numerical simulation of cavitating flow using the upstream finite element method

Numerical simulation of cavitating ¯ow using the upstream®nite 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 Petrov±Galerkin 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 de®ned 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 de®ned 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. Ó 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 forpredicting 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 singularitymethod [3±5], and the panel method [6]. The ¯ow around hydrofoil [1,4] and within a centrifugalimpeller [2,5] could be calculated using these inviscid ¯ow models. Experimental observationshave 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) 235±250

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

0307-904X/98/$19.00 Ó 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

to provide highly accurate calculations. In the viscous ¯ow models, the Navier±Stokes equationincluding cavitation bubble is solved in conjunction with Rayleigh's equation governing thechange in the bubble radius. Kubota et al. [7] analyzed the ¯ows around a hydrofoil by the ®nitedi�erence 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 fractionof 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 forthe ®nite element method to be employed in such analysis [9,10]. This is because the methodcan 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 ¯owanalysis.

In this paper, a ®nite element method for cavitating ¯ows is proposed. The analytical model,which was used for the ®nite di�erence analysis of the bubbly ¯ows in a rotating straight channelin a previous paper [11], is modi®ed 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 anexponential 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 alsocalculated 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 regionscalculated almost coincide with those observed in the experiment. Thus, the method is foundto 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 di�erence method using an analytical model proposed by Matsumoto et al. [13]. In the pre-sent study, the model is modi®ed and applied to the ®nite element analysis of cavitating ¯ow. Theassumptions employed are as follows:1. Cavitating ¯ow is a bubbly ¯ow, in which cavitation bubbles disperse uniformly and there is no

slip velocity between the bubble and the liquid. This no slip assumption yields an appropriateapproximation when the bubbles are su�ciently small.

2. The mass and momentum of the bubble are very small and negligible compared with those ofthe 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-condensablegas obeys the perfect gas law, and the mass is conserved. These assumptions mean that theliquid and the bubbles ¯ow isothermally without phase change.

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

6. Neither fragmentation nor coalescence of the bubble occurs.

236 T. Uchiyama / Appl. Math. Modelling 22 (1998) 235±250

2.2. Governing equations

The conservation equations for the mass and momentum of the cavitating ¯ow are expressedby the following under assumptions (1)±(3).

oot�1ÿ a� � o

oxi��1ÿ a�ui� � 0; �1�

oui

ot� uj

oui

oxj� ÿ 1

�1ÿ a�qopoxi� 1

qosij

oxj; �2�

where

sij � loui

oxj� ouj

oxiÿ 2

3dij

oum

oxm

� �:

Here the void fraction a is de®ned with the use of a bubble radius r and a number density of bub-ble nb,

a � �4=3�pr3nb; �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 p is expressed bythe following equation when neglecting the e�ects of surface tension and viscous damping for sim-plicity from the same viewpoint as Kubota's study [7] under assumptions Eqs. (3)±(5).

rD2rDt2� 3

2

DrDt

� �2

� 1

qpv � r0

r

� �3

pg0 ÿ p� �

; �4�

D

Dt� o

ot� uj

ooxj

;

where r0 and pg0 are the bubble radius and the pressure of non-condensable gas inside the bubbleon the boundary upstream of the calculation domain S. The calculations with considering the ef-fects of surface tension and viscous damping were also carried out. The numerical results, such asthe void fraction distributions, were almost the same as those obtained by neglecting the e�ects.

The boundary of S is postulated to consist of the inlet �C0�, the wall �C1� and the outlet �C2�.The boundary conditions are assumed to be given as:

u � �u on C0 and C1;cj�dijp ÿ sij� � 0 on C2;

�5�where the overbar denotes a known value, and cj is the direction cosine of the unit vector normalto the boundary with respect to the xj axis.

3. Numerical method

3.1. Time-integration method

The governing equations, except for the algebraic equation (3), are solved by a ®nite elementmethod. In this section, the di�erence equations are shown to outline the time-integration method.

