NUMERICAL ANALYSIS OF UNSTEADY VAPOROUS CAVITATING...

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1 Cav03-OS-1-006 Fifth International Symposium on Cavitation (cav2003) Osaka, Japan, November 1-4, 2003 NUMERICAL ANALYSIS OF UNSTEADY VAPOROUS CAVITATING FLOW AROUND A HYDROFOIL Yoshinori SAITO, Institute of Fluid Science, Graduate School of Tohoku University, [email protected] Ichiro NAKAMORI, Institute of Fluid Science, Tohoku University, [email protected] Toshiaki IKOHAGI, Institute of Fluid Science, Tohoku University, [email protected] ABSTRACT In this paper, numerical results of turbulent cavitating flow simulation around a hydrofoil using a phase change model were presented. A locally homogeneous model was proposed for the gas-liquid two-phase medium, and the Navier-Stokes equations along with the mass transfer equation of the vapor were solved. Our numerical method employed the alternate direction implicit (ADI) method and TVD-MUSCL scheme. To obtain the turbulent eddy viscosity, the Baldwin-Lomax model with the Degani-Schiff modification was used. To assess our numerical method, vaporous cavitating flows over a hemisphere / cylinder geometry were first computed and the numerical results were compared with experimental results of Rouse and McNown. Then, the cavitating flows around the CAV2003 hydrofoil were numerically simulated, and the cavity break-off phenomenon was discussed. INTRODUCTION Cavitation is a rapid phase change phenomenon, which often occurs in the high-speed fluid machineries, and it is well known that the cavitating flows raise up the vibration, the noise and the erosion. Therefore, the improvement of the prediction for the cavitating flow is of great interest for many researchers. A lot of researchers have analyzed numerically the cavitating flow using various physical models [1]. Kunz et al. [2] analyzed the cavitating flow by UNCLE-M that is a fully implicit, pre-conditioned and Reynolds-averaged Navier-Stokes solver. Their results agree with the experimental results [3]. Matsumoto et al. [4] simulated the cavitating flow using bubbly flow model that transfers a great number of bubbles throughout the flowfield and takes account of the radial motions based on the Rayleigh-Presset equation. On the other hand, Okuda, et al. [5] considered the apparent compressibility of the gas-liquid two-phase medium, and then introduced the equation of state for the medium, which combined the equation of state for ideal gas with that for the liquid by Tamman [6]. In this model, the gas-liquid two- phase medium in the cavity region was treated as a locally homogeneous single-phase medium. The features of the model are described as, (1) The interfaces were treated as a contact surface, which permits the density jump. (2) The interaction of propagating waves can be captured. By using this model, Iga, et al. [7] numerically analyzed the cavitation instabilities in a cascade hydrofoil with the explicit McCormack scheme. Nakamori et al. [8] computed the 3-D turbulent cavitating flow around a sphere with the phase change model, the turbulence model and the implicit time-stepping method. The purpose of this study is to perform numerical simulation and to clarify the transient behavior of vaporous cavitating flow within the flow-time scale. In our method [8], the modeling of the gas-liquid two-phase medium is based on the locally homogeneous model [5], which takes the phase change into account. In particular, this method is applied to the unsteady cavitating flow around the CAV2003 hydrofoil. NOMENCLATURE Symbols r : mixture density w v u , , : velocity components e : total mixture energy per unit volume H : total mixture enthalpy per unit mass p : pressure Y : quality T : temperature a : void fraction m : viscosity coefficient t m : turbulent eddy viscosity coefficient - m m & & , : mass transfer rates s : cavitation number c : chord length D : diameter Subscripts l : liquid phase v : vapor phase : free stream value

Transcript of NUMERICAL ANALYSIS OF UNSTEADY VAPOROUS CAVITATING...

Page 1: NUMERICAL ANALYSIS OF UNSTEADY VAPOROUS CAVITATING …flow.me.es.osaka-u.ac.jp/cav2003/Papers/Cav03-OS-1-006.pdf · Fifth International Symposium on Cavitation (cav2003) Osaka, Japan,

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Cav03-OS-1-006 Fifth International Symposium on Cavitation (cav2003)

Osaka, Japan, November 1-4, 2003

NUMERICAL ANALYSIS OF UNSTEADY VAPOROUS

CAVITATING FLOW AROUND A HYDROFOIL

Yoshinori SAITO, Institute of Fluid Science, Graduate School of Tohoku University, [email protected]

