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Numerical simulation of cavitating ow using the upstreamnite element method

Tomomi Uchiyama 1

School of Informatics and Sciences, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan

Received 5 November 1996; received in revised form 12 January 1998; accepted 3 March 1998

Abstract

A nite element method is proposed to predict cavitating ows in arbitrarily shaped channels. An upwind scheme,

based on the PetrovGalerkin method using an exponential weighting function, is employed to eliminate the numerical

instability due to the advection term. The solution algorithm is parallel to a fractional step method. The calculation

domain is divided into quadrilateral elements. The pressure is dened at the centroid of the element and assumed to

be constant within the element. The other variables, such as the velocity and void fraction, are dened on the nodes.

Cavitating ows around a circular cylinder are simulated by the present nite element method. The cavitation occur-

rence relates closely to the vortex motion of the water in the sheared layer in accordance with experimental observa-

tions, and the cavitation regions almost coincide with the regions where cavitation bubbles are observed frequently

in experiments. This indicates that the present method is indeed applicable to the prediction of cavitating

ows. 1998 Elsevier Science Inc. All rights reserved.

Keywords: Computational uid dynamics; Cavitation; Upstream nite element method; Bubbly ow; Vortex

shedding

1. Introduction

Cavitation is one of the phenomena that must be taken into account when designing and op-

erating hydraulic machinery. Since cavitation in hydraulic machinery results not only in poor per-

formance but also noise, vibration and erosion, much attention has been devoted to methods for

predicting cavitating ow, especially numerical methods.

Based on the assumption that the ow is inviscid, various numerical methods have been thusfar proposed to simulate cavitating ows; the conformal mapping method [1,2], the singularity

method [35], and the panel method [6]. The ow around hydrofoil [1,4] and within a centrifugal

impeller [2,5] could be calculated using these inviscid ow models. Experimental observations

have revealed that the cavitation appearance relates closely to the viscous phenomena of the liq-

uid-phase, such as the boundary layer and the vortex motion. Recently, viscous ow models,

which regard the cavitating ow as the bubbly ow containing spherical bubbles, were introduced

Applied Mathematical Modelling 22 (1998) 235250

1 Fax: 81 52 789 5187; e-mail: [email protected].

0307-904X/98/$19.00 1998 Elsevier Science Inc. All rights reserved.

PII: S 0 3 0 7 - 9 0 4 X ( 9 8 ) 1 0 0 0 3 - 3

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to provide highly accurate calculations. In the viscous ow models, the NavierStokes equationincluding cavitation bubble is solved in conjunction with Rayleigh's equation governing the

change in the bubble radius. Kubota et al. [7] analyzed the ows around a hydrofoil by the nite

dierence method, and Shimada et al. [8] calculated the ow in a fuel injection pump for dieselengines by the control volume method. The predominating regions of high volumetric fraction

of bubbles obtained by these methods agree well with the cavitation regions observed experimen-

tally.

On the other hand, the analyses of ows in arbitrarily shaped channels are frequently nec-

essary to solve practical problems in many engineering elds. There is a growing tendency for

the nite element method to be employed in such analysis [9,10]. This is because the method

can represent precisely the geometry of the calculation domain and one can construct locally

ner computational meshes. The nite element method promises to analyze successfully cavita-

ting ows in an arbitrarily shaped channel, but it has been rarely applied to cavitating ow

analysis.

In this paper, a nite element method for cavitating ows is proposed. The analytical model,which was used for the nite dierence analysis of the bubbly ows in a rotating straight channel

in a previous paper [11], is modied and applied in the present method. This model largely cor-

responds to the aforementioned viscous ow model. The governing equations are solved by a -

nite element method based on a fractional step method [9]. An upwind scheme using an

exponential weighting function is employed in the nite element formulation to eliminate the nu-

merical instability due to the advection term. The cavitating ow around a circular cylinder is also

calculated by the present method. The appearance of cavitation relates closely to the vortex mo-

tion of the liquid in accordance with experimental observations [12], and the cavitation regions

calculated almost coincide with those observed in the experiment. Thus, the method is found

to be indeed applicable to the prediction of cavitating ows.

