the centre of the test

section.

Measuring tanqentlal and axial velocities

In Fig. 2 a diagram of the total LDV setup is given. The laser used in the experiments is a HeNe laser with a wavelength ? = 632.8 nm, a beam diameter of 1.0 mm and a measuring control volume of 0.lxl.0 mm.

To measure the axial as well as the tangential

velocity

component with a 1D laser, the rotating device is used. This device can rotate the two parallel beams over 90°. In this way

the x- and z-component of the velocity distribution can be measured.

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2

7 6 2

1-o O

I

1

- U

.

,

o , o

Vt calculated standard deviation VI 8 7 U)

>3

2 o o U 0.005 0.01 rEm] Vt calculated o standard deviation Vt

U. I

00, 0 0.015 0.02

Figure 5. Tangential velocity distribution with correct centre position

Data reduction

In

a non cavitating

vortex, the tangential velocity

distribution can be directly measured by measuring the vertical velocity on the horizontal axis through the core of the vortex,

see Fig. 2. But when the core of the vortex is cavitating, it

disturbs the forward scattered light. Hence the vertical velocity

component is measured off-axis and the tangential velocity

must be calculated. To do this, the vertical velocity component, the position of the measuring volume and the position of the centre of the vortex must be known. Also the assumptions must

be made that the real

tangential velocity distribution is

axisymmetric and that there are no radial velocities.

The mean position of the centre of the vortex is difficult to determine visually, due to the wandering of the vortex core. Therefore vertical velocity measurements were made on

trajectories parallel to the y-axis, at about one maximum core diameter above and below the vortex core.

The exact position of the centre of the vortex is found by

visual check of the appearance of the tangential velocity distribution, calculated with a chosen centre position. The correct centre position is defined as the position where the

chosen coordinates of the centre position give a smooth curve for every tangential velocity distribution, compare Figs. 4 and 5.

7 Figure 4. TangentIal velocity distribution with incorrect centre position

4

e

Figure 3. Typical cavitatlng core surfaces

From the position of the measuring control volume, the

position of the centre of the vortex and from the measured

vertical velocity component, the tangential velocity distribution can be derived.

To examine the velocity distribution accurately, the whole LDV setup can be traversed in z and y direction with steps of 0.1 mm.

The forward scattered light

is focussed on a photo

detector, the signal of the detector is processed by a tracker and

the analog output of the tracker is used as the input for a

measurement pc. Most measurements are done at a sample rate

of 16 kHz with a number of samples of 8000. After visual check of the signal, the mean value of the velocity and it's

standard deviation are calculated.

Measurinq the cavitating core diameter

Observation of the cavitating core with the naked eye

using stroboscopic light showed substantial disturbances on the

surface of the cavitating core, see Fig.

3. Therefore the

measurements of the cavitating core diameter where done by

making stroboscopic video recordings with 50 frames per

second to have some redundacy in

the number of the

recordings.

As a reference, recordings of a ruler placed in the test

section were made to determine the real diameter of the

cavitating core.

To observe the dynamics of the disturbances, high speed video recordings were also made. Figure 3 shows eight stages which are placed in order by high speed video observation at 27 kHz. These observations show that even though a diameter can be determined

at many times (frames

i to 3), significant

disturbances occurred, which were moving with the flow.

I

6

0.015 0.02

0.005 0.01

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0.102 £ 0.114

o

.0.645 * 0.653

Figure 6. Measured tangential velocity distributions at several cavitation numbers

Experimental results

In total 6 tangential velocity distributions were measured

at 6 different cavitation numbers, the Reynoldsnumber was

kept constant at 6:5. iO5. The gas concentration was between 3.9 and 5.3' ppm. The cavitation number is defined as:

PoPv

wherein P0 is the static pressure at the test section wall, Pv is the vapor pressure and U is the mean axial velocity measured

by the LDV.

The Reynoldsnumber is defined as

UD

Re

V

wherein D is the diameter of the contraction and y is the

kinematic viscosity. Figure 6 shows that the ambient pressure

(and thus the thIckness of the cavitating core) has

little influence on the tangential velocity distribution. Figure 7 shows the region close to the core in detail. Arrows indicate

the radius of the cavitating core a

at several cavitation

numbers and some indicative lines are given.