Ichiro NAKAMORI, Institute of Fluid Science, Tohoku University, [email protected]

Toshiaki IKOHAGI, Institute of Fluid Science, Tohoku University, [email protected]

ABSTRACT

In this paper, numerical results of turbulent cavitating flow simulation around a hydrofoil using a phase change model were presented. A locally homogeneous model was proposed for the gas-liquid two-phase medium, and the Navier-Stokes equations along with the mass transfer equation of the vapor were solved. Our numerical method employed the alternate direction implicit (ADI) method and TVD-MUSCL scheme. To obtain the turbulent eddy viscosity, the Baldwin-Lomax model with the Degani-Schiff modification was used.

To assess our numerical method, vaporous cavitating flows over a hemisphere / cylinder geometry were first computed and the numerical results were compared with experimental results of Rouse and McNown. Then, the cavitating flows around the CAV2003 hydrofoil were numerically simulated, and the cavity break-off phenomenon was discussed.

INTRODUCTION

Cavitation is a rapid phase change phenomenon, which often occurs in the high-speed fluid machineries, and it is well known that the cavitating flows raise up the vibration, the noise and the erosion. Therefore, the improvement of the prediction for the cavitating flow is of great interest for many researchers.

A lot of researchers have analyzed numerically the cavitating flow using various physical models [1]. Kunz et al. [2] analyzed the cavitating flow by UNCLE-M that is a fully implicit, pre-conditioned and Reynolds-averaged Navier-Stokes solver. Their results agree with the experimental results [3]. Matsumoto et al. [4] simulated the cavitating flow using bubbly flow model that transfers a great number of bubbles throughout the flowfield and takes account of the radial motions based on the Rayleigh-Presset equation.

On the other hand, Okuda, et al. [5] considered the apparent compressibility of the gas-liquid two-phase medium, and then introduced the equation of state for the medium, which combined the equation of state for ideal gas with that for the liquid by Tamman [6]. In this model, the gas-liquid two-phase medium in the cavity region was treated as a locally homogeneous single-phase medium. The features of the model are described as,

(1) The interfaces were treated as a contact surface, which permits the density jump.

(2) The interaction of propagating waves can be captured. By using this model, Iga, et al. [7] numerically analyzed the cavitation instabilities in a cascade hydrofoil with the explicit McCormack scheme. Nakamori et al. [8] computed the 3-D turbulent cavitating flow around a sphere with the phase change model, the turbulence model and the implicit time-stepping method.

The purpose of this study is to perform numerical simulation and to clarify the transient behavior of vaporous cavitating flow within the flow-time scale. In our method [8], the modeling of the gas-liquid two-phase medium is based on the locally homogeneous model [5], which takes the phase change into account. In particular, this method is applied to the unsteady cavitating flow around the CAV2003 hydrofoil.

NOMENCLATURE Symbols

ρ : mixture density wvu ,, : velocity components

e : total mixture energy per unit volume H : total mixture enthalpy per unit mass p : pressure Y : quality T : temperature α : void fraction µ : viscosity coefficient

tµ : turbulent eddy viscosity coefficient −+ mm && , : mass transfer rates

σ : cavitation number c : chord length D : diameter

Subscripts l : liquid phase v : vapor phase ∞ : free stream value

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NUMERICAL METHOD Physical Modeling

In the present physical model, by approximating the vapor-liquid mixture of any finite bubbles existing in each control volume to that of infinite number of infinitesimal bubbles, the local mixture condition in the cavity is specified in each computational cell having the same void fraction. Since this modeling can be simultaneously applied to the inside mixture phase of cavity and the outside liquid phase, it is possible to simulate macroscopically the strong unsteady and complex cavity flows. The equation of state for a locally homogeneous vapor-liquid two-phase medium can be expressed as follows by using quality Y (mass fraction of gas phase) [5]:

)1(,)()()1(

)(0 TppRYTTpYK

pppc

c

+++−+

where 0,Tpc are the pressure and temperature constants of liquid, and KR, are the vapor and liquid constants, respectively. Also, the following relation is satisfied between void fraction α and quality Y :

)2(.,)1()1( vl YY αρρραρ =−=− And, the apparent viscosity coefficient is approximated as follows:

)3(.)5.21)(1( vl αµµααµ ++−= Governing Equations and Phase Change Modeling

The governing equations are the gas-liquid two-phase compressible Navier-Stokes equations, and are written by:

)4(,)()()(

SzGG

yFF

xEE

tQ vvv =

∂−∂

+∂−∂

+∂−∂

+∂∂

where vvv GFEGFEQ ,,,,,, and S denote the solution vectors, the flux vectors in x, y and z directions, and the source term including the modeling term of evaporation and condensation. These terms are written as follows:

,

00000

,

0

0

,

0

0

,

0

0

)5(

,

)(

,

)(

,

)(

, 2

2

2

=

=

=

=

++

=

+

+=

+

+

=

=

m

SGFE

Ywwpepw

vwuww

G

Yvvpe

vwpv

uvv

F

Yuupe

uwuv

puu

E

Yewvu

Q

z

zz

zy

zx

v

y

yz

yy

yx

v

x

xz

xy

xx

v

&βτττ

βτττ

βτττ

ρ

ρρρρ

ρ

ρρ

ρρ

ρ

ρρ

ρρ

ρ

ρρρρ

where wvuq xzxyxxxx τττβ +++= . And, the stress tensor ijτ and the heat fluxe

iq are given by: )6(),)(( ijjitij xuxu ∂∂+∂∂+= µµτ )7(,)( iti xTq ∂∂+= κκ

where κ is the blended heat conductivity coefficient, which is given by:

)8(.)1( vl ακκακ +−= In this study, the non-equilibrium mass transfer modeling

[9] was employed to describe phase change between water and vapor. In order to close the above equations, we applied the mass transfer model based on the theory of evaporation/condensation on a plane surface.

)12(,)1(

)11(,2

)1(

)10(,2

)1(

)9(,else

if

*

*

*

ααπ

αα

πρρ

αα

−=

−−=

−=

<

=

+

+

a

s

vc

s

v

v

le

v

CART

ppACm

RT

ppACm

mppm

m

&

&

&&

&

where sT is the saturation temperature, A denotes the interfacial area concentration in the vapor-liquid mixture.

acae CCCCC =≡* is an empirical model constant, whose value was set to 0.1m-1 in this study. The saturation vapor pressure of water *

vp is given by the empirical formula as:

)13(]}.)16.483)(10787.410152.1(

21379.7)[31.6471exp{(1013.22295

6*

−×−×

+−×=−− TT

Tpv

By using the equation of state (1), the density ratio vl ρρ in eq.(10) is given by:

( )( ) )14(.

0TTpKRTpp c

v

l

++

=ρρ

Numerical Scheme

Explicit time stepping requires a very small time step, so that an implicit time stepping was used to obtain the unsteady fluid motion in the flow-time scale as quickly as possible. To achieve feasible computing efficiency, the alternate direction implicit (ADI) approximate factorization scheme was employed for the time integration. The cell centered finite volume formulation was used to discrete the governing equations. The first derivatives involved by the viscous fluxes were estimated by using Gauss’ theorem. To compose the elements of inviscid flux-vector, the AUSM [10] type scheme was used. To obtain third-order accuracy in space, the conservative variables were interpolated at the cell interfaces by using the MUSCL method with a minmod limiter. [8]

RESULTS AND DISCUSSION Validation of Present Method

The present numerical method is considered relevant to the analysis of complex cavity flows, because it can treat the whole flow fields at once. However our method includes an empirical model constant C*m-1, so we had to validate our method by comparing results between computed and measured of surface pressure distributions for a 3-D hemisphere/cylinder geometry by Rouse and McNown [7]. Their experimental results have been widely used to assess the phase change model in prediction of cavitating region. The experimental flow

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conditions are as follows: the Reynolds number based on the hemisphere diameter D is 51036.1 × , the cavitation numbers σ are 0.8, 0.4, 0.3 and 0.2, and water temperature is assumed to be 300K. For all cases, the computational domain contains

4064256 ×× cells in the streamwise ( ξ ), radial ( η ) and azimuthal (ζ ) directions, respectively. The grids are clustered to the wall boundary. The minimum grid spacing minη∆ near the wall is set to 5

min 100.1/ −×=∆ Dη . The time step t∆ was fixed at T /1000, where T is the flow-time defined by

∞= uDT / . Here, ∞u is the free stream velocity. The time-averaged values were computed over a range of 60T. And turbulent flow was computed by using the Baldwin-Lomax model with the Degani-Schiff modification to determine the turbulent eddy viscosity coefficient tµ . The uniform flow condition was used for the far-field treatment of boundaries, and both the non-slip and adiabatic conditions were applied to the treatment of solid wall boundary.