2. Basic equations

2.1. Assumptions

In a previous paper [11], the bubbly ows in a rotating straight channel were analyzed by the

nite dierence method using an analytical model proposed by Matsumoto et al. [13]. In the pre-

sent study, the model is modied and applied to the nite element analysis of cavitating ow. The

assumptions employed are as follows:

1. Cavitating ow is a bubbly ow, in which cavitation bubbles disperse uniformly and there is noslip velocity between the bubble and the liquid. This no slip assumption yields an appropriate

approximation when the bubbles are suciently small.2. The mass and momentum of the bubble are very small and negligible compared with those of

the liquid.

3. The liquid is incompressible. This assumption is parallel to that of the Kubota's study [7].

4. The gases inside the bubble are composed of a vapor and a non-condensable gas. The bubbles

change isothermally in volume, so the pressure of the vapor is constant. The non-condensable

gas obeys the perfect gas law, and the mass is conserved. These assumptions mean that the

liquid and the bubbles ow isothermally without phase change.

5. The bubbles maintain their spherical shape. This assumption is appropriate when the bubbles

are small.

6. Neither fragmentation nor coalescence of the bubble occurs.

236 T. Uchiyama / Appl. Math. Modelling 22 (1998) 235250

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2.2. Governing equations

The conservation equations for the mass and momentum of the cavitating ow are expressed

by the following under assumptions (1)(3).

o

ot1 a

o

oi1 ai 0Y 1

oi

ot j

oi

oj

1

1 aq

o

oi

1

q

osij

ojY 2

where

sij loi

ojoj

oi

2

3dij

om

om

X

Here the void fraction a is dened with the use of a bubble radius r and a number density of bub-

ble nb,

a 4a3pr3nbY 3

where nb is constant all over the ow eld under assumptions (1) and (6).

The relationship between the bubble radius r and the static pressure of bubble is expressed by

the following equation when neglecting the eects of surface tension and viscous damping for sim-

plicity from the same viewpoint as Kubota's study [7] under assumptions Eqs. (3)(5).

rD2r

Dt2

3

2

Dr

Dt

2

1

qv

r0

r

3g0

!Y 4

D

Dt

o

ot j

o

ojY

where r0 and g0 are the bubble radius and the pressure of non-condensable gas inside the bubble

on the boundary upstream of the calculation domain . The calculations with considering the ef-

fects of surface tension and viscous damping were also carried out. The numerical results, such as

the void fraction distributions, were almost the same as those obtained by neglecting the eects.

The boundary of is postulated to consist of the inlet g0, the wall g1 and the outlet g2.The boundary conditions are assumed to be given as:

u "u on g0 and g1Ycjdij sij 0 on g2Y

5

where the overbar denotes a known value, and cj is the direction cosine of the unit vector normal

to the boundary with respect to the j axis.

3. Numerical method

3.1. Time-integration method

The governing equations, except for the algebraic equation (3), are solved by a nite element

method. In this section, the dierence equations are shown to outline the time-integration method.

The conservation equation of mass, Eq. (1):

1 an1 1 an

Dt

o

oi1 ann1i 0X 6

T. Uchiyama / Appl. Math. Modelling 22 (1998) 235250 237

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The conservation equation of momentum, Eq. (2):

n1i ni Dt

nj

oni

oj

1

1 anq

on1

oi

1

q

osnij

oj !X 7

The equation governing the change in the bubble radius, Eq. (4):

rnfn1 fn

Dt nj

ofn

oj

3

2fn

2

1

qv

r0

rn

3g0

n1

!Y 8

where

fn1 rn1 rn

Dt nj

orn

ojX 9

The void fraction a is obtained from the following equation

an1 4a3prn13

nbX 10

When the ow at a time step t nDt is known, the solution at the next time step t n 1Dtcan be calculated by solving Eqs. (6)(10) simultaneously. For the simultaneous calculation ofEqs. (6) and (7), the following two-step procedure based on a fractional step method [9] is em-ployed.