For O. I <a <0.4 'the gradient of the tangential velocity 'is already very high at r '2a. Due to the wandering of the vortex core in this range of cavitation numbers, no tangential velocity

measurements could be made for r < 2a,

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Figure 7. Comparison of measured tangential velocity distributions

Figure 8. Measured axial velocity distributions

For a = 0.645 the tangential velocity distribution seems

to remain constant and data are found even inside the

cavitating core. This is due to the high intermittancy of the

cavitating core, see frame 5 of Fig. 3.

The axial velocity distributiöns show the opposite; At cavitation numbers of 0.1 to 0.2 the axial velocities seem to

remain constant. But at cavitation numbers of 0.4 to 0.65 the axial velocities near the core are up to 13% higher, see Fig. 8.

The diameter of the cavitating core is measured as a

function of the cavitation number, mainly at a constant

Reynolds number of 6.510, with different gas concentrations

of the water (Cg = 3.9, 4.1, 5.3, 6.5, 8.5 and 10.8 ppm), see

Fig. 9. The diameter measurements were done in a a interval

which was limited on one side (at small cavitation numbers

and large core diameters) by vortex breakdown. On the other side (at high cavitation numbers and small core diameters) the

a interval was limited by absence of cavitation. The largest

value of a at which core diameter measurements were carried out was at the condition where a minimum of 10 cylindrical

181 17 16 19 15 V VF -VA V - V

..ii'

7

i.

D. IS..

0.102 ,o 0.205 'A 0.409 y 0.645

Standard DeviatiOn (Va)

4

05. o

.

f pU2

(i)

o 0.002 0.004 0.006 0.008 r Em]

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5 4.5 E 3.5 8

o,3

C (0 2.5 w w 1.5 E w 0.5 o OD a * A A

..

$

.

s I

O s'

a Cg =3.9 ppm Cg=4.1 ppm a Cg=53 ppm a Cg=85 ppm e Cg=8.5 ppm a Cg =10.8 ppm Stamiard dedation«)) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 11 o o

Figure 9. Diameter of the cavitating core as a function of the cavitation number

bubbles in

20 sec

of video

recording were observed

(desinence).

With a high gas concentration in the water, inception

occurred at larger values of the cavitation number than with a

low gas concentration in the water. This influence on the

cavitation inception number by the concentration of dissolved gas in the water can bedescribedby

cY.=Cpmin+

¡3Gg

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The difference in a is calculated (using Henry's constant

6700 Pa/ppm and U = 164 ms) to be 0.34. This agrees well with the measured difference in a1 between the maximum and minimum gas concentration, see Fig. 9. Since in this testsetup the vortex existed for some time, diffusion effects can indeed

be present.

MODELING A CAVITATING VORTEX

Kûiper (198'!) tised aspirai vortex approach with three regions to describe a cavitating vortax. Two regions are distinguished in the present analysis:

i. A vapor core, treated as a solid core, rotating with an

outer 'velocity equal to the fluid velocity. The pressure at

this cavitating core radius a may be assumed to be the

vapor pressure P.

2. A rolled-up vorticity region, in which the tangential'

velocity distribution is' assumed to have the form

where,R is the radius of the test section.

The parameters a and b in this velocity distribution and' the cavitating core radius a are assumed to be a function of the

pressure P0 at r = R.,But since no forces, act on the vortex when

changing the pressure, there must be conservation of angular momentum and kinetic energy.

The pressure distribution in the vortex can be found by integrating the momentum equation in radial direction

¿IP

v2(r)

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'r

Integrateting this equation using the tangential velocity

distributiOn of Eq. 4 over the region between r = a and R, gives the relation between the cavitation number and the

radius of the cavitating core:'

bU2 a a2

(1 (aC\2b

'R'

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The angular momentum per

unit length contained

between the radii a. and' R is

L =

2,rpjv(r)r2dr

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The kinetic energy of the tangential velocity distribution per unit length contained between the radii a and R is

E =

lrpjv2(r)rdr

₍₈₎

Out of the measurements, five conditions are chosen at which the tangential' velocity distribution (Eq. 5) is determined with it's core radius. These' conditions are listediñ Tab. i.

This leads to three equations (Eqs. 5,7 and 8) with three unknowns'(a, b, as).