Figure 1 shows the time-averaged surface pressure distributions for each cavitation numbers. As the cavitation number decreases, the pressure in the hemispherical expansion region becomes below the saturation vapor pressure, and the cavitation occurs there. In the cavitating conditions, the pressure abruptly recovers through the condensation effect, which causes the collapse of cavity near the termination. In addition, the temperature drop was at most about 0.5K within the cavitating region. Overall the present computational results show good agreement with the experimental results. It becomes obvious that the appropriate prediction for the turbulent cavitating flow can be obtained, as mentioned before, by using the present numerical method with 1.0* =C m-1.

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8

Cp

543210

x / D

Exp. Presentσ = 0.8 0.4 0.3 0.2

Fig.1 Time-averaged surface pressure distributions of a

hemisphere / cylinder geometry CAV2003 Hydrofoil Noncavitating Condition

Next, unsteady turbulent cavitating flows around a CAV2003 hydrofoil were analyzed. Figure 2 illustrates the geometry and boundary conditions for the simulation. The uniform flow condition was used at the inlet boundary, and both the non-slip and adiabatic conditions were applied to the

treatment of the hydrofoil and upper/lower wall boundaries. The hydrofoil is set at the middle of a channel of length 10c and height 4c. Here c is the chord length of 0.1m. The angle of attack is °7 . And in the frame of reference Oxy of the foil, the upper side is defined analytically by the following equation,

.02207.0,07272.0,00593.0,02972.0,11858.0

43

210

4

4

3

3

2

210

−=−==−==

+

+

++=

aaaaa

cx

acx

acx

acx

acx

acy

2c

5c 5c

c = 0.1m ( chord length)

2c

p 8

lower wall

upper wall

x

y

OU 8 = 6 m/s

0.5c

0.5c

: pressure monitoring point

A

B

Fig.2 CAV2003 hydrofoil geometry and boundary conditions

The numerical flow conditions are as follows: the Reynolds number based on the chord length is 5109.5 × , the cavitation numbers σ based on the downstream pressure are 3.5(no cavitation), 1.5, 1.2, 1.1, 1.0, 0.9, 0.8, 0.4 and 0.2, and water temperature is 293.15K. Then we carried out the grid sensitivity studies in noncavitating condition. We employed a C-type grid system, in which grid sizes were 4365× (coarse mesh),

85129× (medium mesh), 85257× (fine mesh) and 85512× (extra fine mesh) in the streamwise (ξ ) and perpendicular (η ) directions, respectively. The grids were clustered to the hydrofoil boundary. The minimum grid spacing minη∆ near the hydrofoil surface is set to 5

min 100.1/ −×=∆ cη . In the noncavitating condition, the time step t∆ was fixed at T/2000, where T is the flow-time defined by ∞= ucT / =16.67ms. And, the time-averaged values were computed over a range of 100T. Figure 3 shows the grid sensitivity to the surface pressure distribution and demonstrates that the differences between the fine and the extra fine meshes are small. In addition, we carried out the computation by using the incompressible flow solver (FLUENT6). In the FLUENT6, the Spalart-Allmaras model was used to obtain the eddy viscosity. The flow field converged to quasi-steady state in both numerical solutions. The grid sensitivity to the time-averaged lift coefficient between our method and FLUENT6 is shown in figure 4. The difference between fine mesh and extra fine mesh with our method is less than 1.5%. Therefore, the fine mesh was used for all subsequent computations presented here. Then, figure 5 shows the time-averaged surface pressure distributions to confirm the effect of RANS model in comparison our method and FLUENT6. From these comparisons, our numerical result with RANS model agrees well with the FLUENT6, though it is seen that our result with RANS slightly overestimates the surface pressure distribution near the trailing edge. On the other hand, the

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periodic separation bubbles shed on the suction surface in our result without RANS model, and such a result disagrees with the result of FLUENT6 throughout the chord.