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

�1ÿ an�1� ÿ �1ÿ an�Dt

� ooxi��1ÿ an�un�1

i � � 0: �6�

T. Uchiyama / Appl. Math. Modelling 22 (1998) 235±250 237

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

un�1i � un

i ÿ Dt unj

ouni

oxj� 1

�1ÿ an�qopn�1

oxiÿ 1

q

osnij

oxj

� �: �7�

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

rn f n�1 ÿ f n

Dt� un

j

of n

oxj

� �� 3

2�f n�2 � 1

qpv � r0

rn

� �3

pg0 ÿ pn�1

� �; �8�

where

f n�1 � rn�1 ÿ rn

Dt� un

j

orn

oxj: �9�

The void fraction a is obtained from the following equation

an�1 � �4=3�p�rn�1�3nb: �10�When the ¯ow at a time step t � nDt is known, the solution at the next time step t � �n� 1�Dt

can 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 ~ui is estimated by the following

~ui � uni ÿ Dt un

j

ouni

oxjÿ 1

q

osnij

oxj

� �: �11�

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

un�1i � ~ui ÿ 1

�1ÿ an�qo/oxi

; �12�where / is a function satisfying

pn�1 � /=Dt: �13�In order to calculate /, the following Poisson equation is derived by substituting Eq. (12) intoEq. (6):

ooxi

o/oxi

� �� q

an ÿ an�1

Dt� o

oxi��1ÿ an�~ui�

� �; �14�

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

o/=oc � 0 on C0 and C1;/ � 0 on C2:

�15�In the second step, un�1

i and pn�1 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 de®nedat the center of each element and assumed to be constant within the element. The other variablesare de®ned on the vertices (nodes) of the element, and their values in the element are interpolatedusing the shape function Nb �b � 1 � 4�: Nb is expressed in local coordinates ni �i � 1; 2� as shownin Fig. 1 as follows:


Nb � �1� n1n1b��1� n2n2b�=4 �b � 1; 2; 3; 4�; �16�where ni is de®ned in a region ÿ16 ni6 1; and nib �b � 1 � 4� denotes the ni coordinate of a nodeb:

When applying the Galerkin method to the di�erence equations (Eqs. (8), (9), (12)±(14)), thefollowing ®nite element equations for each element are obtained:

Mabf n�1b � Mabf n

b ÿ F n1aDt; �17�

Mabrn�1b � Mabrn

b ÿ F n2aDt; �18�

Mabun�1ib � Mab~uib ÿ F n

3ia; �19�pn�1 � Sa/a=�SeDt�; �20�Kab/b � ÿF n

4a; �21�where

F n1a � Cabcjun

jbf nc �Mab

3�f nb �2

2rnb

ÿ 1

qpv

rnb

� r30pg0

�rnb�4ÿ pn�1

rnb

!" #;

F n2a � Cabcjun

jbrnc ÿMabf n�1

b ; F n3ia � Cabci

/c

�1ÿ anb�q

;

F n4a � Mab�an

b ÿ an�1b �q=Dt � Rabi�1ÿ an

b�q~uib;

Mab �Z

NaNb dS; Sa �Z

Na dS; Cabcj �Z

NaNboNc

oxjdS; Kab �

ZoNa

oxj

oNb

oxjdS:

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

ab � �4=3�p�rn�1b �3nb: �22�

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

Wb � eÿa1�n1ÿn1b�ÿa2�n2ÿn2b�Nb; �23�where a1 and a2 are given by the following, with constants j1 and j2,

a1 � j1un1=juj; a2 � j2un2

=juj: �24�

Fig. 1. Quadrilateral element and local coordinates.


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

Gab~uib � Gabunib ÿ F n

5iaDt; �25�where

F n5ia � Aabcjun

jbunic �

1

ql Dabun

ib � Diabju

njb ÿ

2

3Dj

abiunib

� �ÿ Qabtn

ib

� �;

Gab �Z

WaNb dS; Aabcj �Z

WaNboNc

oxjdS; Dab �

ZoWa

oxj

oNb

oxjdS;

Diabj �

ZoWa

oxj

oNb

oxidS; Qab �

ZWaNb dC; tn

ib � cjsnijb:

The void fraction an�1 and the pressure pn�1 are obtained from Eqs. (10) and (20), respectively.The other variables are calculated by the following equations, which are derived by assemblingthe ®nite element equations for each element over the whole domain:

Mf n�1 �Mf n ÿ Fn1Dt; �26�

Mrn�1 �Mrn ÿ Fn2Dt; �27�

Mun�1i �M~ui ÿ Fn

3i; �28�K/ � ÿFn

4; �29�G~ui � Gun

i ÿ Fn5iDt; �30�

where

M �X

Mab; K �X

Kab; G �X

Gab;

Fn1 �

XF n

1a; Fn2 �

XF n

2a; Fn3i �

XF n

3ia; Fn4 �

XF n

4a; Fn5i �

XF n

5ia:

HereP

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� 1; pn�1, to be equal to that at the step n; pn.2. Calculate the bubble radius rn�1 from Eqs. (26) and (27) with use of pn�1.3. Calculate the void fraction an�1 from Eq. (10) with use of rn�1.4. 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 an�1 and ~u.6. Calculate pn�1 and the liquid velocity un�1 from Eqs. (20) and (28), respectively, with use of /.7. Calculate the bubble radius ~rn�1 from Eqs. (26) and (27) with use of pn�1.8. When a condition rn�1 � ~rn�1 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 pn�1 is increased in the element if rn�1 > ~rn�1 or decreased in the elementif rn�1 < ~rn�1. Then, the calculations from (1) to (7) are iterated until the condition rn�1 � ~rn�1

is achieved in all elements. The criterion of convergence is taken asj�rn�1 ÿ ~rn�1�=rn�1j6 0:1� 10ÿ2.The matrices in the ®nite element equations, such as Mab and Cabcj, are calculated by Gaussian

quadrature, where 2� 2 Gauss points are used. The matrices Mab and Gab are lumped into diag-onal ones in order to save computer memory [15]. An LU decomposition method is used to solveEq. (29).


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 and10 mm are used for the ¯ows of Re � 1:52� 104 and 2:87� 104, respectively, where Re is theReynolds number de®ned as qu0D=l, and u0 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 set10D, and the inlet and outlet boundaries are located 5D upstream and 13D downstream of thecylinder, respectively. The number of elements is 3528, and the radial dimension of the elementson the cylinder surface is 0.005D. The dimensionless time increment u0Dt=D is 2:5� 10ÿ4, and theconstants 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 Kakudaand 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, @//@c� 0

Outlet boundary Traction free cj(dijp ) sij)� 0, /� 0

Cylinder surface No slip u1� u2� 0, @//@c� 0

Channel lateral wall Full slip u2� 0, @//@c� 0


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 lateralboundaries of the calculation domain. The distributions of the number density and size of bubblenuclei in the experiment are not clari®ed, but it is assumed that the initial bubble radius r0 is30� 10ÿ6 m and that the number density of bubbles nb is 4:5� 1010 in this calculation.

In order to examine the e�ect of the location of the computational domain's boundaries on thenumerical results, the calculation using a broader domain was also performed, where the width ofthe domain is 15D and the outlet boundary is 20D downstream of the cylinder. The numericalresults, such as the void fraction distributions, were almost the same as those obtained by usingthe domain shown in Fig. 2. This suggests that the domain in Fig. 2 is appropriate for the presentcomputation.

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 thewater velocity u for Re � 1:52� 104 and 2:87� 104 at a time when the ¯ows are su�ciently deve-loped under the non-cavitating condition. The ¯ows separated from the cylinder surface generatevortices 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 coe�cient Cp. The vorticesyield low-pressure regions at their centers. The Strouhal number fD=u0 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.


The generation and shedding of the vortices make the ¯ow around the cylinder unsteady, so thedrag coe�cient CD and the lift coe�cient CL of the cylinder change as functions of time t as shownin Fig. 4. The abscissa is the dimensionless time t�� u0t=D�. The time when the fully developed¯ow is obtained is set to be t�� 0. The frequency of CL coincides with that of the vortex sheddingfor both Reynolds numbers.

Fig. 5 shows the relationship between the time-averaged value of CD and Re. The present re-sults at Re � 1:52� 104 and 2:87� 104 are slightly larger than the measured ones. But, they canbe considered to be satisfactory when compared to the results obtained with a ®nite di�erencemethod [16] and a ®nite element method [14]. This indicates that the value for j1 and j2; 0.4, usedin the present upwind scheme is appropriate.

The distributions of the time-averaged values of Cp on the cylinder surface at Re � 1:52� 104

and 2:87� 104 are indicated in Fig. 6, where h is the azimuthal angle measured from the frontstagnation point of the cylinder. The relation Cp � 1 is satis®ed at h � 0; and Cp decreases mo-notonously with an increment in h: Cp 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 Cp against h agrees approximately with the measured result �Re � 1:33� 104� indicated bythe broken line in Fig. 6, though the calculated values are slightly lower than the measured onesexcept near the front stagnation point.

4.3. Results for cavitating conditions

When the pressure upstream of the cylinder, p0; and hence the cavitation number, r; aredecreased at a constant Reynolds number, Re, a region of high void fraction, a; that is a cavity,

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


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 � 1:52� 104 and r � 3:6. The ¯ow ®eld at t� � 0 corresponds to the fully developed one undernon-cavitating conditions. In this study, the cavity is de®ned as a region where a is more than 0.01and indicated by the contour lines of a. 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 verylarge-scale cavity (darkened area) as seen in Fig. 7(d), where the maximum value of a 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.