In the rst step, the predicted velocity ~i is estimated by the following

~i ni Dt

nj

onioj

1

q

osnij

oj

X 11

When Eq. (11) is subtracted from Eq. (7), the following equation is obtained:

n1i ~i 1

1 anq

o/

oiY 12

where / is a function satisfyingn1 /aDtX 13

In order to calculate /, the following Poisson equation is derived by substituting Eq. (12) into

Eq. (6):

o

oi

o/

oi

q

an an1

Dt

o

oi1 an~i

& 'Y 14

where the boundary conditions for Eq. (14) are given by the following equations derived fromEq. (5):

o/aoc 0 on g0 and g1Y/ 0 on g2X

15

In the second step, n1i and n1 are calculated by substituting / obtained from Eq. (14) intoEqs. (12) and (13), respectively.

3.2. Finite element equations

The calculations in this study correspond to a two-dimensional ow eld. The calculation do-

main S is divided into quadrilateral elements. Fig. 1 shows an element. The pressure p is dened

at the center of each element and assumed to be constant within the element. The other variables

are dened on the vertices (nodes) of the element, and their values in the element are interpolated

using the shape function xb b 1 $ 4X xb is expressed in local coordinates ni i 1Y 2 as shownin Fig. 1 as follows:


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xb 1 n1n1b1 n2n2ba4 b 1Y 2Y 3Y 4Y 16where ni is dened in a region 1T niT 1Y and nib b 1 $ 4 denotes the ni coordinate of a nodebX

When applying the Galerkin method to the dierence equations (Eqs. (8), (9), (12)(14)), the

following nite element equations for each element are obtained:

wabfn1b wabf

nb p

n1aDtY 17

wabrn1b wabr

nb p

n2aDtY 18

wabn1ib wab~ib p

n3iaY 19

n1 a/aaeDtY 20

uab/b pn

4aY 21where

pn1a gabcjn

jbfnc wab

3fnb 2

2rnb

1

q

v

rnb

r30g0

rnb4

n1

rnb

2 34 5Y

pn2a gabcjn

jbrnc wabf

n1b Y p

n3ia gabci

/c

1 anbqY

pn4a wabanb a

n1b qaDt abi1 a

nbq~ibY

wab

xaxb dY a

xa dY gabcj

xaxb

oxc

ojdY uab

oxa

oj

oxb

ojdX

Here the void fraction at the node bY ab is calculated by the following:

ab 4a3prn1b

3nbX 22

The nite element equation for Eq. (11) is derived with the use of an upwind scheme of the

PetrovGalerkin type to eliminate the numerical instability due to the advection term. The up-

wind scheme using the exponential weighting function W proposed by Kakuda and Tosaka

[14] is employed in this study. The function W is expressed by

b e1n1n1b2n2n2bxbY 23

where 1 and 2 are given by the following, with constants j1 and j2,

1 j1n1ajujY 2 j2n2ajujX 24

Fig. 1. Quadrilateral element and local coordinates.


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The nite element equation for Eq. (11) in each element is expressed as:

qab~ib qabnib p

n5iaDtY 25

where

pn5ia eabcjn

jbnic

1

ql hab

nib h

iabj

njb

2

3h

jabi

nib

abt

nib

!Y

qab

axb dY eabcj

axb

oxc

ojdY hab

oa

oj

oxb

ojdY

hiabj

oa

oj

oxb

oidY ab

axb dgY t

nib cjs

nijbX

The void fraction an1 and the pressure n1 are obtained from Eqs. (10) and (20), respectively.