3

2

1

0

-1

-Cp

1.00.80.60.40.20.0

x / c

Extra Fine Fine Medium Coarse

Fig.3 Grid sensitivity to surface pressure distribution for

noncavitating flow over CAV2003 hydrofoil

0.80

0.70

0.60

CL

65x43 129x85 257x85 513x85Number of grid size

Present FLUENT6

Fig.4 Grid sensitivity to lift coefficient for noncavitating flow

over CAV2003 hydrofoil

3

2

1

0

-1

-Cp

1.00.80.60.40.20.0

x / c

Present Present(without RANS) FLUENT6

Fig.5 Comparison of surface pressure distribution around

CAV2003 hydrofoil on fine mesh Cavitating Condition

The unsteady phenomena in cavitating conditions were analyzed. In these cases, a smaller time step t∆ was fixed at

10000/T . And, the time-averaged values were computed over a range of 60T. First, the maximum cavity length cll /maxmax = , the maximum cavity thickness ctt /maxmax = , the position of maximum cavity thickness maxmaxmax / lll tt = and the abscissa of cavity detachment cll dd /= were identified in order to confirm the aspect of the qualitative flow field at each cavitation number. For the identification of those values, the threshold of void fraction was set to 10%. Table 1 shows the summary of the cavitation parameters and figure 6 shows the maximum cavity length and thickness versus the cavitation number. As decreasing the cavitation number, the maximum cavity length become longer and its thickness thicker. On the other hand, the position of maximum cavity thickness is almost constant at 70% of the maximum cavity length except for 2.0=σ . In the case of 2.0=σ , a supercavitating flow fully develops, and the cavitation also appears on the pressure side near the trailing edge. The abscissa of cavity detachment slightly increases near the leading edge with the decrease in cavitation number.

Table 1 Summary of the cavitation parameters

σ maxl maxt maxtl dl LC DC 3.5 - - - - 0.642 0.0173 1.5 0.241 0.0257 0.691 0.00118 0.524 0.0287 1.2 0.368 0.0390 0.746 0.00119 0.500 0.0323 1.1 0.516 0.0563 0.682 0.00132 0.496 0.0397 1.0 0.613 0.0651 0.702 0.00142 0.483 0.0572 0.9 0.692 0.0649 0.680 0.00153 0.475 0.0642 0.8 0.760 0.0879 0.755 0.00174 0.417 0.0638 0.4 1.309 0.1914 0.720 0.00368 0.160 0.0568 0.2 2.608 0.2652 0.483 0.00409 0.108 0.0458

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Max

imum

cav

ity le

ngth

l

max

/ c

1.41.21.00.80.60.40.20.0

σ

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Max

imum

cav

ity th

ickn

ess

t m

ax /

c

maximum cavity length maximum cavity thickness

Fig.6 Maximum cavity length and maximum cavity thickness

versus cavitation number

Figure 7 shows the time-averaged values LC , DC of lift and drag coefficients together with their standard deviations. It is confirmed that as decreasing the cavitation number, the time-averaged lift coefficients decrease, and the time-averaged drag coefficients once increase and then decrease after taking a maximum value at 9.0=σ . In addition, it is indicated that the standard deviations become large in a transient range of

8.0=σ to 1.1.

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0.8

0.6

0.4

0.2

0.0

CL

3.53.02.52.01.51.00.50.0

σ

0.20

0.15

0.10

0.05

0.00

CD

CL CD

Fig.7 Time averaged lift coefficients and drag coefficients with

standard deviations

Then, the unsteady phenomena of cavitating flows were analyzed. Figures 8, 9 and 10 show the instantaneous fields of void fraction, mass flux vector, the evaporation and condensation rates and the surface pressure distributions for 0.1,2.1=σ and 0.8. In the case of 2.1=σ , the comparatively small-scale sheet / cloud cavitation appears. It can be seen that the evaporation appears inside the sheet cavity (see Fig.8-3). There is the local reverse flows accompanied by the circulation in the cavity, and the cavity break-off does not always occur in every reverse flow, though the cavity length oscillates periodically. In the case of 0.1=σ , there is an irregular cavity shedding and the period of the break-off is much longer than that of 2.1=σ , as shown in figure 9. However, in the case of 8.0=σ , the large-scale cloud cavitation with the re-entrant jet occurs (see Fig.10-3). It is observed that the evaporation is considerably generated near the leading edge of the sheet cavity, and that the condensation takes place in the cavity closure region and around the shedding break-off cavities (see Figs.10-1 and 10-5). Then, the pressure waves caused by the impact of re-entrant jet are observed to propagate from the leading edge of the sheet cavity, as seen from the surface pressure distribution of figure 10-4.