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 ofthe vortices. The present numerical result that the appearance of the cavitation relates closelyto 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 � 1:52� 104; r � 3:6� 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 of a P 0.01 appears nearthe separation point on the cylinder surface and behind the cylinder. The maximum value of a is0.067.

Fig. 9(b) shows the distribution of cavitation bubbles observed by Sato [12] using a CCD videocamera (30 frames/s) synchronized with a stroboscopic light (¯ash period� 4 ls), where Re is thesame as in Fig. 9(a) and 3.066 r6 3.61. The value of N is the total number of pictures taken bythe camera, and n denotes the number of pictures in which the bubbles exist. Bubbles are observedin 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 discrepancymay be due to the fact that the value of r in the calculation, r � 3:6, corresponds to the upperboundary of r for the observation (3.066 r6 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 mentionedthat 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 recirculationzone 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).


Fig. 10 shows the distributions of a at Re � 2:87� 104 and r � 3:7. In Figs. 10(a) and (b),cavities appear in the vicinity of the cylinder surface. A large-scale cavity with a high value ofa is observed behind the cylinder in Fig. 10(c). The maximum value of a 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 inFig. 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 of a for the above-mentioned cavitating ¯ow�Re � 2:87� 104; r � 3:7�. A cavity region calculated behind the cylinder almost coincides with

Fig. 10. Distribution of a (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

(3.066 r6 3.61).


the observed region of 2 6 n 6 6 shown in Fig. 12(b), where the observation was conducted atRe � 2:87� 104 and 3:62 6 r 6 4:11.

When the value of r is decreased from 3.7 to 3.4 at the same Reynolds number as in Fig. 12, thetime-averaged value of a is as shown in Fig. 13(a). In comparison with Fig. 12(a), the cavitationregion expands and the value of a increases. The maximum value of a is 0.089. This cavitationregion behind the cylinder is in good agreement with the region where the value of n is quite large�11 6 n 6 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 (3.626 r6 4.11).


5. Conclusions

A ®nite element method for calculating cavitating ¯ow is proposed. The calculation domain isdivided into quadrilateral elements. The pressure is de®ned at the center of each element and as-sumed to be constant within the element. The other variables, such as the velocity and the voidfraction, are de®ned on the nodes. An upwind scheme, based on the Petrov±Galerkin methodusing exponential weighting function, is employed to eliminate the numerical instability due tothe advection term. The solution algorithm uses a fractional step method, and the mass matricesare lumped into diagonal ones to save computer memory.

The present ®nite element method is applied to the ¯ow analyses around a circular cylinder.The cavitation occurrence relates closely to the vortex motion of the water in the sheared layerin accordance with experimental observations, and the cavitation regions almost coincide withthe regions where cavitation bubbles are observed frequently in the experiment. This indicatesthat the present method is indeed applicable to cavitating ¯ow analysis. Especially, the methodis usefully employed for the analyses in hydraulic machinery with complicated geometry. Becauseit can represent precisely the geometry with the use of unstructured computational grids. But, it isnecessary to take account of the bubble deformation, fragmentation, coalescence and slip relativeto the liquid in order to improve the computational accuracy.

Fig. 13. Distribution of time-averaged a (Re� 2.87 ´ 104). (a) Present calculation (r� 3.4), (b) Experimental observa-

tion (2.726 r6 3.62).


References

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¯ows past an oblique ¯at plate, J. Fluid Mech. 13 (1962) 161±181.

[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, 52±480 B (1986) 2954±2962.

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

ASME J. Fluids Eng. 97 (1975) 515±522.

[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. 167±174.

[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. 121±126.

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Nomenclature

C boundary of SCD drag coe�cient of circular cylinder � FD=�qu2

0D=2�CL lift coe�cient of circular cylinder � FL=�qu2

0D=2�Cp pressure coe�cient � �p ÿ p0�=�qu2

0=2�D diameter of circular cylinderFD drag force acting on circular cylinderFL lift force acting on circular cylinderNb shape functionnb number density of bubblesp pressurepv saturated vapor pressurer radius of bubbleRe Reynolds number � qu0D=lS calculation domaint timet� dimensionless time � u0t=Du velocity of liquid-phasex orthogonal coordinatesa void fractionh azimuthal angle from front stagnation point of circular cylinderl viscosity of liquid-phasen local coordinatesq density of liquid-phaser cavitation number � �p0 ÿ pv�=�qu2

0=2�/ function

Subscripts0 upstream boundaryi, j component in direction of xi or xj

Superscriptn time step


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numerical simulation of cavitating flow using the upstream finite element method

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