The other variables are calculated by the following equations, which are derived by assembling

the nite element equations for each element over the whole domain:

Mfn1 Mfn Fn1DtY 26

Mrn1 Mrn Fn2DtY 27

Mun1i M~ui Fn3iY 28

K/ Fn4Y 29

G~ui Guni F

n5iDtY 30

where

M

wabY K

uabY G

qabY

Fn1 pn

1aY Fn2 p

n2aY F

n3i p

n3iaY F

n4 p

n4aY F

n5i p

n5iaX

Here

denotes the assembly over the whole domain.

3.3. Numerical procedure

The numerical procedure is as follows:

1. Suppose the pressure at the time step n 1Yn1, to be equal to that at the step nYn.2. Calculate the bubble radius rn1 from Eqs. (26) and (27) with use of n1.

3. Calculate the void fraction an1 from Eq. (10) with use of rn1.

4. Calculate the predicted velocity of the liquid ~u from Eq. (30) with use of un.

5. Calculate the function / from Eq. (29) with use of an1 and ~u.

6. Calculate n1 and the liquid velocity un1 from Eqs. (20) and (28), respectively, with use of/.

7. Calculate the bubble radius ~rn1

from Eqs. (26) and (27) with use of n1

.8. When a condition rn1 ~rn1 is achieved in all elements, the ow properties at the time step

n 1 have been obtained by the above-mentioned calculations. If this condition is not attained,the estimated pressure n1 is increased in the element ifrn1 b ~rn1 or decreased in the elementif rn1 ` ~rn1. Then, the calculations from (1) to (7) are iterated until the condition rn1 ~rn1

is achieved in all elements. The criterion of convergence is taken as

jrn1 ~rn1arn1jT 0X1 102.The matrices in the nite element equations, such as wab and gabcj, are calculated by Gaussian

quadrature, where 2 2 Gauss points are used. The matrices wab and qab are lumped into diag-onal ones in order to save computer memory [15]. An LU decomposition method is used to solve

Eq. (29).


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4. Numerical results and discussion

4.1. Calculation condition

The cavitating ows around a circular cylinder, which were experimentally observed by Sato

[12], are used for the present calculations. Two kinds of cylinders with diameters D of 5 and

10 mm are used for the ows of Re 1X52 104 and 2X87 104, respectively, where Re is theReynolds number dened as q0hal, and 0 is the velocity of water upstream of the cylinder.

Fig. 2 shows the calculation domain and the nite elements. The width of the domain is set

10D, and the inlet and outlet boundaries are located 5D upstream and 13D downstream of the

cylinder, respectively. The number of elements is 3528, and the radial dimension of the elements

on the cylinder surface is 0.005D. The dimensionless time increment 0Dtah is 2X5 104, and the

constants j1 and j2 in Eq. (24) are set to be 0.4.

The boundary condition is summarized in Table 1. It is parallel to that used by Kakuda

and Tosaka [14] for their calculations of the water ow around a circular cylinder. At the inlet

Fig. 2. Calculation domain and nite elements.

Table 1

Boundary condition

Inlet boundary Uniform ow u1 u0, u2 0, r r0, d//dc 0Outlet boundary Traction free cj(dijp A sij) 0, / 0Cylinder surface No slip u1 u2 0, d//dc 0Channel lateral wall Full slip u2 0, d//dc 0


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boundary, an uniform ow is postulated in due consideration of the experimental condition. Atthe outlet, the uid traction is assumed zero, as mentioned in a previous chapter. A no slip con-

dition is prescribed on the cylinder surface, whereas a full slip condition is assumed on the lateral

boundaries of the calculation domain. The distributions of the number density and size of bubblenuclei in the experiment are not claried, but it is assumed that the initial bubble radius r0 is

30 106 m and that the number density of bubbles nb is 4X5 1010 in this calculation.

In order to examine the eect of the location of the computational domain's boundaries on the

numerical results, the calculation using a broader domain was also performed, where the width of

the domain is 15D and the outlet boundary is 20D downstream of the cylinder. The numerical

results, such as the void fraction distributions, were almost the same as those obtained by using

the domain shown in Fig. 2. This suggests that the domain in Fig. 2 is appropriate for the present

computation.