By regression' analysis, the parameters a and b are

determined using Eq. 6 to fit into the, measured core radii as a function of the cavitation ñumber, see Fig. 9. This resúlts in a

=084 m'' Isec andb = 0.61.

a

r

a < r'<'R (4') 0.8 E E 0.6 w

I

0 0.4 w 0.2

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a iø5o 0645 0A09 0.205 0i'02 a [m] 2.5 lOE4 3.751O 5.75iO 8.5-1 4 1.251'OE3

Table 1. Conditions used ¡n modeling of the cavitating; vortex 14 12 10 E

4->6

4 2 -°-- 0.645 fr 0.409 -4-0205 -.0-0.102 is

-Figure 10. Comparison of tangential veloóity distributions, calculated by exercise 1

Exercise I

In the first exercise these values of a and b are used to

describe the velocity distribution (Eq. 4) of the smallest

cavitating core radius ac. In this condition the energy equals 1.3 J (Eq. 8)and the angular momentum is 0.019 kgin2s' (Eq.

7).

Maintaining this calculated energy and momentum in

combination with another cavitation number o, the new core

radius and the velocity distribution (described by the new

parameters aandb) are now calculated again.

This is done for the four remaining cavitation number&

The results are given in Fig. 10. This Figure shows that the

energy and the angular momentum remain near the core when

the pressure decreases. The calculated values of a

as a

functionof a are given in Fig. 11.

Exercise 2

In the second exercise a similar calculation of the energy

and angular momentum is made, but now based on the

conditiOn with the largest core radius. In this case the energy equals 1.19J and the angular momentum is0.019kgm2s'.

2.50E.03 2.00b03 5.00E-04 0.000.00

-j

s

noasured -e-- exercise 1 -û-- exerce2

-

Fgression fit

Figure 11. Comparison: of calculated values of a with

measured values

Figure 12. Comparison of tangential velocity distributions, calculated by exercise 2

It Should be kept in mind that this redúction in energy

(compared with exercise 1) is caused by the fact that the saine

velocity distribution (the same values a and b) is used at

different pressures..

There is a rather large discrepancy between the measured radius and the radius from the regression line, (for a = OE102),

using these values of a and b (Fig.

11). This causes an overestimate of the cavitation number (a = 0 128) at this largest core diameter.

14 Ici= ]_.o_o.128 12 i -Q-0205 -.6-0409 10 -0- 0.645 --1.06 I 4-0.000 0.005 0.010 0.015 0.020 r (m] 05 1.5 a o 0.000 0.005 0.010 r LrnJ 0.015 0.020

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> 14.00 12.00 10.00 8.00 6.00 4.00 200 0.00 a I b 0=

I

II-O6S3(rm) . .tnd,(Vl) 0- O.ft, ,,g., -+-O.132(,,,,c 3) -fr-O2(.oro 3)

I-°-O4O9(,*, 3) -a-0645(.,erc 3) -.- I OIO(,,,c 3) U O.1(n.) OE1I4(n.) O.2t6(n.) S O.4O9(n.) 84S(n.) 0.000 0.005 0.010 r (nil 0.015 0.020

Figure 13. Comparison of tangential velocity distributions calculated by exercIse 3 withmeasured

tangential velocity distributions

The result of exercise 2 is given in Fig. i2 This figure

shows that the energy and the angulár momentum remain even closerto the core than with exercise 1. Thecalculatedvalues of a as a function of a are given in F:g. 11.

Exercise 3

Neither exercise i nor exercise 2 predict the measured

a,-a rela,-ation very well (Fig. 11). Therefore, a,-a third model ha,-as been tried,

to satisfy the measured values of a and the

measured a value.

From the measurements it is clear that only the velocity

distribution close to the core (inner region) changes as a

function of the pressure. Hence a fixed velocity distribution is

taken for 0.006 < r < 02 m (outer region). A concentric

tangential velocity distribution for both the inner and outer

region is now chosen according toEq. 9.

C

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The parameters c and d for the

outer region are

determined by regression analysis.

14.00 12.00 10.00 ' 8.00 > 6.00 4.00 2.00 0.00

Figure 14. Comparison of tangential velocity distributions, calculated by exercise i with measured

velocity distributions

Table 2. Loss of kinetic energy and angular momentum

The parameters c 'and d for the inner region can now be determined from' the conditions that the velocity distribution at the inner/outer boundary is continuous and the pressure at' the core radius is equal to the vapor pressure. So the measured a-a rela-ation is ma-ainta-ained.

The values c and d in the inner region give the velocity

distribution forthe various pressures.