1

2

3

4

5

6

-Cp

-1.5

-1

-0.5

00.5

1

-Cp

-1.5

-1

-0.5

0

0.5

1

-Cp

-1.5

-1-0.5

0

0.5

1

-Cp

-1.5

-1

-0.5

0

0.51

-Cp

-1.5-1

-0.5

0

0.5

1

x / c

-Cp

0 0.25 0.5 0.75 1-1

0

1

Fig.8 Instantaneous fields of void fraction, mass flux vector, evaporation / condensation rates, and surface pressure distributions

for 2.1=σ , time interval=0.25T

1

6

5

4

3

2

-Cp

-1

0

1

-Cp

-1

0

1

-Cp

-1

0

1

-Cp

-1

0

1

-Cp

-1

0

1

x / c

-Cp

0 0.25 0.5 0.75 1

-1

0

1

Fig.9 Instantaneous fields of void fraction, mass flux vector, evaporation / condensation rates, and surface pressure distributions

for 0.1=σ , time = 1:t0+3.45T, 2: t0+4.35T, 3:t0+6.45T, 4:t0+13.4T, 5:t0+14.25T and 6:t0+15.1T

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1

2

3

4

5

6

-Cp

-1.5

-1

-0.5

0

0.5

-Cp

-1.5

-1

-0.5

0

0.5

-Cp

-1.5

-1

-0.5

0

0.5

-Cp

-1.5

-1

-0.5

0

0.5

-Cp

-1.5

-1

-0.5

0

0.5

x / c

-Cp

0 0.25 0.5 0.75 1-1

-0.5

0

0.5

Fig.10 Instantaneous fields of void fraction, mass flux vector, evaporation / condensation rates, and surface pressure distributions

for 8.0=σ , time = 1:t0+7T, 2: t0+7.5T, 3:t0+8.2T, 4:t0+8.8T, 5:t0+9.5T and 6:t0+10.5T

0.80.60.40.20.0-0.2

Cp , CL

0.30

0.25

0.20

0.15

0.10

0.05

0.00

Tim

e [s

]

12 34 5

6

CL

Cp

σ=1.2

0.80.60.40.20.0-0.2

Cp , CL

0.30

0.25

0.20

0.15

0.10

0.05

0.00

12

3

45

6

σ=1.0

0.80.60.40.20.0-0.2

Cp , CL

0.30

0.25

0.20

0.15

0.10

0.05

0.00

12

34

56

σ=0.8

Fig.11 Time evolutions of the upstream pressure fluctuation at monitoring point A and the lift coefficient fluctuation

Figure 11 shows the time evolutions of the lift coefficients

and the pressure fluctuations monitored at an upstream point A from the leading edge of the hydrofoil, as shown in figure 2. The pressure is presented by using 2)(2 ∞∞∞−= uppC p ρ . The numbered arrows in figure 11 correspond to the each frame number in figures 8, 9 and 10. When the cloud cavity sheds from the leading edge of the hydrofoil, the unsteady lift coefficient tends to drop. Then, the lift recovers, as the suction surface of the hydrofoil is covered with the sheet cavity. Figure 12 shows the DFT (direct Fourier transform) analysis of the lift and the pressure fluctuations. In this figure, fundamental frequencies based on the pressure and the lift coefficient are shown. It is confirmed that the fundamental frequencies based on the lift coefficient obviously correspond with the cavity break-off frequencies. In addition, figure 13 shows the Strouhal numbers St (= ∞ulf /max ) based on the maximum cavity length

and Stc (= ∞ucf / ) based on the chord length, where the blue and red symbols denote the St, Stc based on the pressure fluctuations and the lift coefficient fluctuations, respectively. It is found that there are two types of fluctuations. One type is that the fundamental frequencies based on the pressure correspond to those based on the lift fluctuations, and the values of St are almost 0.2. And another type is that the fundamental frequencies based on the lift are smaller than those based on the pressure for =σ 0.9, 1.0 and 1.1. In this σ region, the period of the cavity break-off is prolonged due to the occurrence of irregular oscillation of the sheet cavity. The lift coefficient fluctuations are influenced directly by the sheet cavity break-off. On the other hand, the pressure fluctuations are also dominated by such intrinsic instabilities as irregular cavity oscillation. Therefore, the Strouhal numbers St and Stc based on the lift are considerably smaller than those based on the pressure.