4.2. Results for non-cavitating conditions

Before cavitating ows were calculated, the numerical accuracy of the present nite elementmethod was evaluated under non-cavitating conditions. Fig. 3 shows the distributions of the

water velocity u for Re 1X52 104 and 2X87 104 at a time when the ows are suciently deve-loped under the non-cavitating condition. The ows separated from the cylinder surface generate

vortices behind the cylinder. The vortices ow downstream of the cylinder. The pressure distribu-

tions are also indicated in Fig. 3 by the contour lines of the pressure coecient gp. The vortices

yield low-pressure regions at their centers. The Strouhal number fha0 estimated from the vortexshedding frequency f is 0.24 for both Reynolds numbers, which is slightly larger than the mea-

sured one (.0.2).

Fig. 3. Distributions of velocity and pressure in water under non-cavitating conditions (interval between contour lines of

Cp is 0.2). (a) Re 1.52 104, (b) Re 2.87 104.


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The generation and shedding of the vortices make the ow around the cylinder unsteady, so thedrag coecient gD and the lift coecient gL of the cylinder change as functions of time tas shownin Fig. 4. The abscissa is the dimensionless time t 0tah. The time when the fully developedow is obtained is set to be t 0. The frequency ofgL coincides with that of the vortex sheddingfor both Reynolds numbers.

Fig. 5 shows the relationship between the time-averaged value of gD and Re. The present re-

sults at Re 1X52 104 and 2X87 104 are slightly larger than the measured ones. But, they canbe considered to be satisfactory when compared to the results obtained with a nite dierence

method [16] and a nite element method [14]. This indicates that the value for j1 and j2Y 0.4, usedin the present upwind scheme is appropriate.

The distributions of the time-averaged values ofgp on the cylinder surface at Re 1X52 104

and 2X87 104 are indicated in Fig. 6, where h is the azimuthal angle measured from the frontstagnation point of the cylinder. The relation gp 1 is satised at h 0Y and gp decreases mo-notonously with an increment in hX gp reaches a minimum at h 74 and remains almost unal-

tered in range of hP 90 due to the ow separation for both Reynolds numbers. The changein gp against h agrees approximately with the measured result Re 1X33 10

4 indicated bythe broken line in Fig. 6, though the calculated values are slightly lower than the measured ones

except near the front stagnation point.

4.3. Results for cavitating conditions

When the pressure upstream of the cylinder, 0Y and hence the cavitation number, rY aredecreased at a constant Reynolds number, Re, a region of high void fraction, aY that is a cavity,

Fig. 4. CD and CL under non-cavitating conditions.


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appears locally. The ow oscillates with almost a constant period under such cavitating condi-

tions.

Fig. 7 shows the distributions of the void fraction a at four times in the oscillation period, forRe 1X52 104 and r 3X6. The ow eld at t 0 corresponds to the fully developed one undernon-cavitating conditions. In this study, the cavity is dened as a region where a is more than 0.01

and indicated by the contour lines ofa. The interval between the contours is 0.02. Cavities appearon the cylinder surface as shown in Fig. 7(a). In Fig. 7(b), one of them grows on the cylinder sur-

face, and the others move away from the cylinder while their a values decrease. A cavity is ob-

served just behind the cylinder as shown in Fig. 7(c). Part of it grows abruptly into a very

large-scale cavity (darkened area) as seen in Fig. 7(d), where the maximum value ofa is 0.81.

Fig. 8 shows the distributions of water velocity at each of the four times shown in Fig. 7. The

ows separated from the cylinder surface generate vortices behind the cylinder. The cavitation re-

gions shown in Fig. 7 almost coincide with the regions in which the vortices occur. This is because

Fig. 6. Distribution of Cp on cylinder surface under non-cavitating conditions.

Fig. 5. Relationship between time-averaged CD and Re under non-cavitating conditions.