The results of this exercise are shown in Fig 13, denoted by

'(exerc 3)' together with the measured tangential velôcities,, denoted by (meas)'.

The subsequent loss of angular momentum and' kinetic

energy can now be cálculated and is shown in Tab 2. The percentages are normalized with respect

to the smallest

diameter (a =2.5lOE4 m).

The loss 'of kinetic energy, shown by Tab. 2, can also be seen in' Figure 13. The inner region velocity distribution of the

largest diameter, (a =0.102 (exerc. 3'))does not have velocities as high as the measured velocities (a =0.102 (meas.)).

IIi

---1

-a AE[%] AL[%] 1.050 0 o 0.645 0.8

005

0.409 3.0

006

0.205

9l

0.32 0.102

i68

0.76 0.000 0.005 0.010 r (mj 0.015 0.020

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DISCUSSION

To develop an alternative criterion for inception it is necessary to know the a-a relation in the inception range.

From Eq. 6 this relatiön can be written as

'C

a26 (10)

From the regression of the data, the value of b was found

:to be 61. Kuiper (1981) found values clOse to 05 for a

propeller tip vortex. The power b depends oñ the tangential

velocity distribution, hence the difference in tangential velocity distributions between the two situations: hub vortex cavitatiòn

and propeller tip vortex cavitation, apparently are not very large.

The results show that

with the given parametric

description of the tangential velocity distribution (Eq. 4), the cavitating core radius can be analytically calculated at various

pressures (Eq. 6). The effects of pressure changes on the

tangential velocity distributions could be estimated using conservation of energy and momentum, within reasonable accuracy.

In exercises 1 and 2, one point on the experimentally determined a- relation was taken and 'the other points were predicted using conservation of eñergy and angular

momentum. The result is given in Fig. i 1, in which also the

experimental data of the cavitating core radius are given with their95% confidence levels. Exercise I, which starts at a large

cavitation number overpredicts the core radius at smaller cavitation numbers Exercise 2 which starts

at a small

cavitation number,. underpredicts the core radius at higher pressures. Within' a limited range of pressure variations the

curves are reasonably well predicted'.

From Figs 10 and 12 it can' be concluded that at decreasing pressures the velocity increase occurs at radii close to the

cavitating core.

The technique

in which the velocity

distribution of various pressures is kept the same for all

pressures (Gowing, 1995) is therefore only slightly wrong in

this case.

In Fig. 14 the calculated velocity distribution from

exercise i (Fig. 10)

is plotted in combination with the

measured points. The shape of the curve is reasonably well

predicted, although the calculated velocities are systematically

lower than the measurements The results of exercise 1 are

slightly closer to the measurements than those of exercise 2.. When the starting point is taken at the measured velocity distribution

in combination with the condition

that the

pressure at a equals the vapor pressure, the velocity

distribution is not well predicted (Fig. 13').

The measured tangential velocities close to the core are

larger than predicted in' exercise 1

(Figs 7 and 14). This

means that even more energy is present around the core than predicted using conservation of energy in tangential direction.

From Fig. 8 it can 'also be seen that axial energy disappears

with decreasing pressure. This could show up in the tangential velocity distribution, but has not been modelled in the present

exercises.

REFERENCES

Arndt, R. E. A., and Keller, A. P., 1992, "Water Quality Effects on Cavitation Inception in a Trailing Vortex", ASME Journal of Fluids Engineering, Vol. 114 ¡pp. 430-438.

Cavitation Committee, 1993, "Report of the Cavitation Committee", International Towing Tank Conference, San Francisco.

Gowing, S., and' Briançon-Marjollet, L., 'and Frechou, 'D.,

'and Godeffroy, V., 1995, "Dissolved Gas. and Nuclei Effects on Tip Vortex Cavitation Inception and Cavitating Core Size International Symposium on Cavitation, Cav95, Deauville,

France.

Kuiper, G., 1981', "Cavitation Inception on Ship Propeller Models Thesis Technical University DeIft

Kuiper, , 1979, "Some Experiments with Distinguished

Types of Cavitation on Ship Propellers", ASME Internatiönal

Symposium on Cavitation Inceptiòn, 'New York.

McCormick jr, B. W., 1962, "On cavitation Produced by a Vortex Trailing from a Lifting Surface",, ASME Journal

of

Basic Engineering, pp. 369-379.