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10-10

10-8

10-6

10-4

10-2

P(f

)

3 4 5 6 710

2 3 4 5 6 7100

2 3 4 5 6 71000

σ = 1.2

Fundamental

10-10

10-8

10-6

10-4

10-2

P(f)

3 4 5 6 710

2 3 4 5 6 7100

2 3 4 5 6 71000

σ = 1.0

Fundamental (CL)

Fundamental (Pressure)

10-10

10-8

10-6

10-4

10-2

P(f)

3 4 5 6 710

2 3 4 5 6 7100

2 3 4 5 6 71000

Frequency [Hz]

σ = 0.8

Fundamental

Pressure CL

Fig.12 Power spectrum of upstream pressure and lift coefficient

fluctuations

1.0

0.8

0.6

0.4

0.2

0.0

St,

St c

1.41.21.00.80.60.40.2σ

St based on pressure Stc based on pressure St based on CL Stc based on CL

Fig.13 Strouhal numbers of lift coefficient and upstream

pressure fluctuation Next, to investigate in detail why the Strouhal numbers

based on the lift coefficient near 0.1=σ are smaller than those based on the pressure, we focused on the momentum of re-entrant jet and the pressure gradient on the suction side of the hydrofoil. Figures 14 and 15 show the momentum of re-entrant jet and the pressure gradient in the reverse flow region whose height from the hydrofoil surface is 0.4mm. Here, the momentum of re-entrant jet is normalized by 22 / ∞∞uu ρρ and

the pressure gradient is defined by ∞−+ −=∆ pppp ii 2)( 11 . The abscissa denotes the time, and the ordinate denotes the foil surface location from the leading edge x/c. As decreasing the cavitation number, the momentum of re-entrant jet increases and the pressure gradient near the cavity closure region increases. In the case of 8.0=σ , it can be reconfirmed that the re-entrant jet plays an important role for the periodical break-off of the sheet cavity near the leading edge. However, in the case of 0.1=σ , these patterns of reverse flow on the suction surface become considerably irregular. In this case, though the reverse flow once occurs together with the sheet cavity contraction, the reverse flow doesn’t reach the leading edge of the sheet cavity. Then, the sheet cavity grows again. After several times of cavity volume oscillation, the sheet cavity becomes unstable and breaks off near the leading edge. Thus, it seems that the dominant factor of the sheet cavity break-off is not the re-entrant jet but the intrinsic cavity instability as suggested by Iga, et al. [10].

Finally, figure 16 shows the void fraction and pressure fields on the collapse of the shedding cloud cavity behind the trailing edge in the case of 8.0=σ and the pressure fluctuation monitored at a downstream point B from the hydrofoil shown in figure 2. This figure demonstrates the generation and propagation of the pressure wave in a short time duration.

2.1=σ

0.1=σ

8.0=σ

Fig.14 Momentum of re-entrant jet on the suction surface of the hydrofoil

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8

2.1=σ

0.1=σ

8.0=σ

Fig.15 Pressure gradient on the suction surface of the hydrofoil

0.6

0.4

0.2

0.0

-0.2

-0.4

Cp

0.300.250.200.150.100.050.00

Time [s]

t0+4.4T

Fig.16 Instantaneous fields of void fraction and pressure for

8.0=σ at a time = t0+4.4T, and the pressure fluctuation at monitoring point B

CONCLUDING REMARKS A numerical method for unsteady turbulent vaporous

cavitating flow based on the locally homogeneous model with

the phase change has been applied to simulate the cavitating flow around the CAV2003 hydrofoil. The following results are obtained: 1) It was confirmed that the present numerical method is

appropriate to predict the cavitating flow, through the comparison with experimental results of surface pressure distributions around a hemisphere / cylinder geometry.

2) It was clarified that there are at least two types of fluctuations against to the cavitation number in the cavitating flow around the CAV2003 hydrofoil.

3) One type is that the re-entrant jet causes the periodic sheet cavity break-off, resulting that the Strouhal number based on the pressure corresponds to that based on the lift.

4) Another type is that in the transient range of 2.18.0 << σ the dominant factor of the cavity break-off is not the re-entrant jet but the intrinsic cavity instability. The Strouhal number based on the lift is much smaller than that based on the pressure. The standard deviations of lift and drag coefficient also become larger.

ACKNOWLEDGMENTS The computation was performed using the ORIGIN2000 in

the Institute of Fluid Science, Tohoku University.

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Progress in Aerospace Sciences, Vol.37(2001), pp.551-581.

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[10] Iga, Y., Nohmi, M., Goto, A., Shin, B. R. and Ikohagi, T., J. Fluids Eng. Trans. ASME, Vol.125(2003), pp.643-651.

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