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the bubble volume expands in the center of the vortex, where the pressure reaches a minimumvalue. It is found that the aforementioned advection of the cavity is caused by the motion of

the vortices. The present numerical result that the appearance of the cavitation relates closely

to the vortex motion of the water in the sheared layer is in good agreement with experimental ob-

servations [12].

The time-averaged distribution of a for the above-mentioned cavitating ow

Re 1X52 104Y r 3X6 is shown in Fig. 9(a), where only the upper half region around the cyl-inder is displayed utilizing the symmetrical distribution. A cavity region ofaP 0.01 appears near

the separation point on the cylinder surface and behind the cylinder. The maximum value ofa is

0.067.

Fig. 9(b) shows the distribution of cavitation bubbles observed by Sato [12] using a CCD video

camera (30 frames/s) synchronized with a stroboscopic light (ash period 4 ls), where Re is thesame as in Fig. 9(a) and 3.06TrT 3.61. The value of x is the total number of pictures taken by

the camera, and n denotes the number of pictures in which the bubbles exist. Bubbles are observed

in the cavitation region calculated in Fig. 9(a). They are also observed downstream of this region,

suggesting that the observed cavitation region is larger than the calculated one. This discrepancy

may be due to the fact that the value ofr in the calculation, r 3X6, corresponds to the upperboundary ofr for the observation (3.06TrT 3.61), and also because the distributions of the ini-

tial radius, r0, and number density, nb, of the bubble are disregarded. It should also be mentioned

that the bubble deformation, fragmentation and coalescence occur behind the cylinder in the Sa-

to's experiment. Thus, it is necessary to take account of such bubble behaviour in the recirculation

zone in order to improve the computational accuracy.

Fig. 7. Distribution of a (Re 1.52 104, r 3.6, interval between contour lines is 0.02).

Fig. 8. Velocity distribution in water (Re 1.52 104, r 3.6).


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Fig. 10 shows the distributions of a at Re 2X87 104 and r 3X7. In Figs. 10(a) and (b),cavities appear in the vicinity of the cylinder surface. A large-scale cavity with a high value of

a is observed behind the cylinder in Fig. 10(c). The maximum value ofa is 0.72. In Fig. 10(d), thiscavity ows downstream with a rapid decrease in a.

The water velocity distributions for Fig. 10 are shown in Fig. 11. The cavitation regions in

Fig. 10 almost coincide with the areas in which vortices occur, just as in Figs. 7 and 8.

Fig. 12(a) shows the time-averaged distribution ofa for the above-mentioned cavitating ow

Re 2X87 104Y r 3X7. A cavity region calculated behind the cylinder almost coincides with

Fig. 10. Distribution ofa (Re 2.87 104, r 3.7, interval between contour lines is 0.02).

Fig. 9. Distribution of time-averaged a (Re 1.52 104). (a) Present calculation (r 3.6), (b) Experimental observation(3.06TrT 3.61).


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the observed region of 2 T n T 6 shown in Fig. 12(b), where the observation was conducted at

Re 2X87 104 and 3X62 T r T 4X11.When the value ofr is decreased from 3.7 to 3.4 at the same Reynolds number as in Fig. 12, the

time-averaged value ofa is as shown in Fig. 13(a). In comparison with Fig. 12(a), the cavitation

region expands and the value of a increases. The maximum value of a is 0.089. This cavitation

region behind the cylinder is in good agreement with the region where the value ofn is quite large

11 T n T 25 shown in Fig. 13(b).

Fig. 11. Velocity distribution in water (Re 2.87 104, r 3.7).

Fig. 12. Distribution of time-averaged a (Re 2.87 104). (a) Present calculation (r 3.7), (b) Experimental observa-tion (3.62TrT 4.11).


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References

[1] T.Y. Wu, A wake model for free-stream ow theory, Part 1: Fully and partially developed wake ows and cavity

ows past an oblique at plate, J. Fluid Mech. 13 (1962) 161181.

[2] Y. Tsujimoto, A.J. Acosta, C.E. Brennen, Analyses of the characteristics of a centrifugal impeller with leading

edge cavitation by mapping methods, Trans. JSME, 52480 B (1986) 29542962.

[3] R.A. Furness, S.P. Hutton, Experimental and technical studies of two-dimensional xed-type cavities, Trans.

ASME J. Fluids Eng. 97 (1975) 515522.

[4] H. Yamaguchi, H. Kato, On application of nonlinear cavity ow theory to thick foil sections, Proceedings of the

Conference on Cavitation, Edinburgh, IME, 1983, pp. 167174.

[5] H. Nishiyama, T. Ota, Hydrodynamic responses of centrifugal impeller with leading edge cavitation to oscillating

inlet ows, Proceedings of the International Symposium on Cavitation, Sendai, 1986, pp. 121126.

[6] H. Lemonnier, A. Rowe, Another approach in modelling of cavitating ows, J. Fluid Mech. 195 (1988) 557580.

Nomenclature

g boundary of

gD drag coecient of circular cylinder pDaq20ha2gL lift coecient of circular cylinder pLaq

20ha2

gp pressure coecient 0aq20a2

h diameter of circular cylinder

pD drag force acting on circular cylinder

pL lift force acting on circular cylinder

xb shape function

nb number density of bubbles

pressure

v saturated vapor pressure

r radius of bubble

Re Reynolds number q0hal calculation domain

t time

t dimensionless time 0tahu velocity of liquid-phase

x orthogonal coordinates

a void fraction

h azimuthal angle from front stagnation point of circular cylinder

l viscosity of liquid-phase

n local coordinates

q density of liquid-phase

r cavitation number 0 vaq20a2

/ function

Subscripts0 upstream boundary

i, j component in direction of i or j

Superscript

n time step


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[7] A. Kubota, H. Kato, H. Yamaguchi, A new modelling of cavitating ows: A numerical study of unsteady

cavitation on a hydrofoil section, J. Fluid Mech. 240 (1992) 5996.

[8] M. Shimada, T. Kobayashi, Y. Matsumoto, Numerical analysis of the ow in the fuel injection pump for diesel

engine, Proceedings of the Cavitation and Multiphase Flow Forum, FED-210, ASME, 1995, pp. 111114.[9] J. Donea, S. Giuliani, H. Laval, L. Quartapelle, Finite element solution of the unsteady NavierStokes equations

by a fractional step method, Comp. Meth. Appl. Mech. Eng. 30 (1982) 5373.

[10] M. Ikegawa, C. Kato, K. Tsuzuki, Three-dimensional turbulent ow analysis in a cleanroom by the nite element

method, FED-66, ASME, 1988, pp. 161167.

[11] T. Uchiyama, K. Minemura, T. Emura, J.C. Wu, Numerical simulation of bubbly ows in a rotating straight

channel, Proceedings of the International Symposium on Fluid Machinery and Fluid Engineering, Beijing, 1996,

pp. 131138.

[12] K. Sato, Inception characteristics of cavitation in a circular cylinder: Especially in subcritical ow range,

Proceedings of the Cavitation and Multiphase Flow Forum, FED109, ASME, 1991, pp. 97100.

[13] Y. Matsumoto, S. Takagi, H. Ohashi, Bubble driven plumes in the aeration tank with free surface, Proceedings of

the International Conference on Multiphase Flows, vol. 2, Tsukuba, (1991) 405408.

[14] K. Kakuda, N. Tosaka, Numerical simulation of high Reynolds number ows by PetrovGalerkin nite element

method, J. Wind Eng. and Ind. Aerodyn. 46 & 47 (1993) 339347.

[15] P.M. Gresho, R.L. Lee, R.L. Sani, Advection-dominated ows, with emphasis on the consequences of masslumping, Finite Elements in Fluids 3 (1978) 335350.

[16] T. Tamura, K. Kuwahara, Direct nite dierence computation of turbulent ow around a circular cylinder,

Numer. Methods in Fluid Dynamics 2 (1989) 645650.


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